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FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9$](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9$](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [$11$](#page-6-3) and [$12$](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [$11$](#page-6-3) compared to the fully decomposed pathway [$12$](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [$8$](#page-6-4) − [$12$](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 $2022$.](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 5: (a) The energy profile for H² formation on the kite-like Fe₂O₃ cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. <span id="page-8-0"></span>![FIG. 1: The most stable structures of (Fe2O3)<sup>n</sup> clusters with n = 1 − 4. The total spin S and the binding energy Eb, of the clusters are shown in inserts.](path) ## <span id="page-8-1"></span>B. Ammonia adsorption on (Fe2O3)<sup>n</sup> clusters Adsorption of ammonia on (Fe2O3)<sup>n</sup> clusters is a crucial initial step in the whole dehydrogenation process. Figure [2](#page-9-0) demonstrates the most stable adsorption configurations of the NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters with n = 1−4. The corresponding free energies of adsorption and Fe−N bond distances are shown in Table [I](#page-10-0) at 0 K. Our calculations show that the adsorption of NH<sup>3</sup> on the smallest Fe2O<sup>3</sup> cluster is the most stable among all cluster sizes considered in this study, with an adsorption free energy of -33.68 kcal/mol. This finding is corroborated by Mulliken charge analysis, which shows that more electrons are shared between the lone pair of the N atom and the 3d orbitals of Fe2<sup>+</sup> for <sup>n</sup> <sup>=</sup> 1. On the other hand, for larger cluster sizes with <sup>n</sup> <sup>=</sup> <sup>2</sup>−4, which primarily contain Fe3+, the electron density is more localized over the bonding region, as also reported by Sierka et al.[55](#page-28-0). Therefore, bonding occurs with the nitrogen lone pair. Our theoretical analysis indicates that the adsorption energy ∆Gads of ammonia on (Fe2O3)<sup>n</sup> clusters decreases from n = 1 to n = 3, followed by a slight increase for n = 4. A similar trend in the change of adsorption energy with cluster size was reported by Shulan Zhou et al.[56](#page-28-1) for Run@CNT systems. We also compared the adsorption energy of NH<sup>3</sup> on different metal and metal oxides in Table [I.](#page-10-0) The obtained NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters are about 10 kcal/mol higher than the data reported by Zhang et al. for the Ru(0001) surface[57](#page-28-2). Moreover, the adsorption of NH<sup>3</sup> and NO<sup>x</sup> on the γ-Fe2O3(111) surface was studied by Wei Huang et al.[58](#page-28-3) using periodic density functional calculations. They calculated adsorption energies on octahedral and tetrahedral sites of γ-Fe2O3(111) to be -2.13 kcal/mol and -21.68 kcal/mol, respectively. Similarly, our calculated NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters for n = 3 and n = 4 are close to the data reported by Wei Huang et al.,[58](#page-28-3) as adsorption of NH<sup>3</sup> on the three-coordinated Fe3<sup>+</sup> site resembles the tetrahedral site of γ-Fe2O3(111), while the adsorption on the four-coordinated Fe3<sup>+</sup> site resembles the octahedral site of γ-Fe2O3. <span id="page-9-0"></span>![FIG. 2: The optimized geometries of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> for n = 1−4; N−Fe distances (Å) are shown in parentheses along with the partial atomic charges on neighboring atoms. The total spin S of the clusters is shown in inserts.](path) As mentioned above, the calculated adsorption energies indicate that the adsorption of an NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters (n = 1−4) weakens as the cluster size increases from n = 1 to n = 3. In industrial processes, the dehydrogenation of ammonia typically occurs at high temperatures, often in the range of 400◦C to 700◦C, depending on the specific catalysts and conditions used. Therefore, it is important to determine the range of temperatures at which ammonia adsorption on (Fe2O3)<sup>n</sup> remains stable. Figure S5 demonstrates the temperature dependence of ∆Gads in the range from 0 K to 1200 K for the most stable adsorption configurations of NH<sup>3</sup> on (Fe2O3)n clusters (n = 1 − 4). The negative values of ∆Gads correspond to stable adsorption. As seen in Fig. S5, NH<sup>3</sup> adsorbed on the smallest Fe2O<sup>3</sup> cluster is stable across the entire temperature range of 0 K to 1200 K. However, for larger cluster sizes, ammonia adsorption becomes energetically unfavorable at temperatures of 1107 (K), 961 (K), and 1000 (K) for n = 2,3, and 4, respectively. ## C. NH<sup>3</sup> decomposition on Fe2O<sup>3</sup> In this section, we discuss the complete NH<sup>3</sup> decomposition and H<sup>2</sup> formation reactions [$7$](#page-6-0) - [$12$](#page-7-0) on the smallest considered cluster, Fe2O3, at room temperature, T=298.15 K, explored by the AFIR method. This method allows for the automatic exploration of the full reaction path network, systematically accounting for the variety of possible isomer structures and adsorption sites. This is an important approach in nanocatalysis because it has been demonstrated that the most stable structures are not always the most reactive. Therefore, a systematic search for reaction pathways that accounts for the contributions of low-energy isomers is required to accurately describe the catalytic properties of clusters at finite temperatures.[49](#page-27-10) To illustrate the isomer and reaction-site effects, we explicitly consider two different isomers of the Fe2O<sup>3</sup> cluster: the most stable kite-like structure with one terminal oxygen atom, and the linear structure isomer with two terminal oxygen atoms which is 6.24 kcal/mol less stable (see Fig. S1). The kite-like structure possess two type of catalytically active metal centers - two-coordinated and three-coordinated Fe sites. Therefore we consider adsorption and decomposition of NH<sup>3</sup> molecule on both of them. Figure [3\(](#page-13-0)a) demonstrates that the adsorption of NH<sup>3</sup> on the kite-like Fe2O<sup>3</sup> cluster is an exothermic reaction, occurring at both the two-coordinated and three-coordinated Fe sites. The adsorption free energies are -26.98 kcal/mol for the two-coordinated Fe site (intermediate I′ 1 1) and -11.29 kcal/mol for the three-coordinated Fe site (intermediate I′′ 1 1), respectively. The optimized structures of all intermediates (I) and transition satates (T) along the reaction pathways are shown in Fig. [3\(](#page-13-0)b) and [4\(](#page-14-0)b), for the kite-like and linear clusters, respectively. Here the lower index corresponds to the cluster size n, while the numbering corresponds to the order of intermedeates (transition states) along the reaction path. As discussed in the previous section, the most stable adsorption site for NH<sup>3</sup> is the two-coordinated Fe site, with an Fe−N bond length of 2.11 Å. In contrast, the Fe−N bond length at the three-coordinated Fe site is 2.16 Å. These findings are supported by the fact that NH<sup>3</sup> adsorption highly depends on the local geometry and electronic structure of the catalyst. In the case of the Fe2O<sup>3</sup> kite-like structure, the first dehydrogenation reaction is the second step in the reaction mechanism, occurring after adsorption with activation barriers of 26.98 kcal/mol and 22.12 kcal/mol through the reaction paths I′ 1 1-T′ 1 1-I′ 1 2 and I′′ 1 1-T′′ 1 1-I′′ 1 2, respectively. The reactions on these two-coodrinated and three-coordinated active sites are exothermic by 16.31 kcal/mol and 7.53 kcal/mol, respectively. However, the first dehydrogenation of NH<sup>3</sup> on the lineartype structure Fig. [4\(](#page-14-0)a) occurs with smaller activation barrier of 16.22 kcal/mol via the reaction path I<sup>L</sup> 1 1 - T<sup>L</sup> 1 1 - I<sup>L</sup> 1 2, demonstrating that the less stable linear isomer is more reactive. The role of Fe2O<sup>3</sup> isomer structure on NH<sup>3</sup> adsorption and first hydrogen atom transfer was previousely studied by Chaoyue Xie et al.[60](#page-28-5) They performed DFT-D3 calculations on the adsorption mechanisms of different molecules (NH3, NO, O2) on activated carbon (AC) supported iron-based catalysts FexOy/AC. The calculated adsorption electronic energies of NH<sup>3</sup> were -37.4 kcal/mol and -53.7 kcal/mol on different isomers of Fe2O3/AC, and the first hydrogen atom transfer had an activation barrier of 15.5 kcal/mol. Similarly, the adsorption and dehydrogenation of ammonia on different metal oxides were investigated by Erdtman and co-workers[62](#page-28-7) for the application of gas sensors. They reported that the adsorption energy of NH<sup>3</sup> on the RuO2(110) surface is -38.24 kcal/mol, and the first N−H bond cleavage had an activation energy barrier of 17.45 kcal/mol. The third step of the NH<sup>3</sup> dehydrogenation reaction [$9$](#page-6-1) involves the dissociation of the adsorbed NH∗ 2 intermediate into NH∗ and H∗ species. In this step, the abstracted hydrogen atom transfers to one of the oxygen atoms in the cluster. Figure [3$](#page-13-0)a) demonstrates, that in the case of the kite-like structure the energy barriers for this step are 43.91 kcal/mol and 34.51 kcal/mol, corresponding to the reaction paths I′ 1 2 - T′ 1 2 - I′ 1 3 and I′′ 1 2 - T′′ 1 2 - I′′ 1 3, respectively. In the fourth step [\(10$](#page-6-2), the adsorbed NH∗ intermediate further dissociates into N∗ and H∗ species as shown in Fig. [3$](#page-13-0)a). The reaction barriers associated with this step are 46.98 kcal/mol and 8.95 kcal/mol for the two-coordinated and three-coordinated reaction paths, respectively. The decomposition of NH<sup>3</sup> on kite-like structures becomes endothermic starting from the third step [\(9$](#page-6-1). Our calculations reveal that NH<sup>3</sup> dehydrogenation has a high energy barrier when the NH<sup>3</sup> molecule is adsorbed at a two-coordinated Fe site, which is the most stable adsorption site. On the other hand, dehydrogenation of the adsorbed NH<sup>3</sup> at a three-coordinated Fe site has a considerably lower activation barrier of 8.95 kcal/mol for the reaction step [$10$](#page-6-2). Overall, for the NH<sup>3</sup> decomposition reaction on the kite-like Fe2O<sup>3</sup> structure, with initial NH<sup>3</sup> adsorption on the two-coordinated Fe atom, the rate-limiting step is the fourth reaction [$10$](#page-6-2), with a barrier of 46.98 kcal/mol. Alternatively, for the less favorable NH<sup>3</sup> adsorption on the threecoordinated Fe atom, the rate-limiting step is the third reaction step [$9$](#page-6-1), with a barrier of 34.51 kcal/mol. The reaction pathway calculated for NH<sup>3</sup> decomposition on the linear-type Fe2O<sup>3</sup> isomer is shown in Fig. [4\(](#page-14-0)a), and respective intermediate and transition state structures are shown in Fig. [4\(](#page-14-0)b). Since this structure consists of two iron atoms connected through a central oxygen, each containing a terminal oxygen, the reaction mechanism differs slightly from that of the kite-like isomer. For instance, in the third step of the reaction, the second hydrogen from the adsorbed NH∗ 2 intermediate is transferred to the second terminal oxygen. The energy barrier for this step on the linear-type structure is 23.8 kcal/mol, as shown in the reaction path (I<sup>L</sup> 1 2 - T<sup>L</sup> 1 2 - I<sup>L</sup> 1 3) in Fig. [4a](#page-14-0). The fourth step on this isomer is not straightforward, involving the central oxygen atom breaking its bond with one of the neighboring iron atoms while forming an Fe − N − Fe bridge. This process leads to two different intermediates: the formation of the adsorbed H2O ∗ and the transfer of a hydrogen atom from one side of the Fe − N − Fe bridge to the other. Subsequently, the final dehydrogenation step from the NH∗ intermediate occurs, with an activation energy barrier of 34.76 kcal/mol. <span id="page-13-0"></span>![FIG. 3: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ reaction path on the kite-like isomer of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-14-0"></span>![FIG. 4: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> + → N ∗ + 3H∗ reaction path on the linear-type isomet of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) As a next step we consider possible H<sup>2</sup> formation via reactions [$11$](#page-6-3) and [$12$](#page-7-0) on the kite-like and linear isomers of Fe2O<sup>3</sup> cluster. The possible pathways for H<sup>2</sup> formation in the case of the most stable ammonia adsorption on the two-coordinated site (I′ intermediates) of the kite-like Fe2O<sup>3</sup> isomer are shown in Fig. [5$](#page-15-0)a), while the corresponding structures of the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [5\(](#page-15-0)b). Note that H<sup>2</sup> formation can occur after partial decomposition of ammonia in reaction [\(11$](#page-6-3), starting from intermediate (I<sup>I</sup> 1 3) via the path I′ 1 3 - T′ 1 6 - I′ 1 7 - T′ 1 7 - I′ 1 8. On the other hand, H<sup>2</sup> formation can occur via full decomposition of ammonia in reaction [$12$](#page-7-0), through the intermediate (II 1 4) via the path I′ 1 4 - T′ 1 4 - I′ 1 5 - T′ 1 5 - I′ 1 6. In both cases, the reaction pathways include breaking one O−H bond and forming an Fe−H bond. The H<sup>2</sup> formation barriers through these intermediates are 89.74 kcal/mol and 92.49 kcal/mol, respectively. From these results, we conclude that H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> structure is more favorable via reaction [$11$](#page-6-3), with the NH<sup>∗</sup> intermediate remaining adsorbed on the cluster. The H<sup>2</sup> formation reaction, starting from (I<sup>I</sup> 1 4), is the rate-limiting step in molecular hydrogen formation on the kite Fe2O<sup>3</sup> cluster. Similarly, the H<sup>2</sup> formation reaction pathways on the linear-type structure of Fe2O<sup>3</sup> are shown in Fig. [6$](#page-16-0)a), while the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [6\(](#page-16-0)b). The H<sup>2</sup> formation through the NH<sup>∗</sup> intermediate (I<sup>L</sup> 1 4) via the reaction path I<sup>L</sup> 1 4 - T<sup>L</sup> 1 8 - I<sup>L</sup> 1 9 - T<sup>L</sup> 1 9 - I<sup>L</sup> 1 10 has an energy barrier of 79.99 kcal/mol. On the other hand H<sup>2</sup> formation through intermediate (I<sup>L</sup> 1 6) via reaction path I<sup>L</sup> 1 6 - T<sup>L</sup> 1 6 - I<sup>L</sup> 1 7 - T<sup>L</sup> 1 7 - I<sup>L</sup> 1 8 has an activation energy of 70.84 kcal/mol, which is about 10 kcal/mol lower energy than reaction path through intermediate (I<sup>L</sup> 1 4). Overall, on the basis of our calculated reaction pathways for H<sup>2</sup> formation show similar pattern for both kite-type and linear-type Fe2O3, where H<sup>2</sup> formation in reactions [\(11$](#page-6-3) and [$12$](#page-7-0) take place via breaking one of O−H bond and forming intermediate Fe−H bond. However, from both thermodynamic and kinetic perspectives, H<sup>2</sup> formation on the two types of Fe2O<sup>3</sup> structures varies. Reaction [$11$](#page-6-3) is more favorable on the kite-like structure, while reaction [$12$](#page-7-0) is more favorable on the linear structure. This highlights that the rate-limiting step for H<sup>2</sup> formation is highly dependent on the catalyst's structure. <span id="page-15-0"></span>![FIG. 5: (a) The energy profile for H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-16-0"></span>![FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [$11$](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [$11$](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [$12$](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [$10$](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8$](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8$](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [$10$](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [$11](#page-6-3) and [12$](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [$11$](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [$11$](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9$](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9$](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [$11$](#page-6-3) and [$12$](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [$11$](#page-6-3) compared to the fully decomposed pathway [$12$](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [$8$](#page-6-4) − [$12$](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 $2022$.](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. 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Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 10: Reaction barrier (∆G ‡ ) for NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions on (Fe2O3)<sup>n</sup> (n = 1−4) clusters.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 $2022$.](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 2: The optimized geometries of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> for n = 1−4; N−Fe distances (Å) are shown in parentheses along with the partial atomic charges on neighboring atoms. The total spin S of the clusters is shown in inserts.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. As mentioned above, the calculated adsorption energies indicate that the adsorption of an NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters (n = 1−4) weakens as the cluster size increases from n = 1 to n = 3. In industrial processes, the dehydrogenation of ammonia typically occurs at high temperatures, often in the range of 400◦C to 700◦C, depending on the specific catalysts and conditions used. Therefore, it is important to determine the range of temperatures at which ammonia adsorption on (Fe2O3)<sup>n</sup> remains stable. Figure S5 demonstrates the temperature dependence of ∆Gads in the range from 0 K to 1200 K for the most stable adsorption configurations of NH<sup>3</sup> on (Fe2O3)n clusters (n = 1 − 4). The negative values of ∆Gads correspond to stable adsorption. As seen in Fig. S5, NH<sup>3</sup> adsorbed on the smallest Fe2O<sup>3</sup> cluster is stable across the entire temperature range of 0 K to 1200 K. However, for larger cluster sizes, ammonia adsorption becomes energetically unfavorable at temperatures of 1107 (K), 961 (K), and 1000 (K) for n = 2,3, and 4, respectively. ## C. NH<sup>3</sup> decomposition on Fe2O<sup>3</sup> In this section, we discuss the complete NH<sup>3</sup> decomposition and H<sup>2</sup> formation reactions [$7$](#page-6-0) - [$12$](#page-7-0) on the smallest considered cluster, Fe2O3, at room temperature, T=298.15 K, explored by the AFIR method. This method allows for the automatic exploration of the full reaction path network, systematically accounting for the variety of possible isomer structures and adsorption sites. This is an important approach in nanocatalysis because it has been demonstrated that the most stable structures are not always the most reactive. Therefore, a systematic search for reaction pathways that accounts for the contributions of low-energy isomers is required to accurately describe the catalytic properties of clusters at finite temperatures.[49](#page-27-10) To illustrate the isomer and reaction-site effects, we explicitly consider two different isomers of the Fe2O<sup>3</sup> cluster: the most stable kite-like structure with one terminal oxygen atom, and the linear structure isomer with two terminal oxygen atoms which is 6.24 kcal/mol less stable (see Fig. S1). The kite-like structure possess two type of catalytically active metal centers - two-coordinated and three-coordinated Fe sites. Therefore we consider adsorption and decomposition of NH<sup>3</sup> molecule on both of them. Figure [3\(](#page-13-0)a) demonstrates that the adsorption of NH<sup>3</sup> on the kite-like Fe2O<sup>3</sup> cluster is an exothermic reaction, occurring at both the two-coordinated and three-coordinated Fe sites. The adsorption free energies are -26.98 kcal/mol for the two-coordinated Fe site (intermediate I′ 1 1) and -11.29 kcal/mol for the three-coordinated Fe site (intermediate I′′ 1 1), respectively. The optimized structures of all intermediates (I) and transition satates (T) along the reaction pathways are shown in Fig. [3\(](#page-13-0)b) and [4\(](#page-14-0)b), for the kite-like and linear clusters, respectively. Here the lower index corresponds to the cluster size n, while the numbering corresponds to the order of intermedeates (transition states) along the reaction path. As discussed in the previous section, the most stable adsorption site for NH<sup>3</sup> is the two-coordinated Fe site, with an Fe−N bond length of 2.11 Å. In contrast, the Fe−N bond length at the three-coordinated Fe site is 2.16 Å. These findings are supported by the fact that NH<sup>3</sup> adsorption highly depends on the local geometry and electronic structure of the catalyst. In the case of the Fe2O<sup>3</sup> kite-like structure, the first dehydrogenation reaction is the second step in the reaction mechanism, occurring after adsorption with activation barriers of 26.98 kcal/mol and 22.12 kcal/mol through the reaction paths I′ 1 1-T′ 1 1-I′ 1 2 and I′′ 1 1-T′′ 1 1-I′′ 1 2, respectively. The reactions on these two-coodrinated and three-coordinated active sites are exothermic by 16.31 kcal/mol and 7.53 kcal/mol, respectively. However, the first dehydrogenation of NH<sup>3</sup> on the lineartype structure Fig. [4\(](#page-14-0)a) occurs with smaller activation barrier of 16.22 kcal/mol via the reaction path I<sup>L</sup> 1 1 - T<sup>L</sup> 1 1 - I<sup>L</sup> 1 2, demonstrating that the less stable linear isomer is more reactive. The role of Fe2O<sup>3</sup> isomer structure on NH<sup>3</sup> adsorption and first hydrogen atom transfer was previousely studied by Chaoyue Xie et al.[60](#page-28-5) They performed DFT-D3 calculations on the adsorption mechanisms of different molecules (NH3, NO, O2) on activated carbon (AC) supported iron-based catalysts FexOy/AC. The calculated adsorption electronic energies of NH<sup>3</sup> were -37.4 kcal/mol and -53.7 kcal/mol on different isomers of Fe2O3/AC, and the first hydrogen atom transfer had an activation barrier of 15.5 kcal/mol. Similarly, the adsorption and dehydrogenation of ammonia on different metal oxides were investigated by Erdtman and co-workers[62](#page-28-7) for the application of gas sensors. They reported that the adsorption energy of NH<sup>3</sup> on the RuO2(110) surface is -38.24 kcal/mol, and the first N−H bond cleavage had an activation energy barrier of 17.45 kcal/mol. The third step of the NH<sup>3</sup> dehydrogenation reaction [$9$](#page-6-1) involves the dissociation of the adsorbed NH∗ 2 intermediate into NH∗ and H∗ species. In this step, the abstracted hydrogen atom transfers to one of the oxygen atoms in the cluster. Figure [3$](#page-13-0)a) demonstrates, that in the case of the kite-like structure the energy barriers for this step are 43.91 kcal/mol and 34.51 kcal/mol, corresponding to the reaction paths I′ 1 2 - T′ 1 2 - I′ 1 3 and I′′ 1 2 - T′′ 1 2 - I′′ 1 3, respectively. In the fourth step [\(10$](#page-6-2), the adsorbed NH∗ intermediate further dissociates into N∗ and H∗ species as shown in Fig. [3$](#page-13-0)a). The reaction barriers associated with this step are 46.98 kcal/mol and 8.95 kcal/mol for the two-coordinated and three-coordinated reaction paths, respectively. The decomposition of NH<sup>3</sup> on kite-like structures becomes endothermic starting from the third step [\(9$](#page-6-1). Our calculations reveal that NH<sup>3</sup> dehydrogenation has a high energy barrier when the NH<sup>3</sup> molecule is adsorbed at a two-coordinated Fe site, which is the most stable adsorption site. On the other hand, dehydrogenation of the adsorbed NH<sup>3</sup> at a three-coordinated Fe site has a considerably lower activation barrier of 8.95 kcal/mol for the reaction step [$10$](#page-6-2). Overall, for the NH<sup>3</sup> decomposition reaction on the kite-like Fe2O<sup>3</sup> structure, with initial NH<sup>3</sup> adsorption on the two-coordinated Fe atom, the rate-limiting step is the fourth reaction [$10$](#page-6-2), with a barrier of 46.98 kcal/mol. Alternatively, for the less favorable NH<sup>3</sup> adsorption on the threecoordinated Fe atom, the rate-limiting step is the third reaction step [$9$](#page-6-1), with a barrier of 34.51 kcal/mol. The reaction pathway calculated for NH<sup>3</sup> decomposition on the linear-type Fe2O<sup>3</sup> isomer is shown in Fig. [4\(](#page-14-0)a), and respective intermediate and transition state structures are shown in Fig. [4\(](#page-14-0)b). Since this structure consists of two iron atoms connected through a central oxygen, each containing a terminal oxygen, the reaction mechanism differs slightly from that of the kite-like isomer. For instance, in the third step of the reaction, the second hydrogen from the adsorbed NH∗ 2 intermediate is transferred to the second terminal oxygen. The energy barrier for this step on the linear-type structure is 23.8 kcal/mol, as shown in the reaction path (I<sup>L</sup> 1 2 - T<sup>L</sup> 1 2 - I<sup>L</sup> 1 3) in Fig. [4a](#page-14-0). The fourth step on this isomer is not straightforward, involving the central oxygen atom breaking its bond with one of the neighboring iron atoms while forming an Fe − N − Fe bridge. This process leads to two different intermediates: the formation of the adsorbed H2O ∗ and the transfer of a hydrogen atom from one side of the Fe − N − Fe bridge to the other. Subsequently, the final dehydrogenation step from the NH∗ intermediate occurs, with an activation energy barrier of 34.76 kcal/mol. <span id="page-13-0"></span>![FIG. 3: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ reaction path on the kite-like isomer of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-14-0"></span>![FIG. 4: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> + → N ∗ + 3H∗ reaction path on the linear-type isomet of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) As a next step we consider possible H<sup>2</sup> formation via reactions [$11$](#page-6-3) and [$12$](#page-7-0) on the kite-like and linear isomers of Fe2O<sup>3</sup> cluster. The possible pathways for H<sup>2</sup> formation in the case of the most stable ammonia adsorption on the two-coordinated site (I′ intermediates) of the kite-like Fe2O<sup>3</sup> isomer are shown in Fig. [5$](#page-15-0)a), while the corresponding structures of the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [5\(](#page-15-0)b). Note that H<sup>2</sup> formation can occur after partial decomposition of ammonia in reaction [\(11$](#page-6-3), starting from intermediate (I<sup>I</sup> 1 3) via the path I′ 1 3 - T′ 1 6 - I′ 1 7 - T′ 1 7 - I′ 1 8. On the other hand, H<sup>2</sup> formation can occur via full decomposition of ammonia in reaction [$12$](#page-7-0), through the intermediate (II 1 4) via the path I′ 1 4 - T′ 1 4 - I′ 1 5 - T′ 1 5 - I′ 1 6. In both cases, the reaction pathways include breaking one O−H bond and forming an Fe−H bond. The H<sup>2</sup> formation barriers through these intermediates are 89.74 kcal/mol and 92.49 kcal/mol, respectively. From these results, we conclude that H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> structure is more favorable via reaction [$11$](#page-6-3), with the NH<sup>∗</sup> intermediate remaining adsorbed on the cluster. The H<sup>2</sup> formation reaction, starting from (I<sup>I</sup> 1 4), is the rate-limiting step in molecular hydrogen formation on the kite Fe2O<sup>3</sup> cluster. Similarly, the H<sup>2</sup> formation reaction pathways on the linear-type structure of Fe2O<sup>3</sup> are shown in Fig. [6$](#page-16-0)a), while the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [6\(](#page-16-0)b). The H<sup>2</sup> formation through the NH<sup>∗</sup> intermediate (I<sup>L</sup> 1 4) via the reaction path I<sup>L</sup> 1 4 - T<sup>L</sup> 1 8 - I<sup>L</sup> 1 9 - T<sup>L</sup> 1 9 - I<sup>L</sup> 1 10 has an energy barrier of 79.99 kcal/mol. On the other hand H<sup>2</sup> formation through intermediate (I<sup>L</sup> 1 6) via reaction path I<sup>L</sup> 1 6 - T<sup>L</sup> 1 6 - I<sup>L</sup> 1 7 - T<sup>L</sup> 1 7 - I<sup>L</sup> 1 8 has an activation energy of 70.84 kcal/mol, which is about 10 kcal/mol lower energy than reaction path through intermediate (I<sup>L</sup> 1 4). Overall, on the basis of our calculated reaction pathways for H<sup>2</sup> formation show similar pattern for both kite-type and linear-type Fe2O3, where H<sup>2</sup> formation in reactions [\(11$](#page-6-3) and [$12$](#page-7-0) take place via breaking one of O−H bond and forming intermediate Fe−H bond. However, from both thermodynamic and kinetic perspectives, H<sup>2</sup> formation on the two types of Fe2O<sup>3</sup> structures varies. Reaction [$11$](#page-6-3) is more favorable on the kite-like structure, while reaction [$12$](#page-7-0) is more favorable on the linear structure. This highlights that the rate-limiting step for H<sup>2</sup> formation is highly dependent on the catalyst's structure. <span id="page-15-0"></span>![FIG. 5: (a) The energy profile for H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-16-0"></span>![FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [$11$](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [$11$](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [$12$](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [$10$](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8$](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8$](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [$10$](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [$11](#page-6-3) and [12$](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [$11$](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [$11$](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9$](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9$](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [$11$](#page-6-3) and [$12$](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [$11$](#page-6-3) compared to the fully decomposed pathway [$12$](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [$8$](#page-6-4) − [$12$](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 $2022$.](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. <span id="page-8-0"></span>![FIG. 1: The most stable structures of (Fe2O3)<sup>n</sup> clusters with n = 1 − 4. The total spin S and the binding energy Eb, of the clusters are shown in inserts.](path) ## <span id="page-8-1"></span>B. Ammonia adsorption on (Fe2O3)<sup>n</sup> clusters Adsorption of ammonia on (Fe2O3)<sup>n</sup> clusters is a crucial initial step in the whole dehydrogenation process. Figure [2](#page-9-0) demonstrates the most stable adsorption configurations of the NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters with n = 1−4. The corresponding free energies of adsorption and Fe−N bond distances are shown in Table [I](#page-10-0) at 0 K. Our calculations show that the adsorption of NH<sup>3</sup> on the smallest Fe2O<sup>3</sup> cluster is the most stable among all cluster sizes considered in this study, with an adsorption free energy of -33.68 kcal/mol. This finding is corroborated by Mulliken charge analysis, which shows that more electrons are shared between the lone pair of the N atom and the 3d orbitals of Fe2<sup>+</sup> for <sup>n</sup> <sup>=</sup> 1. On the other hand, for larger cluster sizes with <sup>n</sup> <sup>=</sup> <sup>2</sup>−4, which primarily contain Fe3+, the electron density is more localized over the bonding region, as also reported by Sierka et al.[55](#page-28-0). Therefore, bonding occurs with the nitrogen lone pair. Our theoretical analysis indicates that the adsorption energy ∆Gads of ammonia on (Fe2O3)<sup>n</sup> clusters decreases from n = 1 to n = 3, followed by a slight increase for n = 4. A similar trend in the change of adsorption energy with cluster size was reported by Shulan Zhou et al.[56](#page-28-1) for Run@CNT systems. We also compared the adsorption energy of NH<sup>3</sup> on different metal and metal oxides in Table [I.](#page-10-0) The obtained NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters are about 10 kcal/mol higher than the data reported by Zhang et al. for the Ru(0001) surface[57](#page-28-2). Moreover, the adsorption of NH<sup>3</sup> and NO<sup>x</sup> on the γ-Fe2O3(111) surface was studied by Wei Huang et al.[58](#page-28-3) using periodic density functional calculations. They calculated adsorption energies on octahedral and tetrahedral sites of γ-Fe2O3(111) to be -2.13 kcal/mol and -21.68 kcal/mol, respectively. Similarly, our calculated NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters for n = 3 and n = 4 are close to the data reported by Wei Huang et al.,[58](#page-28-3) as adsorption of NH<sup>3</sup> on the three-coordinated Fe3<sup>+</sup> site resembles the tetrahedral site of γ-Fe2O3(111), while the adsorption on the four-coordinated Fe3<sup>+</sup> site resembles the octahedral site of γ-Fe2O3. <span id="page-9-0"></span>![FIG. 2: The optimized geometries of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> for n = 1−4; N−Fe distances (Å) are shown in parentheses along with the partial atomic charges on neighboring atoms. The total spin S of the clusters is shown in inserts.](path) As mentioned above, the calculated adsorption energies indicate that the adsorption of an NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters (n = 1−4) weakens as the cluster size increases from n = 1 to n = 3. In industrial processes, the dehydrogenation of ammonia typically occurs at high temperatures, often in the range of 400◦C to 700◦C, depending on the specific catalysts and conditions used. Therefore, it is important to determine the range of temperatures at which ammonia adsorption on (Fe2O3)<sup>n</sup> remains stable. Figure S5 demonstrates the temperature dependence of ∆Gads in the range from 0 K to 1200 K for the most stable adsorption configurations of NH<sup>3</sup> on (Fe2O3)n clusters (n = 1 − 4). The negative values of ∆Gads correspond to stable adsorption. As seen in Fig. S5, NH<sup>3</sup> adsorbed on the smallest Fe2O<sup>3</sup> cluster is stable across the entire temperature range of 0 K to 1200 K. However, for larger cluster sizes, ammonia adsorption becomes energetically unfavorable at temperatures of 1107 (K), 961 (K), and 1000 (K) for n = 2,3, and 4, respectively. ## C. NH<sup>3</sup> decomposition on Fe2O<sup>3</sup> In this section, we discuss the complete NH<sup>3</sup> decomposition and H<sup>2</sup> formation reactions [$7$](#page-6-0) - [$12$](#page-7-0) on the smallest considered cluster, Fe2O3, at room temperature, T=298.15 K, explored by the AFIR method. This method allows for the automatic exploration of the full reaction path network, systematically accounting for the variety of possible isomer structures and adsorption sites. This is an important approach in nanocatalysis because it has been demonstrated that the most stable structures are not always the most reactive. Therefore, a systematic search for reaction pathways that accounts for the contributions of low-energy isomers is required to accurately describe the catalytic properties of clusters at finite temperatures.[49](#page-27-10) To illustrate the isomer and reaction-site effects, we explicitly consider two different isomers of the Fe2O<sup>3</sup> cluster: the most stable kite-like structure with one terminal oxygen atom, and the linear structure isomer with two terminal oxygen atoms which is 6.24 kcal/mol less stable (see Fig. S1). The kite-like structure possess two type of catalytically active metal centers - two-coordinated and three-coordinated Fe sites. Therefore we consider adsorption and decomposition of NH<sup>3</sup> molecule on both of them. Figure [3\(](#page-13-0)a) demonstrates that the adsorption of NH<sup>3</sup> on the kite-like Fe2O<sup>3</sup> cluster is an exothermic reaction, occurring at both the two-coordinated and three-coordinated Fe sites. The adsorption free energies are -26.98 kcal/mol for the two-coordinated Fe site (intermediate I′ 1 1) and -11.29 kcal/mol for the three-coordinated Fe site (intermediate I′′ 1 1), respectively. The optimized structures of all intermediates (I) and transition satates (T) along the reaction pathways are shown in Fig. [3\(](#page-13-0)b) and [4\(](#page-14-0)b), for the kite-like and linear clusters, respectively. Here the lower index corresponds to the cluster size n, while the numbering corresponds to the order of intermedeates (transition states) along the reaction path. As discussed in the previous section, the most stable adsorption site for NH<sup>3</sup> is the two-coordinated Fe site, with an Fe−N bond length of 2.11 Å. In contrast, the Fe−N bond length at the three-coordinated Fe site is 2.16 Å. These findings are supported by the fact that NH<sup>3</sup> adsorption highly depends on the local geometry and electronic structure of the catalyst. In the case of the Fe2O<sup>3</sup> kite-like structure, the first dehydrogenation reaction is the second step in the reaction mechanism, occurring after adsorption with activation barriers of 26.98 kcal/mol and 22.12 kcal/mol through the reaction paths I′ 1 1-T′ 1 1-I′ 1 2 and I′′ 1 1-T′′ 1 1-I′′ 1 2, respectively. The reactions on these two-coodrinated and three-coordinated active sites are exothermic by 16.31 kcal/mol and 7.53 kcal/mol, respectively. However, the first dehydrogenation of NH<sup>3</sup> on the lineartype structure Fig. [4\(](#page-14-0)a) occurs with smaller activation barrier of 16.22 kcal/mol via the reaction path I<sup>L</sup> 1 1 - T<sup>L</sup> 1 1 - I<sup>L</sup> 1 2, demonstrating that the less stable linear isomer is more reactive. The role of Fe2O<sup>3</sup> isomer structure on NH<sup>3</sup> adsorption and first hydrogen atom transfer was previousely studied by Chaoyue Xie et al.[60](#page-28-5) They performed DFT-D3 calculations on the adsorption mechanisms of different molecules (NH3, NO, O2) on activated carbon (AC) supported iron-based catalysts FexOy/AC. The calculated adsorption electronic energies of NH<sup>3</sup> were -37.4 kcal/mol and -53.7 kcal/mol on different isomers of Fe2O3/AC, and the first hydrogen atom transfer had an activation barrier of 15.5 kcal/mol. Similarly, the adsorption and dehydrogenation of ammonia on different metal oxides were investigated by Erdtman and co-workers[62](#page-28-7) for the application of gas sensors. They reported that the adsorption energy of NH<sup>3</sup> on the RuO2(110) surface is -38.24 kcal/mol, and the first N−H bond cleavage had an activation energy barrier of 17.45 kcal/mol. The third step of the NH<sup>3</sup> dehydrogenation reaction [$9$](#page-6-1) involves the dissociation of the adsorbed NH∗ 2 intermediate into NH∗ and H∗ species. In this step, the abstracted hydrogen atom transfers to one of the oxygen atoms in the cluster. Figure [3$](#page-13-0)a) demonstrates, that in the case of the kite-like structure the energy barriers for this step are 43.91 kcal/mol and 34.51 kcal/mol, corresponding to the reaction paths I′ 1 2 - T′ 1 2 - I′ 1 3 and I′′ 1 2 - T′′ 1 2 - I′′ 1 3, respectively. In the fourth step [\(10$](#page-6-2), the adsorbed NH∗ intermediate further dissociates into N∗ and H∗ species as shown in Fig. [3$](#page-13-0)a). The reaction barriers associated with this step are 46.98 kcal/mol and 8.95 kcal/mol for the two-coordinated and three-coordinated reaction paths, respectively. The decomposition of NH<sup>3</sup> on kite-like structures becomes endothermic starting from the third step [\(9$](#page-6-1). Our calculations reveal that NH<sup>3</sup> dehydrogenation has a high energy barrier when the NH<sup>3</sup> molecule is adsorbed at a two-coordinated Fe site, which is the most stable adsorption site. On the other hand, dehydrogenation of the adsorbed NH<sup>3</sup> at a three-coordinated Fe site has a considerably lower activation barrier of 8.95 kcal/mol for the reaction step [$10$](#page-6-2). Overall, for the NH<sup>3</sup> decomposition reaction on the kite-like Fe2O<sup>3</sup> structure, with initial NH<sup>3</sup> adsorption on the two-coordinated Fe atom, the rate-limiting step is the fourth reaction [$10$](#page-6-2), with a barrier of 46.98 kcal/mol. Alternatively, for the less favorable NH<sup>3</sup> adsorption on the threecoordinated Fe atom, the rate-limiting step is the third reaction step [$9$](#page-6-1), with a barrier of 34.51 kcal/mol. The reaction pathway calculated for NH<sup>3</sup> decomposition on the linear-type Fe2O<sup>3</sup> isomer is shown in Fig. [4\(](#page-14-0)a), and respective intermediate and transition state structures are shown in Fig. [4\(](#page-14-0)b). Since this structure consists of two iron atoms connected through a central oxygen, each containing a terminal oxygen, the reaction mechanism differs slightly from that of the kite-like isomer. For instance, in the third step of the reaction, the second hydrogen from the adsorbed NH∗ 2 intermediate is transferred to the second terminal oxygen. The energy barrier for this step on the linear-type structure is 23.8 kcal/mol, as shown in the reaction path (I<sup>L</sup> 1 2 - T<sup>L</sup> 1 2 - I<sup>L</sup> 1 3) in Fig. [4a](#page-14-0). The fourth step on this isomer is not straightforward, involving the central oxygen atom breaking its bond with one of the neighboring iron atoms while forming an Fe − N − Fe bridge. This process leads to two different intermediates: the formation of the adsorbed H2O ∗ and the transfer of a hydrogen atom from one side of the Fe − N − Fe bridge to the other. Subsequently, the final dehydrogenation step from the NH∗ intermediate occurs, with an activation energy barrier of 34.76 kcal/mol. <span id="page-13-0"></span>![FIG. 3: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ reaction path on the kite-like isomer of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-14-0"></span>![FIG. 4: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> + → N ∗ + 3H∗ reaction path on the linear-type isomet of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) As a next step we consider possible H<sup>2</sup> formation via reactions [$11$](#page-6-3) and [$12$](#page-7-0) on the kite-like and linear isomers of Fe2O<sup>3</sup> cluster. The possible pathways for H<sup>2</sup> formation in the case of the most stable ammonia adsorption on the two-coordinated site (I′ intermediates) of the kite-like Fe2O<sup>3</sup> isomer are shown in Fig. [5$](#page-15-0)a), while the corresponding structures of the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [5\(](#page-15-0)b). Note that H<sup>2</sup> formation can occur after partial decomposition of ammonia in reaction [\(11$](#page-6-3), starting from intermediate (I<sup>I</sup> 1 3) via the path I′ 1 3 - T′ 1 6 - I′ 1 7 - T′ 1 7 - I′ 1 8. On the other hand, H<sup>2</sup> formation can occur via full decomposition of ammonia in reaction [$12$](#page-7-0), through the intermediate (II 1 4) via the path I′ 1 4 - T′ 1 4 - I′ 1 5 - T′ 1 5 - I′ 1 6. In both cases, the reaction pathways include breaking one O−H bond and forming an Fe−H bond. The H<sup>2</sup> formation barriers through these intermediates are 89.74 kcal/mol and 92.49 kcal/mol, respectively. From these results, we conclude that H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> structure is more favorable via reaction [$11$](#page-6-3), with the NH<sup>∗</sup> intermediate remaining adsorbed on the cluster. The H<sup>2</sup> formation reaction, starting from (I<sup>I</sup> 1 4), is the rate-limiting step in molecular hydrogen formation on the kite Fe2O<sup>3</sup> cluster. Similarly, the H<sup>2</sup> formation reaction pathways on the linear-type structure of Fe2O<sup>3</sup> are shown in Fig. [6$](#page-16-0)a), while the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [6\(](#page-16-0)b). The H<sup>2</sup> formation through the NH<sup>∗</sup> intermediate (I<sup>L</sup> 1 4) via the reaction path I<sup>L</sup> 1 4 - T<sup>L</sup> 1 8 - I<sup>L</sup> 1 9 - T<sup>L</sup> 1 9 - I<sup>L</sup> 1 10 has an energy barrier of 79.99 kcal/mol. On the other hand H<sup>2</sup> formation through intermediate (I<sup>L</sup> 1 6) via reaction path I<sup>L</sup> 1 6 - T<sup>L</sup> 1 6 - I<sup>L</sup> 1 7 - T<sup>L</sup> 1 7 - I<sup>L</sup> 1 8 has an activation energy of 70.84 kcal/mol, which is about 10 kcal/mol lower energy than reaction path through intermediate (I<sup>L</sup> 1 4). Overall, on the basis of our calculated reaction pathways for H<sup>2</sup> formation show similar pattern for both kite-type and linear-type Fe2O3, where H<sup>2</sup> formation in reactions [\(11$](#page-6-3) and [$12$](#page-7-0) take place via breaking one of O−H bond and forming intermediate Fe−H bond. However, from both thermodynamic and kinetic perspectives, H<sup>2</sup> formation on the two types of Fe2O<sup>3</sup> structures varies. Reaction [$11$](#page-6-3) is more favorable on the kite-like structure, while reaction [$12$](#page-7-0) is more favorable on the linear structure. This highlights that the rate-limiting step for H<sup>2</sup> formation is highly dependent on the catalyst's structure. <span id="page-15-0"></span>![FIG. 5: (a) The energy profile for H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-16-0"></span>![FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [$11$](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [$11$](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [$12$](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [$10$](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8$](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8$](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [$10$](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [$11](#page-6-3) and [12$](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [$11$](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [$11$](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9$](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9$](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [$11$](#page-6-3) and [$12$](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [$11$](#page-6-3) compared to the fully decomposed pathway [$12$](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [$8$](#page-6-4) − [$12$](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 $2022$.](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 4: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> + → N ∗ + 3H∗ reaction path on the linear-type isomet of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. As a next step we consider possible H<sup>2</sup> formation via reactions [$11$](#page-6-3) and [$12$](#page-7-0) on the kite-like and linear isomers of Fe2O<sup>3</sup> cluster. The possible pathways for H<sup>2</sup> formation in the case of the most stable ammonia adsorption on the two-coordinated site (I′ intermediates) of the kite-like Fe2O<sup>3</sup> isomer are shown in Fig. [5$](#page-15-0)a), while the corresponding structures of the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [5\(](#page-15-0)b). Note that H<sup>2</sup> formation can occur after partial decomposition of ammonia in reaction [\(11$](#page-6-3), starting from intermediate (I<sup>I</sup> 1 3) via the path I′ 1 3 - T′ 1 6 - I′ 1 7 - T′ 1 7 - I′ 1 8. On the other hand, H<sup>2</sup> formation can occur via full decomposition of ammonia in reaction [$12$](#page-7-0), through the intermediate (II 1 4) via the path I′ 1 4 - T′ 1 4 - I′ 1 5 - T′ 1 5 - I′ 1 6. In both cases, the reaction pathways include breaking one O−H bond and forming an Fe−H bond. The H<sup>2</sup> formation barriers through these intermediates are 89.74 kcal/mol and 92.49 kcal/mol, respectively. From these results, we conclude that H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> structure is more favorable via reaction [$11$](#page-6-3), with the NH<sup>∗</sup> intermediate remaining adsorbed on the cluster. The H<sup>2</sup> formation reaction, starting from (I<sup>I</sup> 1 4), is the rate-limiting step in molecular hydrogen formation on the kite Fe2O<sup>3</sup> cluster. Similarly, the H<sup>2</sup> formation reaction pathways on the linear-type structure of Fe2O<sup>3</sup> are shown in Fig. [6$](#page-16-0)a), while the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [6\(](#page-16-0)b). The H<sup>2</sup> formation through the NH<sup>∗</sup> intermediate (I<sup>L</sup> 1 4) via the reaction path I<sup>L</sup> 1 4 - T<sup>L</sup> 1 8 - I<sup>L</sup> 1 9 - T<sup>L</sup> 1 9 - I<sup>L</sup> 1 10 has an energy barrier of 79.99 kcal/mol. On the other hand H<sup>2</sup> formation through intermediate (I<sup>L</sup> 1 6) via reaction path I<sup>L</sup> 1 6 - T<sup>L</sup> 1 6 - I<sup>L</sup> 1 7 - T<sup>L</sup> 1 7 - I<sup>L</sup> 1 8 has an activation energy of 70.84 kcal/mol, which is about 10 kcal/mol lower energy than reaction path through intermediate (I<sup>L</sup> 1 4). Overall, on the basis of our calculated reaction pathways for H<sup>2</sup> formation show similar pattern for both kite-type and linear-type Fe2O3, where H<sup>2</sup> formation in reactions [\(11$](#page-6-3) and [$12$](#page-7-0) take place via breaking one of O−H bond and forming intermediate Fe−H bond. However, from both thermodynamic and kinetic perspectives, H<sup>2</sup> formation on the two types of Fe2O<sup>3</sup> structures varies. Reaction [$11$](#page-6-3) is more favorable on the kite-like structure, while reaction [$12$](#page-7-0) is more favorable on the linear structure. This highlights that the rate-limiting step for H<sup>2</sup> formation is highly dependent on the catalyst's structure. <span id="page-15-0"></span>![FIG. 5: (a) The energy profile for H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-16-0"></span>![FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [$11$](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [$11$](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [$12$](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [$10$](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8$](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8$](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [$10$](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [$11](#page-6-3) and [12$](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [$11$](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [$11$](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9$](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9$](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [$11$](#page-6-3) and [$12$](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [$11$](#page-6-3) compared to the fully decomposed pathway [$12$](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [$8$](#page-6-4) − [$12$](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 $2022$.](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [$11$](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [$11$](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [$12$](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [$10$](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8$](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8$](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [$10$](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [$11](#page-6-3) and [12$](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [$11$](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [$11$](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9$](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9$](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [$11$](#page-6-3) and [$12$](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [$11$](#page-6-3) compared to the fully decomposed pathway [$12$](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [$8$](#page-6-4) − [$12$](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 $2022$.](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. 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Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8$](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8$](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [$10$](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [$11](#page-6-3) and [12$](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [$11$](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [$11$](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9$](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9$](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [$11$](#page-6-3) and [$12$](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [$11$](#page-6-3) compared to the fully decomposed pathway [$12$](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [$8$](#page-6-4) − [$12$](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 $2022$.](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [$8$](#page-6-4) − [$12$](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 $2022$.](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 3: (a) The energy profile for NH<sup>3</sup> → NH<sup></sup> 2 + H<sup></sup> → NH<sup></sup> + 2H<sup></sup> → N * + 3H* reaction path on the kite-like isomer of Fe<sub>2</sub>O<sub>3</sub> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. <span id="page-8-0"></span>![FIG. 1: The most stable structures of (Fe2O3)<sup>n</sup> clusters with n = 1 − 4. The total spin S and the binding energy Eb, of the clusters are shown in inserts.](path) ## <span id="page-8-1"></span>B. Ammonia adsorption on (Fe2O3)<sup>n</sup> clusters Adsorption of ammonia on (Fe2O3)<sup>n</sup> clusters is a crucial initial step in the whole dehydrogenation process. Figure [2](#page-9-0) demonstrates the most stable adsorption configurations of the NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters with n = 1−4. The corresponding free energies of adsorption and Fe−N bond distances are shown in Table [I](#page-10-0) at 0 K. Our calculations show that the adsorption of NH<sup>3</sup> on the smallest Fe2O<sup>3</sup> cluster is the most stable among all cluster sizes considered in this study, with an adsorption free energy of -33.68 kcal/mol. This finding is corroborated by Mulliken charge analysis, which shows that more electrons are shared between the lone pair of the N atom and the 3d orbitals of Fe2<sup>+</sup> for <sup>n</sup> <sup>=</sup> 1. On the other hand, for larger cluster sizes with <sup>n</sup> <sup>=</sup> <sup>2</sup>−4, which primarily contain Fe3+, the electron density is more localized over the bonding region, as also reported by Sierka et al.[55](#page-28-0). Therefore, bonding occurs with the nitrogen lone pair. Our theoretical analysis indicates that the adsorption energy ∆Gads of ammonia on (Fe2O3)<sup>n</sup> clusters decreases from n = 1 to n = 3, followed by a slight increase for n = 4. A similar trend in the change of adsorption energy with cluster size was reported by Shulan Zhou et al.[56](#page-28-1) for Run@CNT systems. We also compared the adsorption energy of NH<sup>3</sup> on different metal and metal oxides in Table [I.](#page-10-0) The obtained NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters are about 10 kcal/mol higher than the data reported by Zhang et al. for the Ru(0001) surface[57](#page-28-2). Moreover, the adsorption of NH<sup>3</sup> and NO<sup>x</sup> on the γ-Fe2O3(111) surface was studied by Wei Huang et al.[58](#page-28-3) using periodic density functional calculations. They calculated adsorption energies on octahedral and tetrahedral sites of γ-Fe2O3(111) to be -2.13 kcal/mol and -21.68 kcal/mol, respectively. Similarly, our calculated NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters for n = 3 and n = 4 are close to the data reported by Wei Huang et al.,[58](#page-28-3) as adsorption of NH<sup>3</sup> on the three-coordinated Fe3<sup>+</sup> site resembles the tetrahedral site of γ-Fe2O3(111), while the adsorption on the four-coordinated Fe3<sup>+</sup> site resembles the octahedral site of γ-Fe2O3. <span id="page-9-0"></span>![FIG. 2: The optimized geometries of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> for n = 1−4; N−Fe distances (Å) are shown in parentheses along with the partial atomic charges on neighboring atoms. The total spin S of the clusters is shown in inserts.](path) As mentioned above, the calculated adsorption energies indicate that the adsorption of an NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters (n = 1−4) weakens as the cluster size increases from n = 1 to n = 3. In industrial processes, the dehydrogenation of ammonia typically occurs at high temperatures, often in the range of 400◦C to 700◦C, depending on the specific catalysts and conditions used. Therefore, it is important to determine the range of temperatures at which ammonia adsorption on (Fe2O3)<sup>n</sup> remains stable. Figure S5 demonstrates the temperature dependence of ∆Gads in the range from 0 K to 1200 K for the most stable adsorption configurations of NH<sup>3</sup> on (Fe2O3)n clusters (n = 1 − 4). The negative values of ∆Gads correspond to stable adsorption. As seen in Fig. S5, NH<sup>3</sup> adsorbed on the smallest Fe2O<sup>3</sup> cluster is stable across the entire temperature range of 0 K to 1200 K. However, for larger cluster sizes, ammonia adsorption becomes energetically unfavorable at temperatures of 1107 (K), 961 (K), and 1000 (K) for n = 2,3, and 4, respectively. ## C. NH<sup>3</sup> decomposition on Fe2O<sup>3</sup> In this section, we discuss the complete NH<sup>3</sup> decomposition and H<sup>2</sup> formation reactions [$7$](#page-6-0) - [$12$](#page-7-0) on the smallest considered cluster, Fe2O3, at room temperature, T=298.15 K, explored by the AFIR method. This method allows for the automatic exploration of the full reaction path network, systematically accounting for the variety of possible isomer structures and adsorption sites. This is an important approach in nanocatalysis because it has been demonstrated that the most stable structures are not always the most reactive. Therefore, a systematic search for reaction pathways that accounts for the contributions of low-energy isomers is required to accurately describe the catalytic properties of clusters at finite temperatures.[49](#page-27-10) To illustrate the isomer and reaction-site effects, we explicitly consider two different isomers of the Fe2O<sup>3</sup> cluster: the most stable kite-like structure with one terminal oxygen atom, and the linear structure isomer with two terminal oxygen atoms which is 6.24 kcal/mol less stable (see Fig. S1). The kite-like structure possess two type of catalytically active metal centers - two-coordinated and three-coordinated Fe sites. Therefore we consider adsorption and decomposition of NH<sup>3</sup> molecule on both of them. Figure [3\(](#page-13-0)a) demonstrates that the adsorption of NH<sup>3</sup> on the kite-like Fe2O<sup>3</sup> cluster is an exothermic reaction, occurring at both the two-coordinated and three-coordinated Fe sites. The adsorption free energies are -26.98 kcal/mol for the two-coordinated Fe site (intermediate I′ 1 1) and -11.29 kcal/mol for the three-coordinated Fe site (intermediate I′′ 1 1), respectively. The optimized structures of all intermediates (I) and transition satates (T) along the reaction pathways are shown in Fig. [3\(](#page-13-0)b) and [4\(](#page-14-0)b), for the kite-like and linear clusters, respectively. Here the lower index corresponds to the cluster size n, while the numbering corresponds to the order of intermedeates (transition states) along the reaction path. As discussed in the previous section, the most stable adsorption site for NH<sup>3</sup> is the two-coordinated Fe site, with an Fe−N bond length of 2.11 Å. In contrast, the Fe−N bond length at the three-coordinated Fe site is 2.16 Å. These findings are supported by the fact that NH<sup>3</sup> adsorption highly depends on the local geometry and electronic structure of the catalyst. In the case of the Fe2O<sup>3</sup> kite-like structure, the first dehydrogenation reaction is the second step in the reaction mechanism, occurring after adsorption with activation barriers of 26.98 kcal/mol and 22.12 kcal/mol through the reaction paths I′ 1 1-T′ 1 1-I′ 1 2 and I′′ 1 1-T′′ 1 1-I′′ 1 2, respectively. The reactions on these two-coodrinated and three-coordinated active sites are exothermic by 16.31 kcal/mol and 7.53 kcal/mol, respectively. However, the first dehydrogenation of NH<sup>3</sup> on the lineartype structure Fig. [4\(](#page-14-0)a) occurs with smaller activation barrier of 16.22 kcal/mol via the reaction path I<sup>L</sup> 1 1 - T<sup>L</sup> 1 1 - I<sup>L</sup> 1 2, demonstrating that the less stable linear isomer is more reactive. The role of Fe2O<sup>3</sup> isomer structure on NH<sup>3</sup> adsorption and first hydrogen atom transfer was previousely studied by Chaoyue Xie et al.[60](#page-28-5) They performed DFT-D3 calculations on the adsorption mechanisms of different molecules (NH3, NO, O2) on activated carbon (AC) supported iron-based catalysts FexOy/AC. The calculated adsorption electronic energies of NH<sup>3</sup> were -37.4 kcal/mol and -53.7 kcal/mol on different isomers of Fe2O3/AC, and the first hydrogen atom transfer had an activation barrier of 15.5 kcal/mol. Similarly, the adsorption and dehydrogenation of ammonia on different metal oxides were investigated by Erdtman and co-workers[62](#page-28-7) for the application of gas sensors. They reported that the adsorption energy of NH<sup>3</sup> on the RuO2(110) surface is -38.24 kcal/mol, and the first N−H bond cleavage had an activation energy barrier of 17.45 kcal/mol. The third step of the NH<sup>3</sup> dehydrogenation reaction [$9$](#page-6-1) involves the dissociation of the adsorbed NH∗ 2 intermediate into NH∗ and H∗ species. In this step, the abstracted hydrogen atom transfers to one of the oxygen atoms in the cluster. Figure [3$](#page-13-0)a) demonstrates, that in the case of the kite-like structure the energy barriers for this step are 43.91 kcal/mol and 34.51 kcal/mol, corresponding to the reaction paths I′ 1 2 - T′ 1 2 - I′ 1 3 and I′′ 1 2 - T′′ 1 2 - I′′ 1 3, respectively. In the fourth step [\(10$](#page-6-2), the adsorbed NH∗ intermediate further dissociates into N∗ and H∗ species as shown in Fig. [3$](#page-13-0)a). The reaction barriers associated with this step are 46.98 kcal/mol and 8.95 kcal/mol for the two-coordinated and three-coordinated reaction paths, respectively. The decomposition of NH<sup>3</sup> on kite-like structures becomes endothermic starting from the third step [\(9$](#page-6-1). Our calculations reveal that NH<sup>3</sup> dehydrogenation has a high energy barrier when the NH<sup>3</sup> molecule is adsorbed at a two-coordinated Fe site, which is the most stable adsorption site. On the other hand, dehydrogenation of the adsorbed NH<sup>3</sup> at a three-coordinated Fe site has a considerably lower activation barrier of 8.95 kcal/mol for the reaction step [$10$](#page-6-2). Overall, for the NH<sup>3</sup> decomposition reaction on the kite-like Fe2O<sup>3</sup> structure, with initial NH<sup>3</sup> adsorption on the two-coordinated Fe atom, the rate-limiting step is the fourth reaction [$10$](#page-6-2), with a barrier of 46.98 kcal/mol. Alternatively, for the less favorable NH<sup>3</sup> adsorption on the threecoordinated Fe atom, the rate-limiting step is the third reaction step [$9$](#page-6-1), with a barrier of 34.51 kcal/mol. The reaction pathway calculated for NH<sup>3</sup> decomposition on the linear-type Fe2O<sup>3</sup> isomer is shown in Fig. [4\(](#page-14-0)a), and respective intermediate and transition state structures are shown in Fig. [4\(](#page-14-0)b). Since this structure consists of two iron atoms connected through a central oxygen, each containing a terminal oxygen, the reaction mechanism differs slightly from that of the kite-like isomer. For instance, in the third step of the reaction, the second hydrogen from the adsorbed NH∗ 2 intermediate is transferred to the second terminal oxygen. The energy barrier for this step on the linear-type structure is 23.8 kcal/mol, as shown in the reaction path (I<sup>L</sup> 1 2 - T<sup>L</sup> 1 2 - I<sup>L</sup> 1 3) in Fig. [4a](#page-14-0). The fourth step on this isomer is not straightforward, involving the central oxygen atom breaking its bond with one of the neighboring iron atoms while forming an Fe − N − Fe bridge. This process leads to two different intermediates: the formation of the adsorbed H2O ∗ and the transfer of a hydrogen atom from one side of the Fe − N − Fe bridge to the other. Subsequently, the final dehydrogenation step from the NH∗ intermediate occurs, with an activation energy barrier of 34.76 kcal/mol. <span id="page-13-0"></span>![FIG. 3: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ reaction path on the kite-like isomer of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-14-0"></span>![FIG. 4: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> + → N ∗ + 3H∗ reaction path on the linear-type isomet of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) As a next step we consider possible H<sup>2</sup> formation via reactions [$11$](#page-6-3) and [$12$](#page-7-0) on the kite-like and linear isomers of Fe2O<sup>3</sup> cluster. The possible pathways for H<sup>2</sup> formation in the case of the most stable ammonia adsorption on the two-coordinated site (I′ intermediates) of the kite-like Fe2O<sup>3</sup> isomer are shown in Fig. [5$](#page-15-0)a), while the corresponding structures of the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [5\(](#page-15-0)b). Note that H<sup>2</sup> formation can occur after partial decomposition of ammonia in reaction [\(11$](#page-6-3), starting from intermediate (I<sup>I</sup> 1 3) via the path I′ 1 3 - T′ 1 6 - I′ 1 7 - T′ 1 7 - I′ 1 8. On the other hand, H<sup>2</sup> formation can occur via full decomposition of ammonia in reaction [$12$](#page-7-0), through the intermediate (II 1 4) via the path I′ 1 4 - T′ 1 4 - I′ 1 5 - T′ 1 5 - I′ 1 6. In both cases, the reaction pathways include breaking one O−H bond and forming an Fe−H bond. The H<sup>2</sup> formation barriers through these intermediates are 89.74 kcal/mol and 92.49 kcal/mol, respectively. From these results, we conclude that H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> structure is more favorable via reaction [$11$](#page-6-3), with the NH<sup>∗</sup> intermediate remaining adsorbed on the cluster. The H<sup>2</sup> formation reaction, starting from (I<sup>I</sup> 1 4), is the rate-limiting step in molecular hydrogen formation on the kite Fe2O<sup>3</sup> cluster. Similarly, the H<sup>2</sup> formation reaction pathways on the linear-type structure of Fe2O<sup>3</sup> are shown in Fig. [6$](#page-16-0)a), while the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [6\(](#page-16-0)b). The H<sup>2</sup> formation through the NH<sup>∗</sup> intermediate (I<sup>L</sup> 1 4) via the reaction path I<sup>L</sup> 1 4 - T<sup>L</sup> 1 8 - I<sup>L</sup> 1 9 - T<sup>L</sup> 1 9 - I<sup>L</sup> 1 10 has an energy barrier of 79.99 kcal/mol. On the other hand H<sup>2</sup> formation through intermediate (I<sup>L</sup> 1 6) via reaction path I<sup>L</sup> 1 6 - T<sup>L</sup> 1 6 - I<sup>L</sup> 1 7 - T<sup>L</sup> 1 7 - I<sup>L</sup> 1 8 has an activation energy of 70.84 kcal/mol, which is about 10 kcal/mol lower energy than reaction path through intermediate (I<sup>L</sup> 1 4). Overall, on the basis of our calculated reaction pathways for H<sup>2</sup> formation show similar pattern for both kite-type and linear-type Fe2O3, where H<sup>2</sup> formation in reactions [\(11$](#page-6-3) and [$12$](#page-7-0) take place via breaking one of O−H bond and forming intermediate Fe−H bond. However, from both thermodynamic and kinetic perspectives, H<sup>2</sup> formation on the two types of Fe2O<sup>3</sup> structures varies. Reaction [$11$](#page-6-3) is more favorable on the kite-like structure, while reaction [$12$](#page-7-0) is more favorable on the linear structure. This highlights that the rate-limiting step for H<sup>2</sup> formation is highly dependent on the catalyst's structure. <span id="page-15-0"></span>![FIG. 5: (a) The energy profile for H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-16-0"></span>![FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [$11$](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [$11$](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [$12$](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [$10$](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8$](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8$](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [$10$](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [$11](#page-6-3) and [12$](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [$11$](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [$11$](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9$](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9$](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [$11$](#page-6-3) and [$12$](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [$11$](#page-6-3) compared to the fully decomposed pathway [$12$](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [$8$](#page-6-4) − [$12$](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 $2022$.](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 1: The most stable structures of (Fe2O3)<sup>n</sup> clusters with n = 1 − 4. The total spin S and the binding energy Eb, of the clusters are shown in inserts.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [$1$](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [$3$](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [$5$](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ## <span id="page-8-1"></span>B. Ammonia adsorption on (Fe2O3)<sup>n</sup> clusters Adsorption of ammonia on (Fe2O3)<sup>n</sup> clusters is a crucial initial step in the whole dehydrogenation process. Figure [2](#page-9-0) demonstrates the most stable adsorption configurations of the NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters with n = 1−4. The corresponding free energies of adsorption and Fe−N bond distances are shown in Table [I](#page-10-0) at 0 K. Our calculations show that the adsorption of NH<sup>3</sup> on the smallest Fe2O<sup>3</sup> cluster is the most stable among all cluster sizes considered in this study, with an adsorption free energy of -33.68 kcal/mol. This finding is corroborated by Mulliken charge analysis, which shows that more electrons are shared between the lone pair of the N atom and the 3d orbitals of Fe2<sup>+</sup> for <sup>n</sup> <sup>=</sup> 1. On the other hand, for larger cluster sizes with <sup>n</sup> <sup>=</sup> <sup>2</sup>−4, which primarily contain Fe3+, the electron density is more localized over the bonding region, as also reported by Sierka et al.[55](#page-28-0). Therefore, bonding occurs with the nitrogen lone pair. Our theoretical analysis indicates that the adsorption energy ∆Gads of ammonia on (Fe2O3)<sup>n</sup> clusters decreases from n = 1 to n = 3, followed by a slight increase for n = 4. A similar trend in the change of adsorption energy with cluster size was reported by Shulan Zhou et al.[56](#page-28-1) for Run@CNT systems. We also compared the adsorption energy of NH<sup>3</sup> on different metal and metal oxides in Table [I.](#page-10-0) The obtained NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters are about 10 kcal/mol higher than the data reported by Zhang et al. for the Ru(0001) surface[57](#page-28-2). Moreover, the adsorption of NH<sup>3</sup> and NO<sup>x</sup> on the γ-Fe2O3(111) surface was studied by Wei Huang et al.[58](#page-28-3) using periodic density functional calculations. They calculated adsorption energies on octahedral and tetrahedral sites of γ-Fe2O3(111) to be -2.13 kcal/mol and -21.68 kcal/mol, respectively. Similarly, our calculated NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters for n = 3 and n = 4 are close to the data reported by Wei Huang et al.,[58](#page-28-3) as adsorption of NH<sup>3</sup> on the three-coordinated Fe3<sup>+</sup> site resembles the tetrahedral site of γ-Fe2O3(111), while the adsorption on the four-coordinated Fe3<sup>+</sup> site resembles the octahedral site of γ-Fe2O3. <span id="page-9-0"></span>![FIG. 2: The optimized geometries of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> for n = 1−4; N−Fe distances (Å) are shown in parentheses along with the partial atomic charges on neighboring atoms. The total spin S of the clusters is shown in inserts.](path) As mentioned above, the calculated adsorption energies indicate that the adsorption of an NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters (n = 1−4) weakens as the cluster size increases from n = 1 to n = 3. In industrial processes, the dehydrogenation of ammonia typically occurs at high temperatures, often in the range of 400◦C to 700◦C, depending on the specific catalysts and conditions used. Therefore, it is important to determine the range of temperatures at which ammonia adsorption on (Fe2O3)<sup>n</sup> remains stable. Figure S5 demonstrates the temperature dependence of ∆Gads in the range from 0 K to 1200 K for the most stable adsorption configurations of NH<sup>3</sup> on (Fe2O3)n clusters (n = 1 − 4). The negative values of ∆Gads correspond to stable adsorption. As seen in Fig. S5, NH<sup>3</sup> adsorbed on the smallest Fe2O<sup>3</sup> cluster is stable across the entire temperature range of 0 K to 1200 K. However, for larger cluster sizes, ammonia adsorption becomes energetically unfavorable at temperatures of 1107 (K), 961 (K), and 1000 (K) for n = 2,3, and 4, respectively. ## C. NH<sup>3</sup> decomposition on Fe2O<sup>3</sup> In this section, we discuss the complete NH<sup>3</sup> decomposition and H<sup>2</sup> formation reactions [$7$](#page-6-0) - [$12$](#page-7-0) on the smallest considered cluster, Fe2O3, at room temperature, T=298.15 K, explored by the AFIR method. This method allows for the automatic exploration of the full reaction path network, systematically accounting for the variety of possible isomer structures and adsorption sites. This is an important approach in nanocatalysis because it has been demonstrated that the most stable structures are not always the most reactive. Therefore, a systematic search for reaction pathways that accounts for the contributions of low-energy isomers is required to accurately describe the catalytic properties of clusters at finite temperatures.[49](#page-27-10) To illustrate the isomer and reaction-site effects, we explicitly consider two different isomers of the Fe2O<sup>3</sup> cluster: the most stable kite-like structure with one terminal oxygen atom, and the linear structure isomer with two terminal oxygen atoms which is 6.24 kcal/mol less stable (see Fig. S1). The kite-like structure possess two type of catalytically active metal centers - two-coordinated and three-coordinated Fe sites. Therefore we consider adsorption and decomposition of NH<sup>3</sup> molecule on both of them. Figure [3\(](#page-13-0)a) demonstrates that the adsorption of NH<sup>3</sup> on the kite-like Fe2O<sup>3</sup> cluster is an exothermic reaction, occurring at both the two-coordinated and three-coordinated Fe sites. The adsorption free energies are -26.98 kcal/mol for the two-coordinated Fe site (intermediate I′ 1 1) and -11.29 kcal/mol for the three-coordinated Fe site (intermediate I′′ 1 1), respectively. The optimized structures of all intermediates (I) and transition satates (T) along the reaction pathways are shown in Fig. [3\(](#page-13-0)b) and [4\(](#page-14-0)b), for the kite-like and linear clusters, respectively. Here the lower index corresponds to the cluster size n, while the numbering corresponds to the order of intermedeates (transition states) along the reaction path. As discussed in the previous section, the most stable adsorption site for NH<sup>3</sup> is the two-coordinated Fe site, with an Fe−N bond length of 2.11 Å. In contrast, the Fe−N bond length at the three-coordinated Fe site is 2.16 Å. These findings are supported by the fact that NH<sup>3</sup> adsorption highly depends on the local geometry and electronic structure of the catalyst. In the case of the Fe2O<sup>3</sup> kite-like structure, the first dehydrogenation reaction is the second step in the reaction mechanism, occurring after adsorption with activation barriers of 26.98 kcal/mol and 22.12 kcal/mol through the reaction paths I′ 1 1-T′ 1 1-I′ 1 2 and I′′ 1 1-T′′ 1 1-I′′ 1 2, respectively. The reactions on these two-coodrinated and three-coordinated active sites are exothermic by 16.31 kcal/mol and 7.53 kcal/mol, respectively. However, the first dehydrogenation of NH<sup>3</sup> on the lineartype structure Fig. [4\(](#page-14-0)a) occurs with smaller activation barrier of 16.22 kcal/mol via the reaction path I<sup>L</sup> 1 1 - T<sup>L</sup> 1 1 - I<sup>L</sup> 1 2, demonstrating that the less stable linear isomer is more reactive. The role of Fe2O<sup>3</sup> isomer structure on NH<sup>3</sup> adsorption and first hydrogen atom transfer was previousely studied by Chaoyue Xie et al.[60](#page-28-5) They performed DFT-D3 calculations on the adsorption mechanisms of different molecules (NH3, NO, O2) on activated carbon (AC) supported iron-based catalysts FexOy/AC. The calculated adsorption electronic energies of NH<sup>3</sup> were -37.4 kcal/mol and -53.7 kcal/mol on different isomers of Fe2O3/AC, and the first hydrogen atom transfer had an activation barrier of 15.5 kcal/mol. Similarly, the adsorption and dehydrogenation of ammonia on different metal oxides were investigated by Erdtman and co-workers[62](#page-28-7) for the application of gas sensors. They reported that the adsorption energy of NH<sup>3</sup> on the RuO2(110) surface is -38.24 kcal/mol, and the first N−H bond cleavage had an activation energy barrier of 17.45 kcal/mol. The third step of the NH<sup>3</sup> dehydrogenation reaction [$9$](#page-6-1) involves the dissociation of the adsorbed NH∗ 2 intermediate into NH∗ and H∗ species. In this step, the abstracted hydrogen atom transfers to one of the oxygen atoms in the cluster. Figure [3$](#page-13-0)a) demonstrates, that in the case of the kite-like structure the energy barriers for this step are 43.91 kcal/mol and 34.51 kcal/mol, corresponding to the reaction paths I′ 1 2 - T′ 1 2 - I′ 1 3 and I′′ 1 2 - T′′ 1 2 - I′′ 1 3, respectively. In the fourth step [\(10$](#page-6-2), the adsorbed NH∗ intermediate further dissociates into N∗ and H∗ species as shown in Fig. [3$](#page-13-0)a). The reaction barriers associated with this step are 46.98 kcal/mol and 8.95 kcal/mol for the two-coordinated and three-coordinated reaction paths, respectively. The decomposition of NH<sup>3</sup> on kite-like structures becomes endothermic starting from the third step [\(9$](#page-6-1). Our calculations reveal that NH<sup>3</sup> dehydrogenation has a high energy barrier when the NH<sup>3</sup> molecule is adsorbed at a two-coordinated Fe site, which is the most stable adsorption site. On the other hand, dehydrogenation of the adsorbed NH<sup>3</sup> at a three-coordinated Fe site has a considerably lower activation barrier of 8.95 kcal/mol for the reaction step [$10$](#page-6-2). Overall, for the NH<sup>3</sup> decomposition reaction on the kite-like Fe2O<sup>3</sup> structure, with initial NH<sup>3</sup> adsorption on the two-coordinated Fe atom, the rate-limiting step is the fourth reaction [$10$](#page-6-2), with a barrier of 46.98 kcal/mol. Alternatively, for the less favorable NH<sup>3</sup> adsorption on the threecoordinated Fe atom, the rate-limiting step is the third reaction step [$9$](#page-6-1), with a barrier of 34.51 kcal/mol. The reaction pathway calculated for NH<sup>3</sup> decomposition on the linear-type Fe2O<sup>3</sup> isomer is shown in Fig. [4\(](#page-14-0)a), and respective intermediate and transition state structures are shown in Fig. [4\(](#page-14-0)b). Since this structure consists of two iron atoms connected through a central oxygen, each containing a terminal oxygen, the reaction mechanism differs slightly from that of the kite-like isomer. For instance, in the third step of the reaction, the second hydrogen from the adsorbed NH∗ 2 intermediate is transferred to the second terminal oxygen. The energy barrier for this step on the linear-type structure is 23.8 kcal/mol, as shown in the reaction path (I<sup>L</sup> 1 2 - T<sup>L</sup> 1 2 - I<sup>L</sup> 1 3) in Fig. [4a](#page-14-0). The fourth step on this isomer is not straightforward, involving the central oxygen atom breaking its bond with one of the neighboring iron atoms while forming an Fe − N − Fe bridge. This process leads to two different intermediates: the formation of the adsorbed H2O ∗ and the transfer of a hydrogen atom from one side of the Fe − N − Fe bridge to the other. Subsequently, the final dehydrogenation step from the NH∗ intermediate occurs, with an activation energy barrier of 34.76 kcal/mol. <span id="page-13-0"></span>![FIG. 3: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ reaction path on the kite-like isomer of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-14-0"></span>![FIG. 4: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> + → N ∗ + 3H∗ reaction path on the linear-type isomet of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) As a next step we consider possible H<sup>2</sup> formation via reactions [$11$](#page-6-3) and [$12$](#page-7-0) on the kite-like and linear isomers of Fe2O<sup>3</sup> cluster. The possible pathways for H<sup>2</sup> formation in the case of the most stable ammonia adsorption on the two-coordinated site (I′ intermediates) of the kite-like Fe2O<sup>3</sup> isomer are shown in Fig. [5$](#page-15-0)a), while the corresponding structures of the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [5\(](#page-15-0)b). Note that H<sup>2</sup> formation can occur after partial decomposition of ammonia in reaction [\(11$](#page-6-3), starting from intermediate (I<sup>I</sup> 1 3) via the path I′ 1 3 - T′ 1 6 - I′ 1 7 - T′ 1 7 - I′ 1 8. On the other hand, H<sup>2</sup> formation can occur via full decomposition of ammonia in reaction [$12$](#page-7-0), through the intermediate (II 1 4) via the path I′ 1 4 - T′ 1 4 - I′ 1 5 - T′ 1 5 - I′ 1 6. In both cases, the reaction pathways include breaking one O−H bond and forming an Fe−H bond. The H<sup>2</sup> formation barriers through these intermediates are 89.74 kcal/mol and 92.49 kcal/mol, respectively. From these results, we conclude that H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> structure is more favorable via reaction [$11$](#page-6-3), with the NH<sup>∗</sup> intermediate remaining adsorbed on the cluster. The H<sup>2</sup> formation reaction, starting from (I<sup>I</sup> 1 4), is the rate-limiting step in molecular hydrogen formation on the kite Fe2O<sup>3</sup> cluster. Similarly, the H<sup>2</sup> formation reaction pathways on the linear-type structure of Fe2O<sup>3</sup> are shown in Fig. [6$](#page-16-0)a), while the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [6\(](#page-16-0)b). The H<sup>2</sup> formation through the NH<sup>∗</sup> intermediate (I<sup>L</sup> 1 4) via the reaction path I<sup>L</sup> 1 4 - T<sup>L</sup> 1 8 - I<sup>L</sup> 1 9 - T<sup>L</sup> 1 9 - I<sup>L</sup> 1 10 has an energy barrier of 79.99 kcal/mol. On the other hand H<sup>2</sup> formation through intermediate (I<sup>L</sup> 1 6) via reaction path I<sup>L</sup> 1 6 - T<sup>L</sup> 1 6 - I<sup>L</sup> 1 7 - T<sup>L</sup> 1 7 - I<sup>L</sup> 1 8 has an activation energy of 70.84 kcal/mol, which is about 10 kcal/mol lower energy than reaction path through intermediate (I<sup>L</sup> 1 4). Overall, on the basis of our calculated reaction pathways for H<sup>2</sup> formation show similar pattern for both kite-type and linear-type Fe2O3, where H<sup>2</sup> formation in reactions [\(11$](#page-6-3) and [$12$](#page-7-0) take place via breaking one of O−H bond and forming intermediate Fe−H bond. However, from both thermodynamic and kinetic perspectives, H<sup>2</sup> formation on the two types of Fe2O<sup>3</sup> structures varies. Reaction [$11$](#page-6-3) is more favorable on the kite-like structure, while reaction [$12$](#page-7-0) is more favorable on the linear structure. This highlights that the rate-limiting step for H<sup>2</sup> formation is highly dependent on the catalyst's structure. <span id="page-15-0"></span>![FIG. 5: (a) The energy profile for H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-16-0"></span>![FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [$11$](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [$11$](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [$12$](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [$10$](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8$](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8$](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [$10$](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [$11](#page-6-3) and [12$](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [$11$](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [$11$](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9$](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9$](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [$11$](#page-6-3) and [$12$](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [$11$](#page-6-3) compared to the fully decomposed pathway [$12$](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [$8$](#page-6-4) − [$12$](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 $2022$.](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL^z (a) and spin δS^z (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.	# Light-induced Orbital and Spin Magnetism in 3d, 4d, and 5d Transition Metals Understanding the coherent interplay of light with the magnetization in metals has been a long-standing problem in ultrafast magnetism. While it is known that when laser light acts on a metal it can induce magnetization via the process known as the inverse Faraday effect (IFE), the most basic ingredients of this phenomenon are still largely unexplored. In particular, given a strong recent interest in orbital non-equilibrium dynamics and its role in mediating THz emission in transition metals, the exploration of distinct features in spin and orbital IFE is pertinent. Here, we present a first complete study of the spin and orbital IFE in 3d, 4d and 5d transition metals of groups IV−XI from first-principles. By examining the dependence on the light polarization and frequency, we show that the laser-induced spin and orbital moments may vary significantly both in magnitude and sign. We underpin the interplay between the crystal field splitting and spin-orbit interaction as the key factor which determines the magnitude and key differences between the spin and orbital response. Additionally, we highlight the anisotropy of the effect with respect to the ferromagnetic magnetization and to the crystal structure. The provided complete map of IFE in transition metals is a key reference point in the field of optical magnetism. ### Introduction The demonstration of ultrafast demagnetization in ferromagnets by the application of femtosecond laser pulses [\[1\]](#page-6-0) gave rise to the field of ultrafast spintronics and set the stage for efficient manipulation of magnetism by light. By now it has been rigorously demonstrated that all-optical helicitydependent magnetization switching can be achieved in wide classes of magnetic materials [\[2](#page-6-1)[–10\]](#page-7-0), thus paving the way to contactless ultrafast magnetic recording and information processing. In interpretation of the switching experiments, the inverse Faraday effect (IFE) − i.e. the phenomenon of magnetization induced by a coherent interaction with light acting on a material − has been considered as one of the major underlying mechanisms for the magnetization reversal. Although theoretically predicted and experimentally observed decades ago [\[11–](#page-7-1)[13\]](#page-7-2), a consensus in the theoretical understanding of IFE is still lacking [\[14](#page-7-3)[–19\]](#page-7-4) – however, several microscopic methods have been recently developed for the calculation of this phenomenon in diverse setups [\[20–](#page-7-5)[27\]](#page-7-6). Apart from its established role in the magnetization switching of ferromagnets and ferrimagnents, the impact of IFE in the THz regime has also been studied in antiferromagnets like CrPt [\[28\]](#page-7-7) and Mn2Au [\[29\]](#page-7-8). Moreover, it has been shown that the component of IFE which is perpendicular to the magnetization direction behaves differently with respect to the light helicity than the parallel component [\[21\]](#page-7-9), giving rise to helicity-dependent optical torques [\[30–](#page-7-10)[32\]](#page-7-11) and THz emission [\[33\]](#page-7-12). The impact of the optical torques has been demonstrated to be significant in antiferromagnetic Mn2Au [\[34\]](#page-7-13) where transverse IFE may be responsible for the THz emission [\[35,](#page-7-14) [36\]](#page-7-15), provide an alternative way to switching the magnetization [\[37\]](#page-7-16) as well as to drive domain wall motion [\[38\]](#page-7-17). Recently it has also been shown that light can induce colossal magnetic moments in altermagnets [\[39\]](#page-7-18). Although IFE is conventionally associated with an excitation by circularly polarized light, it can also be activated by linearly-polarized laser pulses [\[25,](#page-7-19) [29,](#page-7-8) [39–](#page-7-18)[42\]](#page-7-20). In solids, two contributions to the magnetization exist: due to spin and orbital moment of electrons. And while the discussion of IFE is normally restricted to the response of spin, since recently, non-equilibrium dynamics of orbital angular momentum started to attract significant attention [\[43](#page-7-21)[–45\]](#page-7-22). Namely, the emergence of orbital currents in the context of the orbital Hall effect [\[46–](#page-7-23)[49\]](#page-8-0), orbital nature of current-induced torques on the magnetization [\[50](#page-8-1)[–52\]](#page-8-2), and current-induced orbital accumulation sizeable even in light materials [\[53–](#page-8-3)[56\]](#page-8-4) have been addressed theoretically and demonstrated experimentally. Notably, in the context of light-induced magnetism, Berritta and co-workers have predicted that the IFE in selected transition metals can exhibit a sizeable orbital component [\[20\]](#page-7-5), with the generality of this observation reaching even into the realm of altermagnets [\[39\]](#page-7-18). At the same time it is also known that within the context of plasmonic IFE the orbital magnetic moment due to electrons excited by laser pulses in small nanoparticles of noble and simple metals can reach atomic values [\[57–](#page-8-5)[59\]](#page-8-6). Despite the fact that very little is known about the interplay of spin and orbital IFE in real materials, it is believed that IFE is a very promising effect in the context of orbitronics − a field which deals with manipulation of the orbital degree of freedom by external perturbations. Although the importance of IFE in mediating the lightinduced magnetism is beyond doubt, a comprehensive indepth material-specific knowledge acquired from microscopic theoretical analysis of this effect is still missing. While acquiring this knowledge is imperative for the field of ultrafast spintronics, since the initial seminal work by Berritta and coworkers [\[20\]](#page-7-5), very little effort has been dedicated to the explorations of this phenomenon from first principles. Here, we ![<span id="page-1-0"></span>FIG. 1. Map of light-induced magnetism in transition metals. (a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) fill this gap by providing a detailed first-principles study of the light-induced spin and orbital magnetism for the 3d, 4d and 5d transition metals of groups IV−XI. By exploring the dependence on the frequency and the polarization of light, we acquire insights into the origin and differences between spin and orbital flavors of IFE, as well as key features of their behavior. Our work provide a solid foundation for further advances in the field of optical magnetism and interaction of light and matter. # Results # Light-induced magnetism in transition metals We begin our discussion by exploring the IFE in a series of 3d, 4d and 5d transition metals of groups IV−XI. IFE is a non-linear optomagnetic effect in which magnetization δO ∝ E × E ∗ is induced as a second-order response to the electric field E of a laser pulse. We calculate the light-induced spin δS and orbital δL magnetic moments by means of the Keldysh formalism, see the section Methods [\[21,](#page-7-9) [39\]](#page-7-18). We focus on light energies hω̵ of 0.25 eV and 1.55 eV which have been routinely used before in our studies within the same formalism [\[29,](#page-7-8) [39,](#page-7-18) [60\]](#page-8-7). Additionally, we choose a lifetime broadening Γ of 25 meV that corresponds to room temperature in order to account for effects of disorder on the electronic states and to provide realistic estimations of the effect. In all calculations we choose the light to be circularly polarized in the xy or yz planes with a light intensity I of 10 GW/cm<sup>2</sup> , while for magnetic materials ferromagnetic magnetization is considered to be along the z-axis. In Fig. [1](#page-1-0) we present the calculated light-induced δL<sup>z</sup> [Fig. [1\(](#page-1-0)a-c)] and δS<sup>z</sup> [Fig. [1\(](#page-1-0)b-d)] moments of the considered transition metals at the Fermi energy. Light is circularly polarized in the xy plane, with energies hω̵ of 0.25 eV [Fig. [1\(](#page-1-0)ab)] and 1.55 eV [Fig. [1\(](#page-1-0)c-d)]. Regarding the 3d magnetic elements Fe, Co, Ni, we analyze both right and left handed circular polarizations of light. The exact values of δL<sup>z</sup> and δS<sup>z</sup> are listed in Tables [II](#page-14-0) and [III](#page-15-0) of the Supplemental Material (SM). Not shown are computed transverse x and y compo- ![<span id="page-2-1"></span>FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.](path) At first sight we notice a strong dependence of the magnitude and sign of both δL<sup>z</sup> and δS<sup>z</sup> on the light frequency. We point out the case of fcc Rh where an colossal δL<sup>z</sup> = −56 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> is predicted at hω̵ = 0.25 eV and which is drastically reduced at the higher frequency, while the spin component remains suppressed in both cases. For the case of magnetic hcp Co we notice a helicity-dependent change of sign for δL<sup>z</sup> at hω̵ = 0.25 eV which is not present for δSz. In general, δL<sup>z</sup> varies stronger with the light helicity than δSz. Moreover, δL<sup>z</sup> is one to two orders of magnitude larger than δS<sup>z</sup> for the non-magnetic elements, while they are of the same magnitude for the magnetic elemental materials. The last two observations are in agreement with the findings of Ref. [\[20\]](#page-7-5) and indicate how the spontaneous magnetization strongly influences the effect in ferromagnets by the time-reversal symmetry breaking. For comparison, in Table [I](#page-2-0) we list the values of the total light-induced moments, defined as the sum of δL<sup>z</sup> and δSz, for the transitional metals studied in [\[20\]](#page-7-5) with hω̵ = 1.55 eV. Our calculated values are of the same order of magnitude, with the exceptions of Au, and of Co for the case of left-handed polarization, which can be attributed to the difference in the computational methods. From Fig. [1\(](#page-1-0)c-d) we get a clear picture of how the induced moments scale with the strength of spin-orbit coupling (SOC). The effect is overall larger for the heavier 5d elements, both in the orbital δL<sup>z</sup> and spin δS<sup>z</sup> channels. However, at the lower frequency of hω̵ = 0.25 eV it is more difficult to draw such a conclusion since the corresponding photon energy falls into the range of SOC strength, which promotes the role of electronic transitions among spin-orbit split bands occurring within a narrow range around the Fermi energy and limited regions in k-space. In contrast, the use of a much larger frequency involves transitions among manifestly orbitally-distinct states which take place over larger portions <span id="page-2-0"></span>TABLE I. Total light-induced magnetic moments for transition metals previously studied in [\[20\]](#page-7-5) in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV. of the reciprocal space with a more uniform impact of the spin-orbit strength on the magnitude of the transition probabilities. From this discussion we have to exclude the ferromagnetic elements since the spontaneous magnetization induces bands splittings on the scale of exchange strength therefore making the effect much more complex. We study the dependence on the light polarization by performing additional calculations for light circularly polarized in the yz-plane, presenting the results for the z and x components of δL and δS respectively in Figs. [6](#page-10-0) and [7](#page-11-0) of the SM for hω̵ = 0.25 eV and hω̵ = 1.55 eV. Our results for δS<sup>z</sup> and δS<sup>x</sup> for the magnetic transition metals reproduce exactly the values of Ref. [\[21\]](#page-7-9), computed with the same method. We further present the values of δL<sup>z</sup> and δLx, where we find the orbital response along the magnetization, δLz, to be by a factor of two larger than the corresponding spin response, when averaged over all magnetic elements. Generally, for the z-component the effect at the magnetic elements is sim- ![<span id="page-3-0"></span>FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level E<sup>F</sup> (dashed gray line). The horizontal dotted red lines at E<sup>F</sup> ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.](path) ilar in magnitude with the case of polarization in the xy-plane shown in Fig. [1,](#page-1-0) and even in the light helicity, while it vanishes for the non-magnetic elements. On the other hand, for the x-component the effect at the magnetic elements becomes one-two orders of magnitude smaller and is odd in the light helicity, although for the non-magnetic elements the values are comparable to the case shown in Fig. [1.](#page-1-0) It is important to note that for light circularly-polarized in the plane containing ferromagnetic magnetization, the light-induced transverse to magnetization induced moments, even though being smaller than the longitudinal ones, are related to light-induced torques that are experimentally demonstrated to lead to helicity-dependent THz emission [\[21,](#page-7-9) [33\]](#page-7-12). In order to get a better understanding of the impact of the time-reversal symmetry breaking on IFE, we consider the cases of ferromagnetic, non-relativistic (i.e. computed without SOC) ferromagnetic, and antiferromagnetic fcc Ni. In Figs. [8](#page-12-0) and [9](#page-13-0) of the SM we present the band filling dependence of the Cartesian components of the light-induced orbital moments δL, for hω̵ = 0.25 eV and hω̵ = 1.55 eV, respectively. In the non-relativistic case only a δL<sup>i</sup> parallel to the light propagation axis is induced which is odd in the helicity, and no δS<sup>i</sup> is induced, exemplifying that the orbital response is the primary non-relativistic one, whereas the spin response is generated through SOC, as was also shown in [\[20\]](#page-7-5). Similarly, for the antiferromagnetic case, only a component parallel to the light propagation axis is induced which is odd in the helicity and remains unchanged under different polarization flavors. While this is the case also for the induced components δL<sup>x</sup> and δL<sup>y</sup> which are transverse to the magnetization in the ferromagnetic case, the situation drastically changes for the induced component δL<sup>z</sup> parallel to the magnetization, as discussed earlier. We note that a tiny, odd in the helicity, δL<sup>x</sup> or δL<sup>y</sup> is additionally induced when light is rotating in a plane containing ferromagnetic magnetization. On the other hand, the induced δL<sup>x</sup> and δLy, developing normal to the polarization plane, serve as a non-relativistic "background" which is independent of the magnetization, with features due to the crystal structure driving the effect over larger regions in energy. The additional band-splittings induced by SOC result in the IFE exhibiting more features with band filling in the relativistic scenario. Remarkably, while the relativistic antiferromagnetic and non-relativistic ferromagnetic cases in principle have similar to each other behavior in energy, a larger signal arises in the antiferromagnetic case by the virtue of flatter bands (see also the discussion for Hf and Pt below). Lastly, we note that a similar behavior has been observed for the in-plane spin IFE in PT-symmetric Mn2Au, however, in the latter case additional out-of-plane moments arise due to linearly polarized light as a result of broken by the magnetization inversion symmetry [\[29\]](#page-7-8). Next, we focus on the case of 5d transition metals where in Fig. [2\(](#page-2-1)a-b) we explore the relation of δL<sup>z</sup> and δSz, respectively, to the band filling, for light circularly polarized in the xy plane and the frequency of hω̵ = 1.55 eV. When going from group IV (hcp Hf) to XI (fcc Au) we observe a smooth ![<span id="page-4-0"></span>FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.](path) variation of δL<sup>z</sup> from positive to negative values, as well as nicely shaped plateaus, where δL<sup>z</sup> remains relatively robust in a wide energy region, for hcp Hf, bcc Ta, bcc W, fcc Ir, and fcc Pt. We note that such plateaus are often characteristic of orbital effects, as witnessed for example in orbital Hall insulators [\[61](#page-8-8)[–63\]](#page-8-9) and orbital Rashba systems [\[60\]](#page-8-7). On the contrary, δS<sup>z</sup> exhibits a very erratic behavior with strong variations for each material, at the same time being much smaller in magnitude than δLz. The above observation is a clear manifestation of how differently the orbital and spin degrees of freedom behave under light excitation, with the orbital angular momentum having its origin in intrinsic structural parameters as manifested in the crystal field splitting, whereas the spin angular momentum is more sensitive to finer details of the electronic structure mediated by SOC. Indeed, the light-induced spin and orbital moments exhibit similarly erratic behavior with band filling once the frequency of the light is drastically reduced to reach the range of spin-orbit interaction, see Fig. [10](#page-13-1) of the SM for hω̵ = 0.25 eV. # Anatomy of IFE in k-space Among the 5d transition metals, hcp Hf and fcc Pt exhibit the largest computed moments at the Fermi energy for hω̵ = 1.55 eV, with the corresponding values of δL<sup>z</sup> = 9.1 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −0.8 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Hf, and δL<sup>z</sup> = −7.4 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −1.1 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Pt. Therefore, we select these two materials and explore the behavior of their light-induced moments in reciprocal space. We present the band-resolved δL<sup>z</sup> and δS<sup>z</sup> for hcp Hf in Fig. [3\(](#page-3-0)a-b), as well as band-resolved δL<sup>z</sup> and δS<sup>z</sup> for fcc Pt in Fig. [3\(](#page-3-0)c-d). For the case of Hf, transitions along the bands near the A-point are the main source of δL<sup>z</sup> and δSz. Light-induced δL<sup>z</sup> consists of hotspot-like negative contributions and secondary, but extended over energy and k-space consistently positive contributions, which are hardly visible. Overall, however, the latter lead to a large positive orbital integrated response. On the other hand, δS<sup>z</sup> For the case of Pt, δL<sup>z</sup> and δS<sup>z</sup> arise from transitions close to X and L high-symmetry points, with both originating in roughly the same regions of (E, k)-space, but often having an opposite sign to each other. Note that the bands in Pt are much more dispersive in the considered energy window, which results in an effective reduction of the regions in (E, k)-space which contribute to the spin and orbital response alike. This is in contrast to Hf, where much flatter bands reside above and below the Fermi energy within the energy window of the laser pulse, providing significant integrated, albeit very small locally, contributions. Moreover, in fcc Pt the Fermi energy cuts through the band edges of the d-states, where the effect of spin-orbit interaction is the strongest, which explains the emergence of strong hotspot-like contributions with a clear correlation in the magnitude of spin and orbital response and the resultant similar behavior of δL<sup>z</sup> and δS<sup>z</sup> with band filling around the Fermi energy. We further scrutinize the reciprocal space distribution of δL<sup>z</sup> and δSz, shown for hcp Hf and fcc Pt in Fig. [3\(](#page-3-0)e-h). For both materials δL<sup>z</sup> distributions consist of large uniform areas of either positive or negative sign. On the contrary, δS<sup>z</sup> distributions are much finer and richer in details with more areas of opposite sign, consistent with the picture we drew above from the band-resolved analysis. A similar behavior of the light-induced magnetism in reciprocal space has been recently reported for rutile altermagnets [\[39\]](#page-7-18). We also observe that, as discussed above, while for Hf the contributions are well spread throughout the Brillouin zone, for Pt the spin and orbital contributions are located at the edges of the considered k<sup>z</sup> − k<sup>y</sup> plane. Overall, this fact indicates that the microscopic behavior of light-induced magnetism varies strongly among transition metals and crucially depends on the crystal structure and position of the Fermi level with respect to the states ![<span id="page-5-0"></span>FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ### Anisotropy of light-induced magnetism We first address the anisotropy of light-induced magnetism with respect to the magnetization by examining the response of magnetic elements under different flavors of circular polarization. In Fig. [4\(](#page-4-0)a-b) we present the computed δL<sup>z</sup> and δS<sup>z</sup> in relation to scattering lifetime Γ, for the cases of magnetic bcc Fe and hcp Co under excitation by light which is circularly polarized in the xy or yz planes at hω̵ = 1.55 eV. As we have already seen in Fig. [1](#page-1-0) for light polarized in the xyplane, the responses behave differently for right (solid lines) or left (dashed lines) polarization. This difference is more pronounced for δLz, for which we even observe a change of sign for Co. Surprisingly, the situation is drastically different for the case of yz-polarization. In this case, for bcc Fe the response is even in the helicity, while for hcp Co δL<sup>z</sup> is almost even and δS<sup>z</sup> is perfectly even in the helicity. Such different behavior with respect to the light helicity between the two magnetic elements can be traced back to the additional anisotropy originating in the crystal structure itself − an effect which we discuss below. Notably, we witness a highly nonlinear behavior with respect to Γ, with the response reaching colossal values in the clean limit. For example, δL<sup>z</sup> and δS<sup>z</sup> can reach as much as 40 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> and 70 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> in Co, respectively, given the scattering lifetime of 1 meV. Besides the potential tunability of the inverse Faraday effect by the degree of the disorder of the samples, one key message that we extract from our observations is that when comparing the values for laser-induced magnetic moments obtained with different methods, special care has to be taken since the implementations of disorder effects, even within a simple constant broadening model, may differ among various approaches. Finally, we analyze the anisotropy that the crystal structure induces in the light-induced magnetism. We select the case of hcp Re and present in Fig. [5\(](#page-5-0)a-b) the x and z components of δL and δS, respectively, in relation to Γ. Due to the inequivalence of x and z axes in the hcp structure, the δL<sup>z</sup> which originates in light circularly polarized in the xy-plane (light blue line) differs from the δL<sup>x</sup> for the case of the yz polarization (brown line). The situation is similar for the case of δS<sup>z</sup> and δSx. On the contrary, as presented in Fig. [11](#page-14-1) of the SM, in the case of fcc Pt we have a perfect match between δL<sup>z</sup> and δL<sup>x</sup> (or δS<sup>z</sup> and δSx) when changing the plane of circular polarization because the x and z axes are equivalent in the fcc structure. The situation is similar for the y component when light is circularly polarized in the xz-plane. As also expected from the symmetry of the crystal structure, we confirm that the light-induced moments are perfectly odd in the helicity for the nonmagnetic elements, which is not the case for the magnetic elements, see Figs. [5](#page-5-0) and [11](#page-14-1) of the SM [\[20\]](#page-7-5). ### Discussion The main goal of our work is to showcase the importance of the orbital component of IFE and its distinct behavior from the spin counterpart. While so far it was mainly the interaction of light with the spin magnetization that was taken into consideration for the interpretation of IFE-related effects, we speculate that the orbital IFE may provide a novel way to coherently induce magnetization and manipulate the magnetic order. For example, in a recent study, different types of optical torques that may arise in ferromagnetic layers were interpreted in terms of the light-induced orbital moment and its interaction with the magnetization through the spin-orbit interaction [\[64\]](#page-8-10). Since it is known that current-induced orbital accumulation and orbital torques exhibit a long-range behavior [\[49,](#page-8-0) [65](#page-8-11)[–67\]](#page-8-12) due to a characteristic small orbital decay, a question arises whether similar behavior can be exhibited by optical torques caused by the orbital IFE. The emergence of long-ranged orbital IFE would come as no surprise given the fact that several recent experiments reported that laser excitation can drive long-range ballistic orbital currents resulting in THz emission [\[68](#page-8-13)[–71\]](#page-8-14). a perturbation by the electric field of the original ground state Hamiltonian, it is common to utilize the angular momentum of light in order to understand the interaction with the magnetic order by the means of transfer of angular momentum. The spin of light through the helicity of circularly polarized pulses plays a crucial role in helicity-dependent all-optical switching scenarios. On the other hand, there is a strong recent interest in utilizing the orbital angular momentum of light via irradiation of matter with e.g. vortex beams or twisted light, in order to probe the magnetization [\[72\]](#page-8-15), generate photocurrents [\[73\]](#page-8-16), drive ultrafast demagnetization [\[74\]](#page-8-17), and induce IFE [\[40,](#page-7-24) [75\]](#page-8-18). Therefore, it is imperative to treat both spin and orbital degrees of freedom on equal footing when exploring the lightmatter interaction. This will not only trigger further advances in the field of ultrafast magnetism and THz spintronics, but also enable a transition to the novel field of attosecond spintronics [\[76,](#page-8-19) [77\]](#page-8-20). Method. In this work we calculate the first-principles electronic structures of 3d, 4d and 5d transition metals by using the full-potential linearized augmented plane wave FLEUR code [\[78\]](#page-8-21). We describe exchange and correlation effects by using the non-relativistic PBE [\[79\]](#page-8-22) functional, while relativistic effects are described by the second-variation scheme [\[80\]](#page-9-0). The parameters of our first-principles calculations, i.e. lattice constants, muffin-tin radii, plane-wave cutoffs, etc. are taken from Table I of Ref. [\[81\]](#page-9-1). Next, we construct maximally-localized Wannier functions (MLWFs) by employing the Wannier90 code [\[82\]](#page-9-2) and its interface with the FLEUR code [\[83\]](#page-9-3). Similarly to [\[81\]](#page-9-1), we choose s, p and d orbitals for the initial projections and disentagle 18 MLWFs out of 36 Bloch states within a frozen window of 5.0 eV above the Fermi energy for each atom in the unit cell. where O<sup>i</sup> is the i-th component of either the orbital angular momentum operator L<sup>i</sup> or of the spin operator S<sup>i</sup> . Moreover, a<sup>0</sup> = 4πϵ0h̵<sup>2</sup> /(mee 2 ) is the Bohr's radius, I = ϵ0cE<sup>2</sup> 0 /2 is the intensity of the pulse, ϵ<sup>0</sup> is the vacuum permittivity, m<sup>e</sup> is the electron mass, e is the elementary charge, h̵ is the reduced Planck constant, c is the light velocity, E<sup>H</sup> = e 2 /(4πϵ0a0) is the Hartree energy, and ϵ<sup>j</sup> is the j-th component of the polarization vector of the pulse. For example, we describe right/left-handedly polarized light in the xy-plane as ϵ = (1,±i, <sup>0</sup>)/<sup>√</sup> 2, respectively, and define its propagation vector to lie along the normal to the polarization plane. A detailed form of the tensor φijk can be seen in Eq.(14) of Ref. [\[21\]](#page-7-9). For the orbital response, the prefactor in Eq. [$1$](#page-6-2) must be multiplied by an additional factor of 2. A 128×128×128 interpolation k-mesh is sufficient to obtain well-converged results. In all calculations the lifetime broadening Γ was set at 25 meV, the light frequency hω̵ at 0.25 eV and 1.55 eV, the intensity of light at 10 GW/cm<sup>2</sup> , and we covered an energy region of [−2.5, 2.5] eV around the Fermi energy level E<sup>F</sup> . Acknowledgements. We thank Frank Freimuth and Maximilian Merte for discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR 173/3 − 268565370 (project A11) and by the K. and A. Wallenberg Foundation (Grants No. 2022.0079 and 2023.0336). We acknowledge support from the EIC Pathfinder OPEN grant 101129641 "OBELIX". We also gratefully acknowledge the Julich Supercomputing Cen- ¨ tre and RWTH Aachen University for providing computational resources under projects jiff40 and jara0062. Author Contributions. T. A. performed numerical calculations and analysed the results. T. A. and Y. M. wrote the manuscript. 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(a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-11-0"></span>FIG. 7. (a-d) Light-induced orbital δL<sup>x</sup> (a, c) and spin δS<sup>x</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-12-0"></span>FIG. 8. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-13-0"></span>FIG. 9. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ![<span id="page-13-1"></span>FIG. 10. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-14-1"></span>FIG. 11. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic fcc Pt. The induced magnetic moments along the x-axis and along the z-axis are presented. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) <span id="page-14-0"></span>TABLE II. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 0.25 eV. <span id="page-15-0"></span>TABLE III. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV.
FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.	# Light-induced Orbital and Spin Magnetism in 3d, 4d, and 5d Transition Metals Understanding the coherent interplay of light with the magnetization in metals has been a long-standing problem in ultrafast magnetism. While it is known that when laser light acts on a metal it can induce magnetization via the process known as the inverse Faraday effect (IFE), the most basic ingredients of this phenomenon are still largely unexplored. In particular, given a strong recent interest in orbital non-equilibrium dynamics and its role in mediating THz emission in transition metals, the exploration of distinct features in spin and orbital IFE is pertinent. Here, we present a first complete study of the spin and orbital IFE in 3d, 4d and 5d transition metals of groups IV−XI from first-principles. By examining the dependence on the light polarization and frequency, we show that the laser-induced spin and orbital moments may vary significantly both in magnitude and sign. We underpin the interplay between the crystal field splitting and spin-orbit interaction as the key factor which determines the magnitude and key differences between the spin and orbital response. Additionally, we highlight the anisotropy of the effect with respect to the ferromagnetic magnetization and to the crystal structure. The provided complete map of IFE in transition metals is a key reference point in the field of optical magnetism. ### Introduction The demonstration of ultrafast demagnetization in ferromagnets by the application of femtosecond laser pulses [\[1\]](#page-6-0) gave rise to the field of ultrafast spintronics and set the stage for efficient manipulation of magnetism by light. By now it has been rigorously demonstrated that all-optical helicitydependent magnetization switching can be achieved in wide classes of magnetic materials [\[2](#page-6-1)[–10\]](#page-7-0), thus paving the way to contactless ultrafast magnetic recording and information processing. In interpretation of the switching experiments, the inverse Faraday effect (IFE) − i.e. the phenomenon of magnetization induced by a coherent interaction with light acting on a material − has been considered as one of the major underlying mechanisms for the magnetization reversal. Although theoretically predicted and experimentally observed decades ago [\[11–](#page-7-1)[13\]](#page-7-2), a consensus in the theoretical understanding of IFE is still lacking [\[14](#page-7-3)[–19\]](#page-7-4) – however, several microscopic methods have been recently developed for the calculation of this phenomenon in diverse setups [\[20–](#page-7-5)[27\]](#page-7-6). Apart from its established role in the magnetization switching of ferromagnets and ferrimagnents, the impact of IFE in the THz regime has also been studied in antiferromagnets like CrPt [\[28\]](#page-7-7) and Mn2Au [\[29\]](#page-7-8). Moreover, it has been shown that the component of IFE which is perpendicular to the magnetization direction behaves differently with respect to the light helicity than the parallel component [\[21\]](#page-7-9), giving rise to helicity-dependent optical torques [\[30–](#page-7-10)[32\]](#page-7-11) and THz emission [\[33\]](#page-7-12). The impact of the optical torques has been demonstrated to be significant in antiferromagnetic Mn2Au [\[34\]](#page-7-13) where transverse IFE may be responsible for the THz emission [\[35,](#page-7-14) [36\]](#page-7-15), provide an alternative way to switching the magnetization [\[37\]](#page-7-16) as well as to drive domain wall motion [\[38\]](#page-7-17). Recently it has also been shown that light can induce colossal magnetic moments in altermagnets [\[39\]](#page-7-18). Although IFE is conventionally associated with an excitation by circularly polarized light, it can also be activated by linearly-polarized laser pulses [\[25,](#page-7-19) [29,](#page-7-8) [39–](#page-7-18)[42\]](#page-7-20). In solids, two contributions to the magnetization exist: due to spin and orbital moment of electrons. And while the discussion of IFE is normally restricted to the response of spin, since recently, non-equilibrium dynamics of orbital angular momentum started to attract significant attention [\[43](#page-7-21)[–45\]](#page-7-22). Namely, the emergence of orbital currents in the context of the orbital Hall effect [\[46–](#page-7-23)[49\]](#page-8-0), orbital nature of current-induced torques on the magnetization [\[50](#page-8-1)[–52\]](#page-8-2), and current-induced orbital accumulation sizeable even in light materials [\[53–](#page-8-3)[56\]](#page-8-4) have been addressed theoretically and demonstrated experimentally. Notably, in the context of light-induced magnetism, Berritta and co-workers have predicted that the IFE in selected transition metals can exhibit a sizeable orbital component [\[20\]](#page-7-5), with the generality of this observation reaching even into the realm of altermagnets [\[39\]](#page-7-18). At the same time it is also known that within the context of plasmonic IFE the orbital magnetic moment due to electrons excited by laser pulses in small nanoparticles of noble and simple metals can reach atomic values [\[57–](#page-8-5)[59\]](#page-8-6). Despite the fact that very little is known about the interplay of spin and orbital IFE in real materials, it is believed that IFE is a very promising effect in the context of orbitronics − a field which deals with manipulation of the orbital degree of freedom by external perturbations. Although the importance of IFE in mediating the lightinduced magnetism is beyond doubt, a comprehensive indepth material-specific knowledge acquired from microscopic theoretical analysis of this effect is still missing. While acquiring this knowledge is imperative for the field of ultrafast spintronics, since the initial seminal work by Berritta and coworkers [\[20\]](#page-7-5), very little effort has been dedicated to the explorations of this phenomenon from first principles. Here, we ![<span id="page-1-0"></span>FIG. 1. Map of light-induced magnetism in transition metals. (a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) fill this gap by providing a detailed first-principles study of the light-induced spin and orbital magnetism for the 3d, 4d and 5d transition metals of groups IV−XI. By exploring the dependence on the frequency and the polarization of light, we acquire insights into the origin and differences between spin and orbital flavors of IFE, as well as key features of their behavior. Our work provide a solid foundation for further advances in the field of optical magnetism and interaction of light and matter. # Results # Light-induced magnetism in transition metals We begin our discussion by exploring the IFE in a series of 3d, 4d and 5d transition metals of groups IV−XI. IFE is a non-linear optomagnetic effect in which magnetization δO ∝ E × E ∗ is induced as a second-order response to the electric field E of a laser pulse. We calculate the light-induced spin δS and orbital δL magnetic moments by means of the Keldysh formalism, see the section Methods [\[21,](#page-7-9) [39\]](#page-7-18). We focus on light energies hω̵ of 0.25 eV and 1.55 eV which have been routinely used before in our studies within the same formalism [\[29,](#page-7-8) [39,](#page-7-18) [60\]](#page-8-7). Additionally, we choose a lifetime broadening Γ of 25 meV that corresponds to room temperature in order to account for effects of disorder on the electronic states and to provide realistic estimations of the effect. In all calculations we choose the light to be circularly polarized in the xy or yz planes with a light intensity I of 10 GW/cm<sup>2</sup> , while for magnetic materials ferromagnetic magnetization is considered to be along the z-axis. In Fig. [1](#page-1-0) we present the calculated light-induced δL<sup>z</sup> [Fig. [1\(](#page-1-0)a-c)] and δS<sup>z</sup> [Fig. [1\(](#page-1-0)b-d)] moments of the considered transition metals at the Fermi energy. Light is circularly polarized in the xy plane, with energies hω̵ of 0.25 eV [Fig. [1\(](#page-1-0)ab)] and 1.55 eV [Fig. [1\(](#page-1-0)c-d)]. Regarding the 3d magnetic elements Fe, Co, Ni, we analyze both right and left handed circular polarizations of light. The exact values of δL<sup>z</sup> and δS<sup>z</sup> are listed in Tables [II](#page-14-0) and [III](#page-15-0) of the Supplemental Material (SM). Not shown are computed transverse x and y compo- ![<span id="page-2-1"></span>FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.](path) At first sight we notice a strong dependence of the magnitude and sign of both δL<sup>z</sup> and δS<sup>z</sup> on the light frequency. We point out the case of fcc Rh where an colossal δL<sup>z</sup> = −56 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> is predicted at hω̵ = 0.25 eV and which is drastically reduced at the higher frequency, while the spin component remains suppressed in both cases. For the case of magnetic hcp Co we notice a helicity-dependent change of sign for δL<sup>z</sup> at hω̵ = 0.25 eV which is not present for δSz. In general, δL<sup>z</sup> varies stronger with the light helicity than δSz. Moreover, δL<sup>z</sup> is one to two orders of magnitude larger than δS<sup>z</sup> for the non-magnetic elements, while they are of the same magnitude for the magnetic elemental materials. The last two observations are in agreement with the findings of Ref. [\[20\]](#page-7-5) and indicate how the spontaneous magnetization strongly influences the effect in ferromagnets by the time-reversal symmetry breaking. For comparison, in Table [I](#page-2-0) we list the values of the total light-induced moments, defined as the sum of δL<sup>z</sup> and δSz, for the transitional metals studied in [\[20\]](#page-7-5) with hω̵ = 1.55 eV. Our calculated values are of the same order of magnitude, with the exceptions of Au, and of Co for the case of left-handed polarization, which can be attributed to the difference in the computational methods. From Fig. [1\(](#page-1-0)c-d) we get a clear picture of how the induced moments scale with the strength of spin-orbit coupling (SOC). The effect is overall larger for the heavier 5d elements, both in the orbital δL<sup>z</sup> and spin δS<sup>z</sup> channels. However, at the lower frequency of hω̵ = 0.25 eV it is more difficult to draw such a conclusion since the corresponding photon energy falls into the range of SOC strength, which promotes the role of electronic transitions among spin-orbit split bands occurring within a narrow range around the Fermi energy and limited regions in k-space. In contrast, the use of a much larger frequency involves transitions among manifestly orbitally-distinct states which take place over larger portions <span id="page-2-0"></span>TABLE I. Total light-induced magnetic moments for transition metals previously studied in [\[20\]](#page-7-5) in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV. of the reciprocal space with a more uniform impact of the spin-orbit strength on the magnitude of the transition probabilities. From this discussion we have to exclude the ferromagnetic elements since the spontaneous magnetization induces bands splittings on the scale of exchange strength therefore making the effect much more complex. We study the dependence on the light polarization by performing additional calculations for light circularly polarized in the yz-plane, presenting the results for the z and x components of δL and δS respectively in Figs. [6](#page-10-0) and [7](#page-11-0) of the SM for hω̵ = 0.25 eV and hω̵ = 1.55 eV. Our results for δS<sup>z</sup> and δS<sup>x</sup> for the magnetic transition metals reproduce exactly the values of Ref. [\[21\]](#page-7-9), computed with the same method. We further present the values of δL<sup>z</sup> and δLx, where we find the orbital response along the magnetization, δLz, to be by a factor of two larger than the corresponding spin response, when averaged over all magnetic elements. Generally, for the z-component the effect at the magnetic elements is sim- ![<span id="page-3-0"></span>FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level E<sup>F</sup> (dashed gray line). The horizontal dotted red lines at E<sup>F</sup> ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.](path) ilar in magnitude with the case of polarization in the xy-plane shown in Fig. [1,](#page-1-0) and even in the light helicity, while it vanishes for the non-magnetic elements. On the other hand, for the x-component the effect at the magnetic elements becomes one-two orders of magnitude smaller and is odd in the light helicity, although for the non-magnetic elements the values are comparable to the case shown in Fig. [1.](#page-1-0) It is important to note that for light circularly-polarized in the plane containing ferromagnetic magnetization, the light-induced transverse to magnetization induced moments, even though being smaller than the longitudinal ones, are related to light-induced torques that are experimentally demonstrated to lead to helicity-dependent THz emission [\[21,](#page-7-9) [33\]](#page-7-12). In order to get a better understanding of the impact of the time-reversal symmetry breaking on IFE, we consider the cases of ferromagnetic, non-relativistic (i.e. computed without SOC) ferromagnetic, and antiferromagnetic fcc Ni. In Figs. [8](#page-12-0) and [9](#page-13-0) of the SM we present the band filling dependence of the Cartesian components of the light-induced orbital moments δL, for hω̵ = 0.25 eV and hω̵ = 1.55 eV, respectively. In the non-relativistic case only a δL<sup>i</sup> parallel to the light propagation axis is induced which is odd in the helicity, and no δS<sup>i</sup> is induced, exemplifying that the orbital response is the primary non-relativistic one, whereas the spin response is generated through SOC, as was also shown in [\[20\]](#page-7-5). Similarly, for the antiferromagnetic case, only a component parallel to the light propagation axis is induced which is odd in the helicity and remains unchanged under different polarization flavors. While this is the case also for the induced components δL<sup>x</sup> and δL<sup>y</sup> which are transverse to the magnetization in the ferromagnetic case, the situation drastically changes for the induced component δL<sup>z</sup> parallel to the magnetization, as discussed earlier. We note that a tiny, odd in the helicity, δL<sup>x</sup> or δL<sup>y</sup> is additionally induced when light is rotating in a plane containing ferromagnetic magnetization. On the other hand, the induced δL<sup>x</sup> and δLy, developing normal to the polarization plane, serve as a non-relativistic "background" which is independent of the magnetization, with features due to the crystal structure driving the effect over larger regions in energy. The additional band-splittings induced by SOC result in the IFE exhibiting more features with band filling in the relativistic scenario. Remarkably, while the relativistic antiferromagnetic and non-relativistic ferromagnetic cases in principle have similar to each other behavior in energy, a larger signal arises in the antiferromagnetic case by the virtue of flatter bands (see also the discussion for Hf and Pt below). Lastly, we note that a similar behavior has been observed for the in-plane spin IFE in PT-symmetric Mn2Au, however, in the latter case additional out-of-plane moments arise due to linearly polarized light as a result of broken by the magnetization inversion symmetry [\[29\]](#page-7-8). Next, we focus on the case of 5d transition metals where in Fig. [2\(](#page-2-1)a-b) we explore the relation of δL<sup>z</sup> and δSz, respectively, to the band filling, for light circularly polarized in the xy plane and the frequency of hω̵ = 1.55 eV. When going from group IV (hcp Hf) to XI (fcc Au) we observe a smooth ![<span id="page-4-0"></span>FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.](path) variation of δL<sup>z</sup> from positive to negative values, as well as nicely shaped plateaus, where δL<sup>z</sup> remains relatively robust in a wide energy region, for hcp Hf, bcc Ta, bcc W, fcc Ir, and fcc Pt. We note that such plateaus are often characteristic of orbital effects, as witnessed for example in orbital Hall insulators [\[61](#page-8-8)[–63\]](#page-8-9) and orbital Rashba systems [\[60\]](#page-8-7). On the contrary, δS<sup>z</sup> exhibits a very erratic behavior with strong variations for each material, at the same time being much smaller in magnitude than δLz. The above observation is a clear manifestation of how differently the orbital and spin degrees of freedom behave under light excitation, with the orbital angular momentum having its origin in intrinsic structural parameters as manifested in the crystal field splitting, whereas the spin angular momentum is more sensitive to finer details of the electronic structure mediated by SOC. Indeed, the light-induced spin and orbital moments exhibit similarly erratic behavior with band filling once the frequency of the light is drastically reduced to reach the range of spin-orbit interaction, see Fig. [10](#page-13-1) of the SM for hω̵ = 0.25 eV. # Anatomy of IFE in k-space Among the 5d transition metals, hcp Hf and fcc Pt exhibit the largest computed moments at the Fermi energy for hω̵ = 1.55 eV, with the corresponding values of δL<sup>z</sup> = 9.1 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −0.8 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Hf, and δL<sup>z</sup> = −7.4 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −1.1 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Pt. Therefore, we select these two materials and explore the behavior of their light-induced moments in reciprocal space. We present the band-resolved δL<sup>z</sup> and δS<sup>z</sup> for hcp Hf in Fig. [3\(](#page-3-0)a-b), as well as band-resolved δL<sup>z</sup> and δS<sup>z</sup> for fcc Pt in Fig. [3\(](#page-3-0)c-d). For the case of Hf, transitions along the bands near the A-point are the main source of δL<sup>z</sup> and δSz. Light-induced δL<sup>z</sup> consists of hotspot-like negative contributions and secondary, but extended over energy and k-space consistently positive contributions, which are hardly visible. Overall, however, the latter lead to a large positive orbital integrated response. On the other hand, δS<sup>z</sup> For the case of Pt, δL<sup>z</sup> and δS<sup>z</sup> arise from transitions close to X and L high-symmetry points, with both originating in roughly the same regions of (E, k)-space, but often having an opposite sign to each other. Note that the bands in Pt are much more dispersive in the considered energy window, which results in an effective reduction of the regions in (E, k)-space which contribute to the spin and orbital response alike. This is in contrast to Hf, where much flatter bands reside above and below the Fermi energy within the energy window of the laser pulse, providing significant integrated, albeit very small locally, contributions. Moreover, in fcc Pt the Fermi energy cuts through the band edges of the d-states, where the effect of spin-orbit interaction is the strongest, which explains the emergence of strong hotspot-like contributions with a clear correlation in the magnitude of spin and orbital response and the resultant similar behavior of δL<sup>z</sup> and δS<sup>z</sup> with band filling around the Fermi energy. We further scrutinize the reciprocal space distribution of δL<sup>z</sup> and δSz, shown for hcp Hf and fcc Pt in Fig. [3\(](#page-3-0)e-h). For both materials δL<sup>z</sup> distributions consist of large uniform areas of either positive or negative sign. On the contrary, δS<sup>z</sup> distributions are much finer and richer in details with more areas of opposite sign, consistent with the picture we drew above from the band-resolved analysis. A similar behavior of the light-induced magnetism in reciprocal space has been recently reported for rutile altermagnets [\[39\]](#page-7-18). We also observe that, as discussed above, while for Hf the contributions are well spread throughout the Brillouin zone, for Pt the spin and orbital contributions are located at the edges of the considered k<sup>z</sup> − k<sup>y</sup> plane. Overall, this fact indicates that the microscopic behavior of light-induced magnetism varies strongly among transition metals and crucially depends on the crystal structure and position of the Fermi level with respect to the states ![<span id="page-5-0"></span>FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ### Anisotropy of light-induced magnetism We first address the anisotropy of light-induced magnetism with respect to the magnetization by examining the response of magnetic elements under different flavors of circular polarization. In Fig. [4\(](#page-4-0)a-b) we present the computed δL<sup>z</sup> and δS<sup>z</sup> in relation to scattering lifetime Γ, for the cases of magnetic bcc Fe and hcp Co under excitation by light which is circularly polarized in the xy or yz planes at hω̵ = 1.55 eV. As we have already seen in Fig. [1](#page-1-0) for light polarized in the xyplane, the responses behave differently for right (solid lines) or left (dashed lines) polarization. This difference is more pronounced for δLz, for which we even observe a change of sign for Co. Surprisingly, the situation is drastically different for the case of yz-polarization. In this case, for bcc Fe the response is even in the helicity, while for hcp Co δL<sup>z</sup> is almost even and δS<sup>z</sup> is perfectly even in the helicity. Such different behavior with respect to the light helicity between the two magnetic elements can be traced back to the additional anisotropy originating in the crystal structure itself − an effect which we discuss below. Notably, we witness a highly nonlinear behavior with respect to Γ, with the response reaching colossal values in the clean limit. For example, δL<sup>z</sup> and δS<sup>z</sup> can reach as much as 40 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> and 70 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> in Co, respectively, given the scattering lifetime of 1 meV. Besides the potential tunability of the inverse Faraday effect by the degree of the disorder of the samples, one key message that we extract from our observations is that when comparing the values for laser-induced magnetic moments obtained with different methods, special care has to be taken since the implementations of disorder effects, even within a simple constant broadening model, may differ among various approaches. Finally, we analyze the anisotropy that the crystal structure induces in the light-induced magnetism. We select the case of hcp Re and present in Fig. [5\(](#page-5-0)a-b) the x and z components of δL and δS, respectively, in relation to Γ. Due to the inequivalence of x and z axes in the hcp structure, the δL<sup>z</sup> which originates in light circularly polarized in the xy-plane (light blue line) differs from the δL<sup>x</sup> for the case of the yz polarization (brown line). The situation is similar for the case of δS<sup>z</sup> and δSx. On the contrary, as presented in Fig. [11](#page-14-1) of the SM, in the case of fcc Pt we have a perfect match between δL<sup>z</sup> and δL<sup>x</sup> (or δS<sup>z</sup> and δSx) when changing the plane of circular polarization because the x and z axes are equivalent in the fcc structure. The situation is similar for the y component when light is circularly polarized in the xz-plane. As also expected from the symmetry of the crystal structure, we confirm that the light-induced moments are perfectly odd in the helicity for the nonmagnetic elements, which is not the case for the magnetic elements, see Figs. [5](#page-5-0) and [11](#page-14-1) of the SM [\[20\]](#page-7-5). ### Discussion The main goal of our work is to showcase the importance of the orbital component of IFE and its distinct behavior from the spin counterpart. While so far it was mainly the interaction of light with the spin magnetization that was taken into consideration for the interpretation of IFE-related effects, we speculate that the orbital IFE may provide a novel way to coherently induce magnetization and manipulate the magnetic order. For example, in a recent study, different types of optical torques that may arise in ferromagnetic layers were interpreted in terms of the light-induced orbital moment and its interaction with the magnetization through the spin-orbit interaction [\[64\]](#page-8-10). Since it is known that current-induced orbital accumulation and orbital torques exhibit a long-range behavior [\[49,](#page-8-0) [65](#page-8-11)[–67\]](#page-8-12) due to a characteristic small orbital decay, a question arises whether similar behavior can be exhibited by optical torques caused by the orbital IFE. The emergence of long-ranged orbital IFE would come as no surprise given the fact that several recent experiments reported that laser excitation can drive long-range ballistic orbital currents resulting in THz emission [\[68](#page-8-13)[–71\]](#page-8-14). a perturbation by the electric field of the original ground state Hamiltonian, it is common to utilize the angular momentum of light in order to understand the interaction with the magnetic order by the means of transfer of angular momentum. The spin of light through the helicity of circularly polarized pulses plays a crucial role in helicity-dependent all-optical switching scenarios. On the other hand, there is a strong recent interest in utilizing the orbital angular momentum of light via irradiation of matter with e.g. vortex beams or twisted light, in order to probe the magnetization [\[72\]](#page-8-15), generate photocurrents [\[73\]](#page-8-16), drive ultrafast demagnetization [\[74\]](#page-8-17), and induce IFE [\[40,](#page-7-24) [75\]](#page-8-18). Therefore, it is imperative to treat both spin and orbital degrees of freedom on equal footing when exploring the lightmatter interaction. This will not only trigger further advances in the field of ultrafast magnetism and THz spintronics, but also enable a transition to the novel field of attosecond spintronics [\[76,](#page-8-19) [77\]](#page-8-20). Method. In this work we calculate the first-principles electronic structures of 3d, 4d and 5d transition metals by using the full-potential linearized augmented plane wave FLEUR code [\[78\]](#page-8-21). We describe exchange and correlation effects by using the non-relativistic PBE [\[79\]](#page-8-22) functional, while relativistic effects are described by the second-variation scheme [\[80\]](#page-9-0). The parameters of our first-principles calculations, i.e. lattice constants, muffin-tin radii, plane-wave cutoffs, etc. are taken from Table I of Ref. [\[81\]](#page-9-1). Next, we construct maximally-localized Wannier functions (MLWFs) by employing the Wannier90 code [\[82\]](#page-9-2) and its interface with the FLEUR code [\[83\]](#page-9-3). Similarly to [\[81\]](#page-9-1), we choose s, p and d orbitals for the initial projections and disentagle 18 MLWFs out of 36 Bloch states within a frozen window of 5.0 eV above the Fermi energy for each atom in the unit cell. where O<sup>i</sup> is the i-th component of either the orbital angular momentum operator L<sup>i</sup> or of the spin operator S<sup>i</sup> . Moreover, a<sup>0</sup> = 4πϵ0h̵<sup>2</sup> /(mee 2 ) is the Bohr's radius, I = ϵ0cE<sup>2</sup> 0 /2 is the intensity of the pulse, ϵ<sup>0</sup> is the vacuum permittivity, m<sup>e</sup> is the electron mass, e is the elementary charge, h̵ is the reduced Planck constant, c is the light velocity, E<sup>H</sup> = e 2 /(4πϵ0a0) is the Hartree energy, and ϵ<sup>j</sup> is the j-th component of the polarization vector of the pulse. For example, we describe right/left-handedly polarized light in the xy-plane as ϵ = (1,±i, <sup>0</sup>)/<sup>√</sup> 2, respectively, and define its propagation vector to lie along the normal to the polarization plane. A detailed form of the tensor φijk can be seen in Eq.(14) of Ref. [\[21\]](#page-7-9). For the orbital response, the prefactor in Eq. [$1$](#page-6-2) must be multiplied by an additional factor of 2. A 128×128×128 interpolation k-mesh is sufficient to obtain well-converged results. In all calculations the lifetime broadening Γ was set at 25 meV, the light frequency hω̵ at 0.25 eV and 1.55 eV, the intensity of light at 10 GW/cm<sup>2</sup> , and we covered an energy region of [−2.5, 2.5] eV around the Fermi energy level E<sup>F</sup> . Acknowledgements. We thank Frank Freimuth and Maximilian Merte for discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR 173/3 − 268565370 (project A11) and by the K. and A. Wallenberg Foundation (Grants No. 2022.0079 and 2023.0336). We acknowledge support from the EIC Pathfinder OPEN grant 101129641 "OBELIX". We also gratefully acknowledge the Julich Supercomputing Cen- ¨ tre and RWTH Aachen University for providing computational resources under projects jiff40 and jara0062. Author Contributions. T. A. performed numerical calculations and analysed the results. T. A. and Y. M. wrote the manuscript. 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(a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-11-0"></span>FIG. 7. (a-d) Light-induced orbital δL<sup>x</sup> (a, c) and spin δS<sup>x</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-12-0"></span>FIG. 8. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-13-0"></span>FIG. 9. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ![<span id="page-13-1"></span>FIG. 10. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-14-1"></span>FIG. 11. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic fcc Pt. The induced magnetic moments along the x-axis and along the z-axis are presented. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) <span id="page-14-0"></span>TABLE II. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 0.25 eV. <span id="page-15-0"></span>TABLE III. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV.
<span id="page-1-0"></span>FIG. 1. Map of light-induced magnetism in transition metals. (a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).	# Light-induced Orbital and Spin Magnetism in 3d, 4d, and 5d Transition Metals Understanding the coherent interplay of light with the magnetization in metals has been a long-standing problem in ultrafast magnetism. While it is known that when laser light acts on a metal it can induce magnetization via the process known as the inverse Faraday effect (IFE), the most basic ingredients of this phenomenon are still largely unexplored. In particular, given a strong recent interest in orbital non-equilibrium dynamics and its role in mediating THz emission in transition metals, the exploration of distinct features in spin and orbital IFE is pertinent. Here, we present a first complete study of the spin and orbital IFE in 3d, 4d and 5d transition metals of groups IV−XI from first-principles. By examining the dependence on the light polarization and frequency, we show that the laser-induced spin and orbital moments may vary significantly both in magnitude and sign. We underpin the interplay between the crystal field splitting and spin-orbit interaction as the key factor which determines the magnitude and key differences between the spin and orbital response. Additionally, we highlight the anisotropy of the effect with respect to the ferromagnetic magnetization and to the crystal structure. The provided complete map of IFE in transition metals is a key reference point in the field of optical magnetism. ### Introduction The demonstration of ultrafast demagnetization in ferromagnets by the application of femtosecond laser pulses [\[1\]](#page-6-0) gave rise to the field of ultrafast spintronics and set the stage for efficient manipulation of magnetism by light. By now it has been rigorously demonstrated that all-optical helicitydependent magnetization switching can be achieved in wide classes of magnetic materials [\[2](#page-6-1)[–10\]](#page-7-0), thus paving the way to contactless ultrafast magnetic recording and information processing. In interpretation of the switching experiments, the inverse Faraday effect (IFE) − i.e. the phenomenon of magnetization induced by a coherent interaction with light acting on a material − has been considered as one of the major underlying mechanisms for the magnetization reversal. Although theoretically predicted and experimentally observed decades ago [\[11–](#page-7-1)[13\]](#page-7-2), a consensus in the theoretical understanding of IFE is still lacking [\[14](#page-7-3)[–19\]](#page-7-4) – however, several microscopic methods have been recently developed for the calculation of this phenomenon in diverse setups [\[20–](#page-7-5)[27\]](#page-7-6). Apart from its established role in the magnetization switching of ferromagnets and ferrimagnents, the impact of IFE in the THz regime has also been studied in antiferromagnets like CrPt [\[28\]](#page-7-7) and Mn2Au [\[29\]](#page-7-8). Moreover, it has been shown that the component of IFE which is perpendicular to the magnetization direction behaves differently with respect to the light helicity than the parallel component [\[21\]](#page-7-9), giving rise to helicity-dependent optical torques [\[30–](#page-7-10)[32\]](#page-7-11) and THz emission [\[33\]](#page-7-12). The impact of the optical torques has been demonstrated to be significant in antiferromagnetic Mn2Au [\[34\]](#page-7-13) where transverse IFE may be responsible for the THz emission [\[35,](#page-7-14) [36\]](#page-7-15), provide an alternative way to switching the magnetization [\[37\]](#page-7-16) as well as to drive domain wall motion [\[38\]](#page-7-17). Recently it has also been shown that light can induce colossal magnetic moments in altermagnets [\[39\]](#page-7-18). Although IFE is conventionally associated with an excitation by circularly polarized light, it can also be activated by linearly-polarized laser pulses [\[25,](#page-7-19) [29,](#page-7-8) [39–](#page-7-18)[42\]](#page-7-20). In solids, two contributions to the magnetization exist: due to spin and orbital moment of electrons. And while the discussion of IFE is normally restricted to the response of spin, since recently, non-equilibrium dynamics of orbital angular momentum started to attract significant attention [\[43](#page-7-21)[–45\]](#page-7-22). Namely, the emergence of orbital currents in the context of the orbital Hall effect [\[46–](#page-7-23)[49\]](#page-8-0), orbital nature of current-induced torques on the magnetization [\[50](#page-8-1)[–52\]](#page-8-2), and current-induced orbital accumulation sizeable even in light materials [\[53–](#page-8-3)[56\]](#page-8-4) have been addressed theoretically and demonstrated experimentally. Notably, in the context of light-induced magnetism, Berritta and co-workers have predicted that the IFE in selected transition metals can exhibit a sizeable orbital component [\[20\]](#page-7-5), with the generality of this observation reaching even into the realm of altermagnets [\[39\]](#page-7-18). At the same time it is also known that within the context of plasmonic IFE the orbital magnetic moment due to electrons excited by laser pulses in small nanoparticles of noble and simple metals can reach atomic values [\[57–](#page-8-5)[59\]](#page-8-6). Despite the fact that very little is known about the interplay of spin and orbital IFE in real materials, it is believed that IFE is a very promising effect in the context of orbitronics − a field which deals with manipulation of the orbital degree of freedom by external perturbations. Although the importance of IFE in mediating the lightinduced magnetism is beyond doubt, a comprehensive indepth material-specific knowledge acquired from microscopic theoretical analysis of this effect is still missing. While acquiring this knowledge is imperative for the field of ultrafast spintronics, since the initial seminal work by Berritta and coworkers [\[20\]](#page-7-5), very little effort has been dedicated to the explorations of this phenomenon from first principles. Here, we fill this gap by providing a detailed first-principles study of the light-induced spin and orbital magnetism for the 3d, 4d and 5d transition metals of groups IV−XI. By exploring the dependence on the frequency and the polarization of light, we acquire insights into the origin and differences between spin and orbital flavors of IFE, as well as key features of their behavior. Our work provide a solid foundation for further advances in the field of optical magnetism and interaction of light and matter. # Results # Light-induced magnetism in transition metals We begin our discussion by exploring the IFE in a series of 3d, 4d and 5d transition metals of groups IV−XI. IFE is a non-linear optomagnetic effect in which magnetization δO ∝ E × E ∗ is induced as a second-order response to the electric field E of a laser pulse. We calculate the light-induced spin δS and orbital δL magnetic moments by means of the Keldysh formalism, see the section Methods [\[21,](#page-7-9) [39\]](#page-7-18). We focus on light energies hω̵ of 0.25 eV and 1.55 eV which have been routinely used before in our studies within the same formalism [\[29,](#page-7-8) [39,](#page-7-18) [60\]](#page-8-7). Additionally, we choose a lifetime broadening Γ of 25 meV that corresponds to room temperature in order to account for effects of disorder on the electronic states and to provide realistic estimations of the effect. In all calculations we choose the light to be circularly polarized in the xy or yz planes with a light intensity I of 10 GW/cm<sup>2</sup> , while for magnetic materials ferromagnetic magnetization is considered to be along the z-axis. In Fig. [1](#page-1-0) we present the calculated light-induced δL<sup>z</sup> [Fig. [1\(](#page-1-0)a-c)] and δS<sup>z</sup> [Fig. [1\(](#page-1-0)b-d)] moments of the considered transition metals at the Fermi energy. Light is circularly polarized in the xy plane, with energies hω̵ of 0.25 eV [Fig. [1\(](#page-1-0)ab)] and 1.55 eV [Fig. [1\(](#page-1-0)c-d)]. Regarding the 3d magnetic elements Fe, Co, Ni, we analyze both right and left handed circular polarizations of light. The exact values of δL<sup>z</sup> and δS<sup>z</sup> are listed in Tables [II](#page-14-0) and [III](#page-15-0) of the Supplemental Material (SM). Not shown are computed transverse x and y compo- ![<span id="page-2-1"></span>FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.](path) At first sight we notice a strong dependence of the magnitude and sign of both δL<sup>z</sup> and δS<sup>z</sup> on the light frequency. We point out the case of fcc Rh where an colossal δL<sup>z</sup> = −56 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> is predicted at hω̵ = 0.25 eV and which is drastically reduced at the higher frequency, while the spin component remains suppressed in both cases. For the case of magnetic hcp Co we notice a helicity-dependent change of sign for δL<sup>z</sup> at hω̵ = 0.25 eV which is not present for δSz. In general, δL<sup>z</sup> varies stronger with the light helicity than δSz. Moreover, δL<sup>z</sup> is one to two orders of magnitude larger than δS<sup>z</sup> for the non-magnetic elements, while they are of the same magnitude for the magnetic elemental materials. The last two observations are in agreement with the findings of Ref. [\[20\]](#page-7-5) and indicate how the spontaneous magnetization strongly influences the effect in ferromagnets by the time-reversal symmetry breaking. For comparison, in Table [I](#page-2-0) we list the values of the total light-induced moments, defined as the sum of δL<sup>z</sup> and δSz, for the transitional metals studied in [\[20\]](#page-7-5) with hω̵ = 1.55 eV. Our calculated values are of the same order of magnitude, with the exceptions of Au, and of Co for the case of left-handed polarization, which can be attributed to the difference in the computational methods. From Fig. [1\(](#page-1-0)c-d) we get a clear picture of how the induced moments scale with the strength of spin-orbit coupling (SOC). The effect is overall larger for the heavier 5d elements, both in the orbital δL<sup>z</sup> and spin δS<sup>z</sup> channels. However, at the lower frequency of hω̵ = 0.25 eV it is more difficult to draw such a conclusion since the corresponding photon energy falls into the range of SOC strength, which promotes the role of electronic transitions among spin-orbit split bands occurring within a narrow range around the Fermi energy and limited regions in k-space. In contrast, the use of a much larger frequency involves transitions among manifestly orbitally-distinct states which take place over larger portions <span id="page-2-0"></span>TABLE I. Total light-induced magnetic moments for transition metals previously studied in [\[20\]](#page-7-5) in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV. of the reciprocal space with a more uniform impact of the spin-orbit strength on the magnitude of the transition probabilities. From this discussion we have to exclude the ferromagnetic elements since the spontaneous magnetization induces bands splittings on the scale of exchange strength therefore making the effect much more complex. We study the dependence on the light polarization by performing additional calculations for light circularly polarized in the yz-plane, presenting the results for the z and x components of δL and δS respectively in Figs. [6](#page-10-0) and [7](#page-11-0) of the SM for hω̵ = 0.25 eV and hω̵ = 1.55 eV. Our results for δS<sup>z</sup> and δS<sup>x</sup> for the magnetic transition metals reproduce exactly the values of Ref. [\[21\]](#page-7-9), computed with the same method. We further present the values of δL<sup>z</sup> and δLx, where we find the orbital response along the magnetization, δLz, to be by a factor of two larger than the corresponding spin response, when averaged over all magnetic elements. Generally, for the z-component the effect at the magnetic elements is sim- ![<span id="page-3-0"></span>FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level E<sup>F</sup> (dashed gray line). The horizontal dotted red lines at E<sup>F</sup> ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.](path) ilar in magnitude with the case of polarization in the xy-plane shown in Fig. [1,](#page-1-0) and even in the light helicity, while it vanishes for the non-magnetic elements. On the other hand, for the x-component the effect at the magnetic elements becomes one-two orders of magnitude smaller and is odd in the light helicity, although for the non-magnetic elements the values are comparable to the case shown in Fig. [1.](#page-1-0) It is important to note that for light circularly-polarized in the plane containing ferromagnetic magnetization, the light-induced transverse to magnetization induced moments, even though being smaller than the longitudinal ones, are related to light-induced torques that are experimentally demonstrated to lead to helicity-dependent THz emission [\[21,](#page-7-9) [33\]](#page-7-12). In order to get a better understanding of the impact of the time-reversal symmetry breaking on IFE, we consider the cases of ferromagnetic, non-relativistic (i.e. computed without SOC) ferromagnetic, and antiferromagnetic fcc Ni. In Figs. [8](#page-12-0) and [9](#page-13-0) of the SM we present the band filling dependence of the Cartesian components of the light-induced orbital moments δL, for hω̵ = 0.25 eV and hω̵ = 1.55 eV, respectively. In the non-relativistic case only a δL<sup>i</sup> parallel to the light propagation axis is induced which is odd in the helicity, and no δS<sup>i</sup> is induced, exemplifying that the orbital response is the primary non-relativistic one, whereas the spin response is generated through SOC, as was also shown in [\[20\]](#page-7-5). Similarly, for the antiferromagnetic case, only a component parallel to the light propagation axis is induced which is odd in the helicity and remains unchanged under different polarization flavors. While this is the case also for the induced components δL<sup>x</sup> and δL<sup>y</sup> which are transverse to the magnetization in the ferromagnetic case, the situation drastically changes for the induced component δL<sup>z</sup> parallel to the magnetization, as discussed earlier. We note that a tiny, odd in the helicity, δL<sup>x</sup> or δL<sup>y</sup> is additionally induced when light is rotating in a plane containing ferromagnetic magnetization. On the other hand, the induced δL<sup>x</sup> and δLy, developing normal to the polarization plane, serve as a non-relativistic "background" which is independent of the magnetization, with features due to the crystal structure driving the effect over larger regions in energy. The additional band-splittings induced by SOC result in the IFE exhibiting more features with band filling in the relativistic scenario. Remarkably, while the relativistic antiferromagnetic and non-relativistic ferromagnetic cases in principle have similar to each other behavior in energy, a larger signal arises in the antiferromagnetic case by the virtue of flatter bands (see also the discussion for Hf and Pt below). Lastly, we note that a similar behavior has been observed for the in-plane spin IFE in PT-symmetric Mn2Au, however, in the latter case additional out-of-plane moments arise due to linearly polarized light as a result of broken by the magnetization inversion symmetry [\[29\]](#page-7-8). Next, we focus on the case of 5d transition metals where in Fig. [2\(](#page-2-1)a-b) we explore the relation of δL<sup>z</sup> and δSz, respectively, to the band filling, for light circularly polarized in the xy plane and the frequency of hω̵ = 1.55 eV. When going from group IV (hcp Hf) to XI (fcc Au) we observe a smooth ![<span id="page-4-0"></span>FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.](path) variation of δL<sup>z</sup> from positive to negative values, as well as nicely shaped plateaus, where δL<sup>z</sup> remains relatively robust in a wide energy region, for hcp Hf, bcc Ta, bcc W, fcc Ir, and fcc Pt. We note that such plateaus are often characteristic of orbital effects, as witnessed for example in orbital Hall insulators [\[61](#page-8-8)[–63\]](#page-8-9) and orbital Rashba systems [\[60\]](#page-8-7). On the contrary, δS<sup>z</sup> exhibits a very erratic behavior with strong variations for each material, at the same time being much smaller in magnitude than δLz. The above observation is a clear manifestation of how differently the orbital and spin degrees of freedom behave under light excitation, with the orbital angular momentum having its origin in intrinsic structural parameters as manifested in the crystal field splitting, whereas the spin angular momentum is more sensitive to finer details of the electronic structure mediated by SOC. Indeed, the light-induced spin and orbital moments exhibit similarly erratic behavior with band filling once the frequency of the light is drastically reduced to reach the range of spin-orbit interaction, see Fig. [10](#page-13-1) of the SM for hω̵ = 0.25 eV. # Anatomy of IFE in k-space Among the 5d transition metals, hcp Hf and fcc Pt exhibit the largest computed moments at the Fermi energy for hω̵ = 1.55 eV, with the corresponding values of δL<sup>z</sup> = 9.1 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −0.8 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Hf, and δL<sup>z</sup> = −7.4 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −1.1 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Pt. Therefore, we select these two materials and explore the behavior of their light-induced moments in reciprocal space. We present the band-resolved δL<sup>z</sup> and δS<sup>z</sup> for hcp Hf in Fig. [3\(](#page-3-0)a-b), as well as band-resolved δL<sup>z</sup> and δS<sup>z</sup> for fcc Pt in Fig. [3\(](#page-3-0)c-d). For the case of Hf, transitions along the bands near the A-point are the main source of δL<sup>z</sup> and δSz. Light-induced δL<sup>z</sup> consists of hotspot-like negative contributions and secondary, but extended over energy and k-space consistently positive contributions, which are hardly visible. Overall, however, the latter lead to a large positive orbital integrated response. On the other hand, δS<sup>z</sup> For the case of Pt, δL<sup>z</sup> and δS<sup>z</sup> arise from transitions close to X and L high-symmetry points, with both originating in roughly the same regions of (E, k)-space, but often having an opposite sign to each other. Note that the bands in Pt are much more dispersive in the considered energy window, which results in an effective reduction of the regions in (E, k)-space which contribute to the spin and orbital response alike. This is in contrast to Hf, where much flatter bands reside above and below the Fermi energy within the energy window of the laser pulse, providing significant integrated, albeit very small locally, contributions. Moreover, in fcc Pt the Fermi energy cuts through the band edges of the d-states, where the effect of spin-orbit interaction is the strongest, which explains the emergence of strong hotspot-like contributions with a clear correlation in the magnitude of spin and orbital response and the resultant similar behavior of δL<sup>z</sup> and δS<sup>z</sup> with band filling around the Fermi energy. We further scrutinize the reciprocal space distribution of δL<sup>z</sup> and δSz, shown for hcp Hf and fcc Pt in Fig. [3\(](#page-3-0)e-h). For both materials δL<sup>z</sup> distributions consist of large uniform areas of either positive or negative sign. On the contrary, δS<sup>z</sup> distributions are much finer and richer in details with more areas of opposite sign, consistent with the picture we drew above from the band-resolved analysis. A similar behavior of the light-induced magnetism in reciprocal space has been recently reported for rutile altermagnets [\[39\]](#page-7-18). We also observe that, as discussed above, while for Hf the contributions are well spread throughout the Brillouin zone, for Pt the spin and orbital contributions are located at the edges of the considered k<sup>z</sup> − k<sup>y</sup> plane. Overall, this fact indicates that the microscopic behavior of light-induced magnetism varies strongly among transition metals and crucially depends on the crystal structure and position of the Fermi level with respect to the states ![<span id="page-5-0"></span>FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ### Anisotropy of light-induced magnetism We first address the anisotropy of light-induced magnetism with respect to the magnetization by examining the response of magnetic elements under different flavors of circular polarization. In Fig. [4\(](#page-4-0)a-b) we present the computed δL<sup>z</sup> and δS<sup>z</sup> in relation to scattering lifetime Γ, for the cases of magnetic bcc Fe and hcp Co under excitation by light which is circularly polarized in the xy or yz planes at hω̵ = 1.55 eV. As we have already seen in Fig. [1](#page-1-0) for light polarized in the xyplane, the responses behave differently for right (solid lines) or left (dashed lines) polarization. This difference is more pronounced for δLz, for which we even observe a change of sign for Co. Surprisingly, the situation is drastically different for the case of yz-polarization. In this case, for bcc Fe the response is even in the helicity, while for hcp Co δL<sup>z</sup> is almost even and δS<sup>z</sup> is perfectly even in the helicity. Such different behavior with respect to the light helicity between the two magnetic elements can be traced back to the additional anisotropy originating in the crystal structure itself − an effect which we discuss below. Notably, we witness a highly nonlinear behavior with respect to Γ, with the response reaching colossal values in the clean limit. For example, δL<sup>z</sup> and δS<sup>z</sup> can reach as much as 40 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> and 70 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> in Co, respectively, given the scattering lifetime of 1 meV. Besides the potential tunability of the inverse Faraday effect by the degree of the disorder of the samples, one key message that we extract from our observations is that when comparing the values for laser-induced magnetic moments obtained with different methods, special care has to be taken since the implementations of disorder effects, even within a simple constant broadening model, may differ among various approaches. Finally, we analyze the anisotropy that the crystal structure induces in the light-induced magnetism. We select the case of hcp Re and present in Fig. [5\(](#page-5-0)a-b) the x and z components of δL and δS, respectively, in relation to Γ. Due to the inequivalence of x and z axes in the hcp structure, the δL<sup>z</sup> which originates in light circularly polarized in the xy-plane (light blue line) differs from the δL<sup>x</sup> for the case of the yz polarization (brown line). The situation is similar for the case of δS<sup>z</sup> and δSx. On the contrary, as presented in Fig. [11](#page-14-1) of the SM, in the case of fcc Pt we have a perfect match between δL<sup>z</sup> and δL<sup>x</sup> (or δS<sup>z</sup> and δSx) when changing the plane of circular polarization because the x and z axes are equivalent in the fcc structure. The situation is similar for the y component when light is circularly polarized in the xz-plane. As also expected from the symmetry of the crystal structure, we confirm that the light-induced moments are perfectly odd in the helicity for the nonmagnetic elements, which is not the case for the magnetic elements, see Figs. [5](#page-5-0) and [11](#page-14-1) of the SM [\[20\]](#page-7-5). ### Discussion The main goal of our work is to showcase the importance of the orbital component of IFE and its distinct behavior from the spin counterpart. While so far it was mainly the interaction of light with the spin magnetization that was taken into consideration for the interpretation of IFE-related effects, we speculate that the orbital IFE may provide a novel way to coherently induce magnetization and manipulate the magnetic order. For example, in a recent study, different types of optical torques that may arise in ferromagnetic layers were interpreted in terms of the light-induced orbital moment and its interaction with the magnetization through the spin-orbit interaction [\[64\]](#page-8-10). Since it is known that current-induced orbital accumulation and orbital torques exhibit a long-range behavior [\[49,](#page-8-0) [65](#page-8-11)[–67\]](#page-8-12) due to a characteristic small orbital decay, a question arises whether similar behavior can be exhibited by optical torques caused by the orbital IFE. The emergence of long-ranged orbital IFE would come as no surprise given the fact that several recent experiments reported that laser excitation can drive long-range ballistic orbital currents resulting in THz emission [\[68](#page-8-13)[–71\]](#page-8-14). a perturbation by the electric field of the original ground state Hamiltonian, it is common to utilize the angular momentum of light in order to understand the interaction with the magnetic order by the means of transfer of angular momentum. The spin of light through the helicity of circularly polarized pulses plays a crucial role in helicity-dependent all-optical switching scenarios. On the other hand, there is a strong recent interest in utilizing the orbital angular momentum of light via irradiation of matter with e.g. vortex beams or twisted light, in order to probe the magnetization [\[72\]](#page-8-15), generate photocurrents [\[73\]](#page-8-16), drive ultrafast demagnetization [\[74\]](#page-8-17), and induce IFE [\[40,](#page-7-24) [75\]](#page-8-18). Therefore, it is imperative to treat both spin and orbital degrees of freedom on equal footing when exploring the lightmatter interaction. This will not only trigger further advances in the field of ultrafast magnetism and THz spintronics, but also enable a transition to the novel field of attosecond spintronics [\[76,](#page-8-19) [77\]](#page-8-20). Method. In this work we calculate the first-principles electronic structures of 3d, 4d and 5d transition metals by using the full-potential linearized augmented plane wave FLEUR code [\[78\]](#page-8-21). We describe exchange and correlation effects by using the non-relativistic PBE [\[79\]](#page-8-22) functional, while relativistic effects are described by the second-variation scheme [\[80\]](#page-9-0). The parameters of our first-principles calculations, i.e. lattice constants, muffin-tin radii, plane-wave cutoffs, etc. are taken from Table I of Ref. [\[81\]](#page-9-1). Next, we construct maximally-localized Wannier functions (MLWFs) by employing the Wannier90 code [\[82\]](#page-9-2) and its interface with the FLEUR code [\[83\]](#page-9-3). Similarly to [\[81\]](#page-9-1), we choose s, p and d orbitals for the initial projections and disentagle 18 MLWFs out of 36 Bloch states within a frozen window of 5.0 eV above the Fermi energy for each atom in the unit cell. where O<sup>i</sup> is the i-th component of either the orbital angular momentum operator L<sup>i</sup> or of the spin operator S<sup>i</sup> . Moreover, a<sup>0</sup> = 4πϵ0h̵<sup>2</sup> /(mee 2 ) is the Bohr's radius, I = ϵ0cE<sup>2</sup> 0 /2 is the intensity of the pulse, ϵ<sup>0</sup> is the vacuum permittivity, m<sup>e</sup> is the electron mass, e is the elementary charge, h̵ is the reduced Planck constant, c is the light velocity, E<sup>H</sup> = e 2 /(4πϵ0a0) is the Hartree energy, and ϵ<sup>j</sup> is the j-th component of the polarization vector of the pulse. For example, we describe right/left-handedly polarized light in the xy-plane as ϵ = (1,±i, <sup>0</sup>)/<sup>√</sup> 2, respectively, and define its propagation vector to lie along the normal to the polarization plane. A detailed form of the tensor φijk can be seen in Eq.(14) of Ref. [\[21\]](#page-7-9). For the orbital response, the prefactor in Eq. [$1$](#page-6-2) must be multiplied by an additional factor of 2. A 128×128×128 interpolation k-mesh is sufficient to obtain well-converged results. In all calculations the lifetime broadening Γ was set at 25 meV, the light frequency hω̵ at 0.25 eV and 1.55 eV, the intensity of light at 10 GW/cm<sup>2</sup> , and we covered an energy region of [−2.5, 2.5] eV around the Fermi energy level E<sup>F</sup> . Acknowledgements. We thank Frank Freimuth and Maximilian Merte for discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR 173/3 − 268565370 (project A11) and by the K. and A. Wallenberg Foundation (Grants No. 2022.0079 and 2023.0336). We acknowledge support from the EIC Pathfinder OPEN grant 101129641 "OBELIX". We also gratefully acknowledge the Julich Supercomputing Cen- ¨ tre and RWTH Aachen University for providing computational resources under projects jiff40 and jara0062. Author Contributions. T. A. performed numerical calculations and analysed the results. T. A. and Y. M. wrote the manuscript. 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(a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-11-0"></span>FIG. 7. (a-d) Light-induced orbital δL<sup>x</sup> (a, c) and spin δS<sup>x</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-12-0"></span>FIG. 8. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-13-0"></span>FIG. 9. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ![<span id="page-13-1"></span>FIG. 10. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-14-1"></span>FIG. 11. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic fcc Pt. The induced magnetic moments along the x-axis and along the z-axis are presented. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) <span id="page-14-0"></span>TABLE II. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 0.25 eV. <span id="page-15-0"></span>TABLE III. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV.
FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL⁽ᶻ⁾ (a, c) and spin δS⁽ᶻ⁾ (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level Eᴮ (dashed gray line). The horizontal dotted red lines at Eᴮ ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.	# Light-induced Orbital and Spin Magnetism in 3d, 4d, and 5d Transition Metals Understanding the coherent interplay of light with the magnetization in metals has been a long-standing problem in ultrafast magnetism. While it is known that when laser light acts on a metal it can induce magnetization via the process known as the inverse Faraday effect (IFE), the most basic ingredients of this phenomenon are still largely unexplored. In particular, given a strong recent interest in orbital non-equilibrium dynamics and its role in mediating THz emission in transition metals, the exploration of distinct features in spin and orbital IFE is pertinent. Here, we present a first complete study of the spin and orbital IFE in 3d, 4d and 5d transition metals of groups IV−XI from first-principles. By examining the dependence on the light polarization and frequency, we show that the laser-induced spin and orbital moments may vary significantly both in magnitude and sign. We underpin the interplay between the crystal field splitting and spin-orbit interaction as the key factor which determines the magnitude and key differences between the spin and orbital response. Additionally, we highlight the anisotropy of the effect with respect to the ferromagnetic magnetization and to the crystal structure. The provided complete map of IFE in transition metals is a key reference point in the field of optical magnetism. ### Introduction The demonstration of ultrafast demagnetization in ferromagnets by the application of femtosecond laser pulses [\[1\]](#page-6-0) gave rise to the field of ultrafast spintronics and set the stage for efficient manipulation of magnetism by light. By now it has been rigorously demonstrated that all-optical helicitydependent magnetization switching can be achieved in wide classes of magnetic materials [\[2](#page-6-1)[–10\]](#page-7-0), thus paving the way to contactless ultrafast magnetic recording and information processing. In interpretation of the switching experiments, the inverse Faraday effect (IFE) − i.e. the phenomenon of magnetization induced by a coherent interaction with light acting on a material − has been considered as one of the major underlying mechanisms for the magnetization reversal. Although theoretically predicted and experimentally observed decades ago [\[11–](#page-7-1)[13\]](#page-7-2), a consensus in the theoretical understanding of IFE is still lacking [\[14](#page-7-3)[–19\]](#page-7-4) – however, several microscopic methods have been recently developed for the calculation of this phenomenon in diverse setups [\[20–](#page-7-5)[27\]](#page-7-6). Apart from its established role in the magnetization switching of ferromagnets and ferrimagnents, the impact of IFE in the THz regime has also been studied in antiferromagnets like CrPt [\[28\]](#page-7-7) and Mn2Au [\[29\]](#page-7-8). Moreover, it has been shown that the component of IFE which is perpendicular to the magnetization direction behaves differently with respect to the light helicity than the parallel component [\[21\]](#page-7-9), giving rise to helicity-dependent optical torques [\[30–](#page-7-10)[32\]](#page-7-11) and THz emission [\[33\]](#page-7-12). The impact of the optical torques has been demonstrated to be significant in antiferromagnetic Mn2Au [\[34\]](#page-7-13) where transverse IFE may be responsible for the THz emission [\[35,](#page-7-14) [36\]](#page-7-15), provide an alternative way to switching the magnetization [\[37\]](#page-7-16) as well as to drive domain wall motion [\[38\]](#page-7-17). Recently it has also been shown that light can induce colossal magnetic moments in altermagnets [\[39\]](#page-7-18). Although IFE is conventionally associated with an excitation by circularly polarized light, it can also be activated by linearly-polarized laser pulses [\[25,](#page-7-19) [29,](#page-7-8) [39–](#page-7-18)[42\]](#page-7-20). In solids, two contributions to the magnetization exist: due to spin and orbital moment of electrons. And while the discussion of IFE is normally restricted to the response of spin, since recently, non-equilibrium dynamics of orbital angular momentum started to attract significant attention [\[43](#page-7-21)[–45\]](#page-7-22). Namely, the emergence of orbital currents in the context of the orbital Hall effect [\[46–](#page-7-23)[49\]](#page-8-0), orbital nature of current-induced torques on the magnetization [\[50](#page-8-1)[–52\]](#page-8-2), and current-induced orbital accumulation sizeable even in light materials [\[53–](#page-8-3)[56\]](#page-8-4) have been addressed theoretically and demonstrated experimentally. Notably, in the context of light-induced magnetism, Berritta and co-workers have predicted that the IFE in selected transition metals can exhibit a sizeable orbital component [\[20\]](#page-7-5), with the generality of this observation reaching even into the realm of altermagnets [\[39\]](#page-7-18). At the same time it is also known that within the context of plasmonic IFE the orbital magnetic moment due to electrons excited by laser pulses in small nanoparticles of noble and simple metals can reach atomic values [\[57–](#page-8-5)[59\]](#page-8-6). Despite the fact that very little is known about the interplay of spin and orbital IFE in real materials, it is believed that IFE is a very promising effect in the context of orbitronics − a field which deals with manipulation of the orbital degree of freedom by external perturbations. Although the importance of IFE in mediating the lightinduced magnetism is beyond doubt, a comprehensive indepth material-specific knowledge acquired from microscopic theoretical analysis of this effect is still missing. While acquiring this knowledge is imperative for the field of ultrafast spintronics, since the initial seminal work by Berritta and coworkers [\[20\]](#page-7-5), very little effort has been dedicated to the explorations of this phenomenon from first principles. Here, we ![<span id="page-1-0"></span>FIG. 1. Map of light-induced magnetism in transition metals. (a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) fill this gap by providing a detailed first-principles study of the light-induced spin and orbital magnetism for the 3d, 4d and 5d transition metals of groups IV−XI. By exploring the dependence on the frequency and the polarization of light, we acquire insights into the origin and differences between spin and orbital flavors of IFE, as well as key features of their behavior. Our work provide a solid foundation for further advances in the field of optical magnetism and interaction of light and matter. # Results # Light-induced magnetism in transition metals We begin our discussion by exploring the IFE in a series of 3d, 4d and 5d transition metals of groups IV−XI. IFE is a non-linear optomagnetic effect in which magnetization δO ∝ E × E ∗ is induced as a second-order response to the electric field E of a laser pulse. We calculate the light-induced spin δS and orbital δL magnetic moments by means of the Keldysh formalism, see the section Methods [\[21,](#page-7-9) [39\]](#page-7-18). We focus on light energies hω̵ of 0.25 eV and 1.55 eV which have been routinely used before in our studies within the same formalism [\[29,](#page-7-8) [39,](#page-7-18) [60\]](#page-8-7). Additionally, we choose a lifetime broadening Γ of 25 meV that corresponds to room temperature in order to account for effects of disorder on the electronic states and to provide realistic estimations of the effect. In all calculations we choose the light to be circularly polarized in the xy or yz planes with a light intensity I of 10 GW/cm<sup>2</sup> , while for magnetic materials ferromagnetic magnetization is considered to be along the z-axis. In Fig. [1](#page-1-0) we present the calculated light-induced δL<sup>z</sup> [Fig. [1\(](#page-1-0)a-c)] and δS<sup>z</sup> [Fig. [1\(](#page-1-0)b-d)] moments of the considered transition metals at the Fermi energy. Light is circularly polarized in the xy plane, with energies hω̵ of 0.25 eV [Fig. [1\(](#page-1-0)ab)] and 1.55 eV [Fig. [1\(](#page-1-0)c-d)]. Regarding the 3d magnetic elements Fe, Co, Ni, we analyze both right and left handed circular polarizations of light. The exact values of δL<sup>z</sup> and δS<sup>z</sup> are listed in Tables [II](#page-14-0) and [III](#page-15-0) of the Supplemental Material (SM). Not shown are computed transverse x and y compo- ![<span id="page-2-1"></span>FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.](path) At first sight we notice a strong dependence of the magnitude and sign of both δL<sup>z</sup> and δS<sup>z</sup> on the light frequency. We point out the case of fcc Rh where an colossal δL<sup>z</sup> = −56 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> is predicted at hω̵ = 0.25 eV and which is drastically reduced at the higher frequency, while the spin component remains suppressed in both cases. For the case of magnetic hcp Co we notice a helicity-dependent change of sign for δL<sup>z</sup> at hω̵ = 0.25 eV which is not present for δSz. In general, δL<sup>z</sup> varies stronger with the light helicity than δSz. Moreover, δL<sup>z</sup> is one to two orders of magnitude larger than δS<sup>z</sup> for the non-magnetic elements, while they are of the same magnitude for the magnetic elemental materials. The last two observations are in agreement with the findings of Ref. [\[20\]](#page-7-5) and indicate how the spontaneous magnetization strongly influences the effect in ferromagnets by the time-reversal symmetry breaking. For comparison, in Table [I](#page-2-0) we list the values of the total light-induced moments, defined as the sum of δL<sup>z</sup> and δSz, for the transitional metals studied in [\[20\]](#page-7-5) with hω̵ = 1.55 eV. Our calculated values are of the same order of magnitude, with the exceptions of Au, and of Co for the case of left-handed polarization, which can be attributed to the difference in the computational methods. From Fig. [1\(](#page-1-0)c-d) we get a clear picture of how the induced moments scale with the strength of spin-orbit coupling (SOC). The effect is overall larger for the heavier 5d elements, both in the orbital δL<sup>z</sup> and spin δS<sup>z</sup> channels. However, at the lower frequency of hω̵ = 0.25 eV it is more difficult to draw such a conclusion since the corresponding photon energy falls into the range of SOC strength, which promotes the role of electronic transitions among spin-orbit split bands occurring within a narrow range around the Fermi energy and limited regions in k-space. In contrast, the use of a much larger frequency involves transitions among manifestly orbitally-distinct states which take place over larger portions <span id="page-2-0"></span>TABLE I. Total light-induced magnetic moments for transition metals previously studied in [\[20\]](#page-7-5) in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV. of the reciprocal space with a more uniform impact of the spin-orbit strength on the magnitude of the transition probabilities. From this discussion we have to exclude the ferromagnetic elements since the spontaneous magnetization induces bands splittings on the scale of exchange strength therefore making the effect much more complex. We study the dependence on the light polarization by performing additional calculations for light circularly polarized in the yz-plane, presenting the results for the z and x components of δL and δS respectively in Figs. [6](#page-10-0) and [7](#page-11-0) of the SM for hω̵ = 0.25 eV and hω̵ = 1.55 eV. Our results for δS<sup>z</sup> and δS<sup>x</sup> for the magnetic transition metals reproduce exactly the values of Ref. [\[21\]](#page-7-9), computed with the same method. We further present the values of δL<sup>z</sup> and δLx, where we find the orbital response along the magnetization, δLz, to be by a factor of two larger than the corresponding spin response, when averaged over all magnetic elements. Generally, for the z-component the effect at the magnetic elements is sim- ![<span id="page-3-0"></span>FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level E<sup>F</sup> (dashed gray line). The horizontal dotted red lines at E<sup>F</sup> ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.](path) ilar in magnitude with the case of polarization in the xy-plane shown in Fig. [1,](#page-1-0) and even in the light helicity, while it vanishes for the non-magnetic elements. On the other hand, for the x-component the effect at the magnetic elements becomes one-two orders of magnitude smaller and is odd in the light helicity, although for the non-magnetic elements the values are comparable to the case shown in Fig. [1.](#page-1-0) It is important to note that for light circularly-polarized in the plane containing ferromagnetic magnetization, the light-induced transverse to magnetization induced moments, even though being smaller than the longitudinal ones, are related to light-induced torques that are experimentally demonstrated to lead to helicity-dependent THz emission [\[21,](#page-7-9) [33\]](#page-7-12). In order to get a better understanding of the impact of the time-reversal symmetry breaking on IFE, we consider the cases of ferromagnetic, non-relativistic (i.e. computed without SOC) ferromagnetic, and antiferromagnetic fcc Ni. In Figs. [8](#page-12-0) and [9](#page-13-0) of the SM we present the band filling dependence of the Cartesian components of the light-induced orbital moments δL, for hω̵ = 0.25 eV and hω̵ = 1.55 eV, respectively. In the non-relativistic case only a δL<sup>i</sup> parallel to the light propagation axis is induced which is odd in the helicity, and no δS<sup>i</sup> is induced, exemplifying that the orbital response is the primary non-relativistic one, whereas the spin response is generated through SOC, as was also shown in [\[20\]](#page-7-5). Similarly, for the antiferromagnetic case, only a component parallel to the light propagation axis is induced which is odd in the helicity and remains unchanged under different polarization flavors. While this is the case also for the induced components δL<sup>x</sup> and δL<sup>y</sup> which are transverse to the magnetization in the ferromagnetic case, the situation drastically changes for the induced component δL<sup>z</sup> parallel to the magnetization, as discussed earlier. We note that a tiny, odd in the helicity, δL<sup>x</sup> or δL<sup>y</sup> is additionally induced when light is rotating in a plane containing ferromagnetic magnetization. On the other hand, the induced δL<sup>x</sup> and δLy, developing normal to the polarization plane, serve as a non-relativistic "background" which is independent of the magnetization, with features due to the crystal structure driving the effect over larger regions in energy. The additional band-splittings induced by SOC result in the IFE exhibiting more features with band filling in the relativistic scenario. Remarkably, while the relativistic antiferromagnetic and non-relativistic ferromagnetic cases in principle have similar to each other behavior in energy, a larger signal arises in the antiferromagnetic case by the virtue of flatter bands (see also the discussion for Hf and Pt below). Lastly, we note that a similar behavior has been observed for the in-plane spin IFE in PT-symmetric Mn2Au, however, in the latter case additional out-of-plane moments arise due to linearly polarized light as a result of broken by the magnetization inversion symmetry [\[29\]](#page-7-8). Next, we focus on the case of 5d transition metals where in Fig. [2\(](#page-2-1)a-b) we explore the relation of δL<sup>z</sup> and δSz, respectively, to the band filling, for light circularly polarized in the xy plane and the frequency of hω̵ = 1.55 eV. When going from group IV (hcp Hf) to XI (fcc Au) we observe a smooth ![<span id="page-4-0"></span>FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.](path) variation of δL<sup>z</sup> from positive to negative values, as well as nicely shaped plateaus, where δL<sup>z</sup> remains relatively robust in a wide energy region, for hcp Hf, bcc Ta, bcc W, fcc Ir, and fcc Pt. We note that such plateaus are often characteristic of orbital effects, as witnessed for example in orbital Hall insulators [\[61](#page-8-8)[–63\]](#page-8-9) and orbital Rashba systems [\[60\]](#page-8-7). On the contrary, δS<sup>z</sup> exhibits a very erratic behavior with strong variations for each material, at the same time being much smaller in magnitude than δLz. The above observation is a clear manifestation of how differently the orbital and spin degrees of freedom behave under light excitation, with the orbital angular momentum having its origin in intrinsic structural parameters as manifested in the crystal field splitting, whereas the spin angular momentum is more sensitive to finer details of the electronic structure mediated by SOC. Indeed, the light-induced spin and orbital moments exhibit similarly erratic behavior with band filling once the frequency of the light is drastically reduced to reach the range of spin-orbit interaction, see Fig. [10](#page-13-1) of the SM for hω̵ = 0.25 eV. # Anatomy of IFE in k-space Among the 5d transition metals, hcp Hf and fcc Pt exhibit the largest computed moments at the Fermi energy for hω̵ = 1.55 eV, with the corresponding values of δL<sup>z</sup> = 9.1 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −0.8 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Hf, and δL<sup>z</sup> = −7.4 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −1.1 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Pt. Therefore, we select these two materials and explore the behavior of their light-induced moments in reciprocal space. We present the band-resolved δL<sup>z</sup> and δS<sup>z</sup> for hcp Hf in Fig. [3\(](#page-3-0)a-b), as well as band-resolved δL<sup>z</sup> and δS<sup>z</sup> for fcc Pt in Fig. [3\(](#page-3-0)c-d). For the case of Hf, transitions along the bands near the A-point are the main source of δL<sup>z</sup> and δSz. Light-induced δL<sup>z</sup> consists of hotspot-like negative contributions and secondary, but extended over energy and k-space consistently positive contributions, which are hardly visible. Overall, however, the latter lead to a large positive orbital integrated response. On the other hand, δS<sup>z</sup> For the case of Pt, δL<sup>z</sup> and δS<sup>z</sup> arise from transitions close to X and L high-symmetry points, with both originating in roughly the same regions of (E, k)-space, but often having an opposite sign to each other. Note that the bands in Pt are much more dispersive in the considered energy window, which results in an effective reduction of the regions in (E, k)-space which contribute to the spin and orbital response alike. This is in contrast to Hf, where much flatter bands reside above and below the Fermi energy within the energy window of the laser pulse, providing significant integrated, albeit very small locally, contributions. Moreover, in fcc Pt the Fermi energy cuts through the band edges of the d-states, where the effect of spin-orbit interaction is the strongest, which explains the emergence of strong hotspot-like contributions with a clear correlation in the magnitude of spin and orbital response and the resultant similar behavior of δL<sup>z</sup> and δS<sup>z</sup> with band filling around the Fermi energy. We further scrutinize the reciprocal space distribution of δL<sup>z</sup> and δSz, shown for hcp Hf and fcc Pt in Fig. [3\(](#page-3-0)e-h). For both materials δL<sup>z</sup> distributions consist of large uniform areas of either positive or negative sign. On the contrary, δS<sup>z</sup> distributions are much finer and richer in details with more areas of opposite sign, consistent with the picture we drew above from the band-resolved analysis. A similar behavior of the light-induced magnetism in reciprocal space has been recently reported for rutile altermagnets [\[39\]](#page-7-18). We also observe that, as discussed above, while for Hf the contributions are well spread throughout the Brillouin zone, for Pt the spin and orbital contributions are located at the edges of the considered k<sup>z</sup> − k<sup>y</sup> plane. Overall, this fact indicates that the microscopic behavior of light-induced magnetism varies strongly among transition metals and crucially depends on the crystal structure and position of the Fermi level with respect to the states ![<span id="page-5-0"></span>FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ### Anisotropy of light-induced magnetism We first address the anisotropy of light-induced magnetism with respect to the magnetization by examining the response of magnetic elements under different flavors of circular polarization. In Fig. [4\(](#page-4-0)a-b) we present the computed δL<sup>z</sup> and δS<sup>z</sup> in relation to scattering lifetime Γ, for the cases of magnetic bcc Fe and hcp Co under excitation by light which is circularly polarized in the xy or yz planes at hω̵ = 1.55 eV. As we have already seen in Fig. [1](#page-1-0) for light polarized in the xyplane, the responses behave differently for right (solid lines) or left (dashed lines) polarization. This difference is more pronounced for δLz, for which we even observe a change of sign for Co. Surprisingly, the situation is drastically different for the case of yz-polarization. In this case, for bcc Fe the response is even in the helicity, while for hcp Co δL<sup>z</sup> is almost even and δS<sup>z</sup> is perfectly even in the helicity. Such different behavior with respect to the light helicity between the two magnetic elements can be traced back to the additional anisotropy originating in the crystal structure itself − an effect which we discuss below. Notably, we witness a highly nonlinear behavior with respect to Γ, with the response reaching colossal values in the clean limit. For example, δL<sup>z</sup> and δS<sup>z</sup> can reach as much as 40 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> and 70 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> in Co, respectively, given the scattering lifetime of 1 meV. Besides the potential tunability of the inverse Faraday effect by the degree of the disorder of the samples, one key message that we extract from our observations is that when comparing the values for laser-induced magnetic moments obtained with different methods, special care has to be taken since the implementations of disorder effects, even within a simple constant broadening model, may differ among various approaches. Finally, we analyze the anisotropy that the crystal structure induces in the light-induced magnetism. We select the case of hcp Re and present in Fig. [5\(](#page-5-0)a-b) the x and z components of δL and δS, respectively, in relation to Γ. Due to the inequivalence of x and z axes in the hcp structure, the δL<sup>z</sup> which originates in light circularly polarized in the xy-plane (light blue line) differs from the δL<sup>x</sup> for the case of the yz polarization (brown line). The situation is similar for the case of δS<sup>z</sup> and δSx. On the contrary, as presented in Fig. [11](#page-14-1) of the SM, in the case of fcc Pt we have a perfect match between δL<sup>z</sup> and δL<sup>x</sup> (or δS<sup>z</sup> and δSx) when changing the plane of circular polarization because the x and z axes are equivalent in the fcc structure. The situation is similar for the y component when light is circularly polarized in the xz-plane. As also expected from the symmetry of the crystal structure, we confirm that the light-induced moments are perfectly odd in the helicity for the nonmagnetic elements, which is not the case for the magnetic elements, see Figs. [5](#page-5-0) and [11](#page-14-1) of the SM [\[20\]](#page-7-5). ### Discussion The main goal of our work is to showcase the importance of the orbital component of IFE and its distinct behavior from the spin counterpart. While so far it was mainly the interaction of light with the spin magnetization that was taken into consideration for the interpretation of IFE-related effects, we speculate that the orbital IFE may provide a novel way to coherently induce magnetization and manipulate the magnetic order. For example, in a recent study, different types of optical torques that may arise in ferromagnetic layers were interpreted in terms of the light-induced orbital moment and its interaction with the magnetization through the spin-orbit interaction [\[64\]](#page-8-10). Since it is known that current-induced orbital accumulation and orbital torques exhibit a long-range behavior [\[49,](#page-8-0) [65](#page-8-11)[–67\]](#page-8-12) due to a characteristic small orbital decay, a question arises whether similar behavior can be exhibited by optical torques caused by the orbital IFE. The emergence of long-ranged orbital IFE would come as no surprise given the fact that several recent experiments reported that laser excitation can drive long-range ballistic orbital currents resulting in THz emission [\[68](#page-8-13)[–71\]](#page-8-14). a perturbation by the electric field of the original ground state Hamiltonian, it is common to utilize the angular momentum of light in order to understand the interaction with the magnetic order by the means of transfer of angular momentum. The spin of light through the helicity of circularly polarized pulses plays a crucial role in helicity-dependent all-optical switching scenarios. On the other hand, there is a strong recent interest in utilizing the orbital angular momentum of light via irradiation of matter with e.g. vortex beams or twisted light, in order to probe the magnetization [\[72\]](#page-8-15), generate photocurrents [\[73\]](#page-8-16), drive ultrafast demagnetization [\[74\]](#page-8-17), and induce IFE [\[40,](#page-7-24) [75\]](#page-8-18). Therefore, it is imperative to treat both spin and orbital degrees of freedom on equal footing when exploring the lightmatter interaction. This will not only trigger further advances in the field of ultrafast magnetism and THz spintronics, but also enable a transition to the novel field of attosecond spintronics [\[76,](#page-8-19) [77\]](#page-8-20). Method. In this work we calculate the first-principles electronic structures of 3d, 4d and 5d transition metals by using the full-potential linearized augmented plane wave FLEUR code [\[78\]](#page-8-21). We describe exchange and correlation effects by using the non-relativistic PBE [\[79\]](#page-8-22) functional, while relativistic effects are described by the second-variation scheme [\[80\]](#page-9-0). The parameters of our first-principles calculations, i.e. lattice constants, muffin-tin radii, plane-wave cutoffs, etc. are taken from Table I of Ref. [\[81\]](#page-9-1). Next, we construct maximally-localized Wannier functions (MLWFs) by employing the Wannier90 code [\[82\]](#page-9-2) and its interface with the FLEUR code [\[83\]](#page-9-3). Similarly to [\[81\]](#page-9-1), we choose s, p and d orbitals for the initial projections and disentagle 18 MLWFs out of 36 Bloch states within a frozen window of 5.0 eV above the Fermi energy for each atom in the unit cell. where O<sup>i</sup> is the i-th component of either the orbital angular momentum operator L<sup>i</sup> or of the spin operator S<sup>i</sup> . Moreover, a<sup>0</sup> = 4πϵ0h̵<sup>2</sup> /(mee 2 ) is the Bohr's radius, I = ϵ0cE<sup>2</sup> 0 /2 is the intensity of the pulse, ϵ<sup>0</sup> is the vacuum permittivity, m<sup>e</sup> is the electron mass, e is the elementary charge, h̵ is the reduced Planck constant, c is the light velocity, E<sup>H</sup> = e 2 /(4πϵ0a0) is the Hartree energy, and ϵ<sup>j</sup> is the j-th component of the polarization vector of the pulse. For example, we describe right/left-handedly polarized light in the xy-plane as ϵ = (1,±i, <sup>0</sup>)/<sup>√</sup> 2, respectively, and define its propagation vector to lie along the normal to the polarization plane. A detailed form of the tensor φijk can be seen in Eq.(14) of Ref. [\[21\]](#page-7-9). For the orbital response, the prefactor in Eq. [$1$](#page-6-2) must be multiplied by an additional factor of 2. A 128×128×128 interpolation k-mesh is sufficient to obtain well-converged results. In all calculations the lifetime broadening Γ was set at 25 meV, the light frequency hω̵ at 0.25 eV and 1.55 eV, the intensity of light at 10 GW/cm<sup>2</sup> , and we covered an energy region of [−2.5, 2.5] eV around the Fermi energy level E<sup>F</sup> . Acknowledgements. We thank Frank Freimuth and Maximilian Merte for discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR 173/3 − 268565370 (project A11) and by the K. and A. Wallenberg Foundation (Grants No. 2022.0079 and 2023.0336). We acknowledge support from the EIC Pathfinder OPEN grant 101129641 "OBELIX". We also gratefully acknowledge the Julich Supercomputing Cen- ¨ tre and RWTH Aachen University for providing computational resources under projects jiff40 and jara0062. Author Contributions. T. A. performed numerical calculations and analysed the results. T. A. and Y. M. wrote the manuscript. 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(a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-11-0"></span>FIG. 7. (a-d) Light-induced orbital δL<sup>x</sup> (a, c) and spin δS<sup>x</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-12-0"></span>FIG. 8. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-13-0"></span>FIG. 9. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ![<span id="page-13-1"></span>FIG. 10. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-14-1"></span>FIG. 11. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic fcc Pt. The induced magnetic moments along the x-axis and along the z-axis are presented. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) <span id="page-14-0"></span>TABLE II. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 0.25 eV. <span id="page-15-0"></span>TABLE III. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV.
FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.	# Light-induced Orbital and Spin Magnetism in 3d, 4d, and 5d Transition Metals Understanding the coherent interplay of light with the magnetization in metals has been a long-standing problem in ultrafast magnetism. While it is known that when laser light acts on a metal it can induce magnetization via the process known as the inverse Faraday effect (IFE), the most basic ingredients of this phenomenon are still largely unexplored. In particular, given a strong recent interest in orbital non-equilibrium dynamics and its role in mediating THz emission in transition metals, the exploration of distinct features in spin and orbital IFE is pertinent. Here, we present a first complete study of the spin and orbital IFE in 3d, 4d and 5d transition metals of groups IV−XI from first-principles. By examining the dependence on the light polarization and frequency, we show that the laser-induced spin and orbital moments may vary significantly both in magnitude and sign. We underpin the interplay between the crystal field splitting and spin-orbit interaction as the key factor which determines the magnitude and key differences between the spin and orbital response. Additionally, we highlight the anisotropy of the effect with respect to the ferromagnetic magnetization and to the crystal structure. The provided complete map of IFE in transition metals is a key reference point in the field of optical magnetism. ### Introduction The demonstration of ultrafast demagnetization in ferromagnets by the application of femtosecond laser pulses [\[1\]](#page-6-0) gave rise to the field of ultrafast spintronics and set the stage for efficient manipulation of magnetism by light. By now it has been rigorously demonstrated that all-optical helicitydependent magnetization switching can be achieved in wide classes of magnetic materials [\[2](#page-6-1)[–10\]](#page-7-0), thus paving the way to contactless ultrafast magnetic recording and information processing. In interpretation of the switching experiments, the inverse Faraday effect (IFE) − i.e. the phenomenon of magnetization induced by a coherent interaction with light acting on a material − has been considered as one of the major underlying mechanisms for the magnetization reversal. Although theoretically predicted and experimentally observed decades ago [\[11–](#page-7-1)[13\]](#page-7-2), a consensus in the theoretical understanding of IFE is still lacking [\[14](#page-7-3)[–19\]](#page-7-4) – however, several microscopic methods have been recently developed for the calculation of this phenomenon in diverse setups [\[20–](#page-7-5)[27\]](#page-7-6). Apart from its established role in the magnetization switching of ferromagnets and ferrimagnents, the impact of IFE in the THz regime has also been studied in antiferromagnets like CrPt [\[28\]](#page-7-7) and Mn2Au [\[29\]](#page-7-8). Moreover, it has been shown that the component of IFE which is perpendicular to the magnetization direction behaves differently with respect to the light helicity than the parallel component [\[21\]](#page-7-9), giving rise to helicity-dependent optical torques [\[30–](#page-7-10)[32\]](#page-7-11) and THz emission [\[33\]](#page-7-12). The impact of the optical torques has been demonstrated to be significant in antiferromagnetic Mn2Au [\[34\]](#page-7-13) where transverse IFE may be responsible for the THz emission [\[35,](#page-7-14) [36\]](#page-7-15), provide an alternative way to switching the magnetization [\[37\]](#page-7-16) as well as to drive domain wall motion [\[38\]](#page-7-17). Recently it has also been shown that light can induce colossal magnetic moments in altermagnets [\[39\]](#page-7-18). Although IFE is conventionally associated with an excitation by circularly polarized light, it can also be activated by linearly-polarized laser pulses [\[25,](#page-7-19) [29,](#page-7-8) [39–](#page-7-18)[42\]](#page-7-20). In solids, two contributions to the magnetization exist: due to spin and orbital moment of electrons. And while the discussion of IFE is normally restricted to the response of spin, since recently, non-equilibrium dynamics of orbital angular momentum started to attract significant attention [\[43](#page-7-21)[–45\]](#page-7-22). Namely, the emergence of orbital currents in the context of the orbital Hall effect [\[46–](#page-7-23)[49\]](#page-8-0), orbital nature of current-induced torques on the magnetization [\[50](#page-8-1)[–52\]](#page-8-2), and current-induced orbital accumulation sizeable even in light materials [\[53–](#page-8-3)[56\]](#page-8-4) have been addressed theoretically and demonstrated experimentally. Notably, in the context of light-induced magnetism, Berritta and co-workers have predicted that the IFE in selected transition metals can exhibit a sizeable orbital component [\[20\]](#page-7-5), with the generality of this observation reaching even into the realm of altermagnets [\[39\]](#page-7-18). At the same time it is also known that within the context of plasmonic IFE the orbital magnetic moment due to electrons excited by laser pulses in small nanoparticles of noble and simple metals can reach atomic values [\[57–](#page-8-5)[59\]](#page-8-6). Despite the fact that very little is known about the interplay of spin and orbital IFE in real materials, it is believed that IFE is a very promising effect in the context of orbitronics − a field which deals with manipulation of the orbital degree of freedom by external perturbations. Although the importance of IFE in mediating the lightinduced magnetism is beyond doubt, a comprehensive indepth material-specific knowledge acquired from microscopic theoretical analysis of this effect is still missing. While acquiring this knowledge is imperative for the field of ultrafast spintronics, since the initial seminal work by Berritta and coworkers [\[20\]](#page-7-5), very little effort has been dedicated to the explorations of this phenomenon from first principles. Here, we ![<span id="page-1-0"></span>FIG. 1. Map of light-induced magnetism in transition metals. (a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) fill this gap by providing a detailed first-principles study of the light-induced spin and orbital magnetism for the 3d, 4d and 5d transition metals of groups IV−XI. By exploring the dependence on the frequency and the polarization of light, we acquire insights into the origin and differences between spin and orbital flavors of IFE, as well as key features of their behavior. Our work provide a solid foundation for further advances in the field of optical magnetism and interaction of light and matter. # Results # Light-induced magnetism in transition metals We begin our discussion by exploring the IFE in a series of 3d, 4d and 5d transition metals of groups IV−XI. IFE is a non-linear optomagnetic effect in which magnetization δO ∝ E × E ∗ is induced as a second-order response to the electric field E of a laser pulse. We calculate the light-induced spin δS and orbital δL magnetic moments by means of the Keldysh formalism, see the section Methods [\[21,](#page-7-9) [39\]](#page-7-18). We focus on light energies hω̵ of 0.25 eV and 1.55 eV which have been routinely used before in our studies within the same formalism [\[29,](#page-7-8) [39,](#page-7-18) [60\]](#page-8-7). Additionally, we choose a lifetime broadening Γ of 25 meV that corresponds to room temperature in order to account for effects of disorder on the electronic states and to provide realistic estimations of the effect. In all calculations we choose the light to be circularly polarized in the xy or yz planes with a light intensity I of 10 GW/cm<sup>2</sup> , while for magnetic materials ferromagnetic magnetization is considered to be along the z-axis. In Fig. [1](#page-1-0) we present the calculated light-induced δL<sup>z</sup> [Fig. [1\(](#page-1-0)a-c)] and δS<sup>z</sup> [Fig. [1\(](#page-1-0)b-d)] moments of the considered transition metals at the Fermi energy. Light is circularly polarized in the xy plane, with energies hω̵ of 0.25 eV [Fig. [1\(](#page-1-0)ab)] and 1.55 eV [Fig. [1\(](#page-1-0)c-d)]. Regarding the 3d magnetic elements Fe, Co, Ni, we analyze both right and left handed circular polarizations of light. The exact values of δL<sup>z</sup> and δS<sup>z</sup> are listed in Tables [II](#page-14-0) and [III](#page-15-0) of the Supplemental Material (SM). Not shown are computed transverse x and y compo- ![<span id="page-2-1"></span>FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.](path) At first sight we notice a strong dependence of the magnitude and sign of both δL<sup>z</sup> and δS<sup>z</sup> on the light frequency. We point out the case of fcc Rh where an colossal δL<sup>z</sup> = −56 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> is predicted at hω̵ = 0.25 eV and which is drastically reduced at the higher frequency, while the spin component remains suppressed in both cases. For the case of magnetic hcp Co we notice a helicity-dependent change of sign for δL<sup>z</sup> at hω̵ = 0.25 eV which is not present for δSz. In general, δL<sup>z</sup> varies stronger with the light helicity than δSz. Moreover, δL<sup>z</sup> is one to two orders of magnitude larger than δS<sup>z</sup> for the non-magnetic elements, while they are of the same magnitude for the magnetic elemental materials. The last two observations are in agreement with the findings of Ref. [\[20\]](#page-7-5) and indicate how the spontaneous magnetization strongly influences the effect in ferromagnets by the time-reversal symmetry breaking. For comparison, in Table [I](#page-2-0) we list the values of the total light-induced moments, defined as the sum of δL<sup>z</sup> and δSz, for the transitional metals studied in [\[20\]](#page-7-5) with hω̵ = 1.55 eV. Our calculated values are of the same order of magnitude, with the exceptions of Au, and of Co for the case of left-handed polarization, which can be attributed to the difference in the computational methods. From Fig. [1\(](#page-1-0)c-d) we get a clear picture of how the induced moments scale with the strength of spin-orbit coupling (SOC). The effect is overall larger for the heavier 5d elements, both in the orbital δL<sup>z</sup> and spin δS<sup>z</sup> channels. However, at the lower frequency of hω̵ = 0.25 eV it is more difficult to draw such a conclusion since the corresponding photon energy falls into the range of SOC strength, which promotes the role of electronic transitions among spin-orbit split bands occurring within a narrow range around the Fermi energy and limited regions in k-space. In contrast, the use of a much larger frequency involves transitions among manifestly orbitally-distinct states which take place over larger portions <span id="page-2-0"></span>TABLE I. Total light-induced magnetic moments for transition metals previously studied in [\[20\]](#page-7-5) in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV. of the reciprocal space with a more uniform impact of the spin-orbit strength on the magnitude of the transition probabilities. From this discussion we have to exclude the ferromagnetic elements since the spontaneous magnetization induces bands splittings on the scale of exchange strength therefore making the effect much more complex. We study the dependence on the light polarization by performing additional calculations for light circularly polarized in the yz-plane, presenting the results for the z and x components of δL and δS respectively in Figs. [6](#page-10-0) and [7](#page-11-0) of the SM for hω̵ = 0.25 eV and hω̵ = 1.55 eV. Our results for δS<sup>z</sup> and δS<sup>x</sup> for the magnetic transition metals reproduce exactly the values of Ref. [\[21\]](#page-7-9), computed with the same method. We further present the values of δL<sup>z</sup> and δLx, where we find the orbital response along the magnetization, δLz, to be by a factor of two larger than the corresponding spin response, when averaged over all magnetic elements. Generally, for the z-component the effect at the magnetic elements is sim- ![<span id="page-3-0"></span>FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level E<sup>F</sup> (dashed gray line). The horizontal dotted red lines at E<sup>F</sup> ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.](path) ilar in magnitude with the case of polarization in the xy-plane shown in Fig. [1,](#page-1-0) and even in the light helicity, while it vanishes for the non-magnetic elements. On the other hand, for the x-component the effect at the magnetic elements becomes one-two orders of magnitude smaller and is odd in the light helicity, although for the non-magnetic elements the values are comparable to the case shown in Fig. [1.](#page-1-0) It is important to note that for light circularly-polarized in the plane containing ferromagnetic magnetization, the light-induced transverse to magnetization induced moments, even though being smaller than the longitudinal ones, are related to light-induced torques that are experimentally demonstrated to lead to helicity-dependent THz emission [\[21,](#page-7-9) [33\]](#page-7-12). In order to get a better understanding of the impact of the time-reversal symmetry breaking on IFE, we consider the cases of ferromagnetic, non-relativistic (i.e. computed without SOC) ferromagnetic, and antiferromagnetic fcc Ni. In Figs. [8](#page-12-0) and [9](#page-13-0) of the SM we present the band filling dependence of the Cartesian components of the light-induced orbital moments δL, for hω̵ = 0.25 eV and hω̵ = 1.55 eV, respectively. In the non-relativistic case only a δL<sup>i</sup> parallel to the light propagation axis is induced which is odd in the helicity, and no δS<sup>i</sup> is induced, exemplifying that the orbital response is the primary non-relativistic one, whereas the spin response is generated through SOC, as was also shown in [\[20\]](#page-7-5). Similarly, for the antiferromagnetic case, only a component parallel to the light propagation axis is induced which is odd in the helicity and remains unchanged under different polarization flavors. While this is the case also for the induced components δL<sup>x</sup> and δL<sup>y</sup> which are transverse to the magnetization in the ferromagnetic case, the situation drastically changes for the induced component δL<sup>z</sup> parallel to the magnetization, as discussed earlier. We note that a tiny, odd in the helicity, δL<sup>x</sup> or δL<sup>y</sup> is additionally induced when light is rotating in a plane containing ferromagnetic magnetization. On the other hand, the induced δL<sup>x</sup> and δLy, developing normal to the polarization plane, serve as a non-relativistic "background" which is independent of the magnetization, with features due to the crystal structure driving the effect over larger regions in energy. The additional band-splittings induced by SOC result in the IFE exhibiting more features with band filling in the relativistic scenario. Remarkably, while the relativistic antiferromagnetic and non-relativistic ferromagnetic cases in principle have similar to each other behavior in energy, a larger signal arises in the antiferromagnetic case by the virtue of flatter bands (see also the discussion for Hf and Pt below). Lastly, we note that a similar behavior has been observed for the in-plane spin IFE in PT-symmetric Mn2Au, however, in the latter case additional out-of-plane moments arise due to linearly polarized light as a result of broken by the magnetization inversion symmetry [\[29\]](#page-7-8). Next, we focus on the case of 5d transition metals where in Fig. [2\(](#page-2-1)a-b) we explore the relation of δL<sup>z</sup> and δSz, respectively, to the band filling, for light circularly polarized in the xy plane and the frequency of hω̵ = 1.55 eV. When going from group IV (hcp Hf) to XI (fcc Au) we observe a smooth ![<span id="page-4-0"></span>FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.](path) variation of δL<sup>z</sup> from positive to negative values, as well as nicely shaped plateaus, where δL<sup>z</sup> remains relatively robust in a wide energy region, for hcp Hf, bcc Ta, bcc W, fcc Ir, and fcc Pt. We note that such plateaus are often characteristic of orbital effects, as witnessed for example in orbital Hall insulators [\[61](#page-8-8)[–63\]](#page-8-9) and orbital Rashba systems [\[60\]](#page-8-7). On the contrary, δS<sup>z</sup> exhibits a very erratic behavior with strong variations for each material, at the same time being much smaller in magnitude than δLz. The above observation is a clear manifestation of how differently the orbital and spin degrees of freedom behave under light excitation, with the orbital angular momentum having its origin in intrinsic structural parameters as manifested in the crystal field splitting, whereas the spin angular momentum is more sensitive to finer details of the electronic structure mediated by SOC. Indeed, the light-induced spin and orbital moments exhibit similarly erratic behavior with band filling once the frequency of the light is drastically reduced to reach the range of spin-orbit interaction, see Fig. [10](#page-13-1) of the SM for hω̵ = 0.25 eV. # Anatomy of IFE in k-space Among the 5d transition metals, hcp Hf and fcc Pt exhibit the largest computed moments at the Fermi energy for hω̵ = 1.55 eV, with the corresponding values of δL<sup>z</sup> = 9.1 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −0.8 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Hf, and δL<sup>z</sup> = −7.4 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −1.1 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Pt. Therefore, we select these two materials and explore the behavior of their light-induced moments in reciprocal space. We present the band-resolved δL<sup>z</sup> and δS<sup>z</sup> for hcp Hf in Fig. [3\(](#page-3-0)a-b), as well as band-resolved δL<sup>z</sup> and δS<sup>z</sup> for fcc Pt in Fig. [3\(](#page-3-0)c-d). For the case of Hf, transitions along the bands near the A-point are the main source of δL<sup>z</sup> and δSz. Light-induced δL<sup>z</sup> consists of hotspot-like negative contributions and secondary, but extended over energy and k-space consistently positive contributions, which are hardly visible. Overall, however, the latter lead to a large positive orbital integrated response. On the other hand, δS<sup>z</sup> For the case of Pt, δL<sup>z</sup> and δS<sup>z</sup> arise from transitions close to X and L high-symmetry points, with both originating in roughly the same regions of (E, k)-space, but often having an opposite sign to each other. Note that the bands in Pt are much more dispersive in the considered energy window, which results in an effective reduction of the regions in (E, k)-space which contribute to the spin and orbital response alike. This is in contrast to Hf, where much flatter bands reside above and below the Fermi energy within the energy window of the laser pulse, providing significant integrated, albeit very small locally, contributions. Moreover, in fcc Pt the Fermi energy cuts through the band edges of the d-states, where the effect of spin-orbit interaction is the strongest, which explains the emergence of strong hotspot-like contributions with a clear correlation in the magnitude of spin and orbital response and the resultant similar behavior of δL<sup>z</sup> and δS<sup>z</sup> with band filling around the Fermi energy. We further scrutinize the reciprocal space distribution of δL<sup>z</sup> and δSz, shown for hcp Hf and fcc Pt in Fig. [3\(](#page-3-0)e-h). For both materials δL<sup>z</sup> distributions consist of large uniform areas of either positive or negative sign. On the contrary, δS<sup>z</sup> distributions are much finer and richer in details with more areas of opposite sign, consistent with the picture we drew above from the band-resolved analysis. A similar behavior of the light-induced magnetism in reciprocal space has been recently reported for rutile altermagnets [\[39\]](#page-7-18). We also observe that, as discussed above, while for Hf the contributions are well spread throughout the Brillouin zone, for Pt the spin and orbital contributions are located at the edges of the considered k<sup>z</sup> − k<sup>y</sup> plane. Overall, this fact indicates that the microscopic behavior of light-induced magnetism varies strongly among transition metals and crucially depends on the crystal structure and position of the Fermi level with respect to the states ![<span id="page-5-0"></span>FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ### Anisotropy of light-induced magnetism We first address the anisotropy of light-induced magnetism with respect to the magnetization by examining the response of magnetic elements under different flavors of circular polarization. In Fig. [4\(](#page-4-0)a-b) we present the computed δL<sup>z</sup> and δS<sup>z</sup> in relation to scattering lifetime Γ, for the cases of magnetic bcc Fe and hcp Co under excitation by light which is circularly polarized in the xy or yz planes at hω̵ = 1.55 eV. As we have already seen in Fig. [1](#page-1-0) for light polarized in the xyplane, the responses behave differently for right (solid lines) or left (dashed lines) polarization. This difference is more pronounced for δLz, for which we even observe a change of sign for Co. Surprisingly, the situation is drastically different for the case of yz-polarization. In this case, for bcc Fe the response is even in the helicity, while for hcp Co δL<sup>z</sup> is almost even and δS<sup>z</sup> is perfectly even in the helicity. Such different behavior with respect to the light helicity between the two magnetic elements can be traced back to the additional anisotropy originating in the crystal structure itself − an effect which we discuss below. Notably, we witness a highly nonlinear behavior with respect to Γ, with the response reaching colossal values in the clean limit. For example, δL<sup>z</sup> and δS<sup>z</sup> can reach as much as 40 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> and 70 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> in Co, respectively, given the scattering lifetime of 1 meV. Besides the potential tunability of the inverse Faraday effect by the degree of the disorder of the samples, one key message that we extract from our observations is that when comparing the values for laser-induced magnetic moments obtained with different methods, special care has to be taken since the implementations of disorder effects, even within a simple constant broadening model, may differ among various approaches. Finally, we analyze the anisotropy that the crystal structure induces in the light-induced magnetism. We select the case of hcp Re and present in Fig. [5\(](#page-5-0)a-b) the x and z components of δL and δS, respectively, in relation to Γ. Due to the inequivalence of x and z axes in the hcp structure, the δL<sup>z</sup> which originates in light circularly polarized in the xy-plane (light blue line) differs from the δL<sup>x</sup> for the case of the yz polarization (brown line). The situation is similar for the case of δS<sup>z</sup> and δSx. On the contrary, as presented in Fig. [11](#page-14-1) of the SM, in the case of fcc Pt we have a perfect match between δL<sup>z</sup> and δL<sup>x</sup> (or δS<sup>z</sup> and δSx) when changing the plane of circular polarization because the x and z axes are equivalent in the fcc structure. The situation is similar for the y component when light is circularly polarized in the xz-plane. As also expected from the symmetry of the crystal structure, we confirm that the light-induced moments are perfectly odd in the helicity for the nonmagnetic elements, which is not the case for the magnetic elements, see Figs. [5](#page-5-0) and [11](#page-14-1) of the SM [\[20\]](#page-7-5). ### Discussion The main goal of our work is to showcase the importance of the orbital component of IFE and its distinct behavior from the spin counterpart. While so far it was mainly the interaction of light with the spin magnetization that was taken into consideration for the interpretation of IFE-related effects, we speculate that the orbital IFE may provide a novel way to coherently induce magnetization and manipulate the magnetic order. For example, in a recent study, different types of optical torques that may arise in ferromagnetic layers were interpreted in terms of the light-induced orbital moment and its interaction with the magnetization through the spin-orbit interaction [\[64\]](#page-8-10). Since it is known that current-induced orbital accumulation and orbital torques exhibit a long-range behavior [\[49,](#page-8-0) [65](#page-8-11)[–67\]](#page-8-12) due to a characteristic small orbital decay, a question arises whether similar behavior can be exhibited by optical torques caused by the orbital IFE. The emergence of long-ranged orbital IFE would come as no surprise given the fact that several recent experiments reported that laser excitation can drive long-range ballistic orbital currents resulting in THz emission [\[68](#page-8-13)[–71\]](#page-8-14). a perturbation by the electric field of the original ground state Hamiltonian, it is common to utilize the angular momentum of light in order to understand the interaction with the magnetic order by the means of transfer of angular momentum. The spin of light through the helicity of circularly polarized pulses plays a crucial role in helicity-dependent all-optical switching scenarios. On the other hand, there is a strong recent interest in utilizing the orbital angular momentum of light via irradiation of matter with e.g. vortex beams or twisted light, in order to probe the magnetization [\[72\]](#page-8-15), generate photocurrents [\[73\]](#page-8-16), drive ultrafast demagnetization [\[74\]](#page-8-17), and induce IFE [\[40,](#page-7-24) [75\]](#page-8-18). Therefore, it is imperative to treat both spin and orbital degrees of freedom on equal footing when exploring the lightmatter interaction. This will not only trigger further advances in the field of ultrafast magnetism and THz spintronics, but also enable a transition to the novel field of attosecond spintronics [\[76,](#page-8-19) [77\]](#page-8-20). Method. In this work we calculate the first-principles electronic structures of 3d, 4d and 5d transition metals by using the full-potential linearized augmented plane wave FLEUR code [\[78\]](#page-8-21). We describe exchange and correlation effects by using the non-relativistic PBE [\[79\]](#page-8-22) functional, while relativistic effects are described by the second-variation scheme [\[80\]](#page-9-0). The parameters of our first-principles calculations, i.e. lattice constants, muffin-tin radii, plane-wave cutoffs, etc. are taken from Table I of Ref. [\[81\]](#page-9-1). Next, we construct maximally-localized Wannier functions (MLWFs) by employing the Wannier90 code [\[82\]](#page-9-2) and its interface with the FLEUR code [\[83\]](#page-9-3). Similarly to [\[81\]](#page-9-1), we choose s, p and d orbitals for the initial projections and disentagle 18 MLWFs out of 36 Bloch states within a frozen window of 5.0 eV above the Fermi energy for each atom in the unit cell. where O<sup>i</sup> is the i-th component of either the orbital angular momentum operator L<sup>i</sup> or of the spin operator S<sup>i</sup> . Moreover, a<sup>0</sup> = 4πϵ0h̵<sup>2</sup> /(mee 2 ) is the Bohr's radius, I = ϵ0cE<sup>2</sup> 0 /2 is the intensity of the pulse, ϵ<sup>0</sup> is the vacuum permittivity, m<sup>e</sup> is the electron mass, e is the elementary charge, h̵ is the reduced Planck constant, c is the light velocity, E<sup>H</sup> = e 2 /(4πϵ0a0) is the Hartree energy, and ϵ<sup>j</sup> is the j-th component of the polarization vector of the pulse. For example, we describe right/left-handedly polarized light in the xy-plane as ϵ = (1,±i, <sup>0</sup>)/<sup>√</sup> 2, respectively, and define its propagation vector to lie along the normal to the polarization plane. A detailed form of the tensor φijk can be seen in Eq.(14) of Ref. [\[21\]](#page-7-9). For the orbital response, the prefactor in Eq. [$1$](#page-6-2) must be multiplied by an additional factor of 2. A 128×128×128 interpolation k-mesh is sufficient to obtain well-converged results. In all calculations the lifetime broadening Γ was set at 25 meV, the light frequency hω̵ at 0.25 eV and 1.55 eV, the intensity of light at 10 GW/cm<sup>2</sup> , and we covered an energy region of [−2.5, 2.5] eV around the Fermi energy level E<sup>F</sup> . Acknowledgements. We thank Frank Freimuth and Maximilian Merte for discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR 173/3 − 268565370 (project A11) and by the K. and A. Wallenberg Foundation (Grants No. 2022.0079 and 2023.0336). We acknowledge support from the EIC Pathfinder OPEN grant 101129641 "OBELIX". We also gratefully acknowledge the Julich Supercomputing Cen- ¨ tre and RWTH Aachen University for providing computational resources under projects jiff40 and jara0062. Author Contributions. T. A. performed numerical calculations and analysed the results. T. A. and Y. M. wrote the manuscript. 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(a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-11-0"></span>FIG. 7. (a-d) Light-induced orbital δL<sup>x</sup> (a, c) and spin δS<sup>x</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-12-0"></span>FIG. 8. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-13-0"></span>FIG. 9. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ![<span id="page-13-1"></span>FIG. 10. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-14-1"></span>FIG. 11. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic fcc Pt. The induced magnetic moments along the x-axis and along the z-axis are presented. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) <span id="page-14-0"></span>TABLE II. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 0.25 eV. <span id="page-15-0"></span>TABLE III. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV.
FIG. 8. (a-i) Cartesian components of the light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.	# Light-induced Orbital and Spin Magnetism in 3d, 4d, and 5d Transition Metals Understanding the coherent interplay of light with the magnetization in metals has been a long-standing problem in ultrafast magnetism. While it is known that when laser light acts on a metal it can induce magnetization via the process known as the inverse Faraday effect (IFE), the most basic ingredients of this phenomenon are still largely unexplored. In particular, given a strong recent interest in orbital non-equilibrium dynamics and its role in mediating THz emission in transition metals, the exploration of distinct features in spin and orbital IFE is pertinent. Here, we present a first complete study of the spin and orbital IFE in 3d, 4d and 5d transition metals of groups IV−XI from first-principles. By examining the dependence on the light polarization and frequency, we show that the laser-induced spin and orbital moments may vary significantly both in magnitude and sign. We underpin the interplay between the crystal field splitting and spin-orbit interaction as the key factor which determines the magnitude and key differences between the spin and orbital response. Additionally, we highlight the anisotropy of the effect with respect to the ferromagnetic magnetization and to the crystal structure. The provided complete map of IFE in transition metals is a key reference point in the field of optical magnetism. ### Introduction The demonstration of ultrafast demagnetization in ferromagnets by the application of femtosecond laser pulses [\[1\]](#page-6-0) gave rise to the field of ultrafast spintronics and set the stage for efficient manipulation of magnetism by light. By now it has been rigorously demonstrated that all-optical helicitydependent magnetization switching can be achieved in wide classes of magnetic materials [\[2](#page-6-1)[–10\]](#page-7-0), thus paving the way to contactless ultrafast magnetic recording and information processing. In interpretation of the switching experiments, the inverse Faraday effect (IFE) − i.e. the phenomenon of magnetization induced by a coherent interaction with light acting on a material − has been considered as one of the major underlying mechanisms for the magnetization reversal. Although theoretically predicted and experimentally observed decades ago [\[11–](#page-7-1)[13\]](#page-7-2), a consensus in the theoretical understanding of IFE is still lacking [\[14](#page-7-3)[–19\]](#page-7-4) – however, several microscopic methods have been recently developed for the calculation of this phenomenon in diverse setups [\[20–](#page-7-5)[27\]](#page-7-6). Apart from its established role in the magnetization switching of ferromagnets and ferrimagnents, the impact of IFE in the THz regime has also been studied in antiferromagnets like CrPt [\[28\]](#page-7-7) and Mn2Au [\[29\]](#page-7-8). Moreover, it has been shown that the component of IFE which is perpendicular to the magnetization direction behaves differently with respect to the light helicity than the parallel component [\[21\]](#page-7-9), giving rise to helicity-dependent optical torques [\[30–](#page-7-10)[32\]](#page-7-11) and THz emission [\[33\]](#page-7-12). The impact of the optical torques has been demonstrated to be significant in antiferromagnetic Mn2Au [\[34\]](#page-7-13) where transverse IFE may be responsible for the THz emission [\[35,](#page-7-14) [36\]](#page-7-15), provide an alternative way to switching the magnetization [\[37\]](#page-7-16) as well as to drive domain wall motion [\[38\]](#page-7-17). Recently it has also been shown that light can induce colossal magnetic moments in altermagnets [\[39\]](#page-7-18). Although IFE is conventionally associated with an excitation by circularly polarized light, it can also be activated by linearly-polarized laser pulses [\[25,](#page-7-19) [29,](#page-7-8) [39–](#page-7-18)[42\]](#page-7-20). In solids, two contributions to the magnetization exist: due to spin and orbital moment of electrons. And while the discussion of IFE is normally restricted to the response of spin, since recently, non-equilibrium dynamics of orbital angular momentum started to attract significant attention [\[43](#page-7-21)[–45\]](#page-7-22). Namely, the emergence of orbital currents in the context of the orbital Hall effect [\[46–](#page-7-23)[49\]](#page-8-0), orbital nature of current-induced torques on the magnetization [\[50](#page-8-1)[–52\]](#page-8-2), and current-induced orbital accumulation sizeable even in light materials [\[53–](#page-8-3)[56\]](#page-8-4) have been addressed theoretically and demonstrated experimentally. Notably, in the context of light-induced magnetism, Berritta and co-workers have predicted that the IFE in selected transition metals can exhibit a sizeable orbital component [\[20\]](#page-7-5), with the generality of this observation reaching even into the realm of altermagnets [\[39\]](#page-7-18). At the same time it is also known that within the context of plasmonic IFE the orbital magnetic moment due to electrons excited by laser pulses in small nanoparticles of noble and simple metals can reach atomic values [\[57–](#page-8-5)[59\]](#page-8-6). Despite the fact that very little is known about the interplay of spin and orbital IFE in real materials, it is believed that IFE is a very promising effect in the context of orbitronics − a field which deals with manipulation of the orbital degree of freedom by external perturbations. Although the importance of IFE in mediating the lightinduced magnetism is beyond doubt, a comprehensive indepth material-specific knowledge acquired from microscopic theoretical analysis of this effect is still missing. While acquiring this knowledge is imperative for the field of ultrafast spintronics, since the initial seminal work by Berritta and coworkers [\[20\]](#page-7-5), very little effort has been dedicated to the explorations of this phenomenon from first principles. Here, we ![<span id="page-1-0"></span>FIG. 1. Map of light-induced magnetism in transition metals. (a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) fill this gap by providing a detailed first-principles study of the light-induced spin and orbital magnetism for the 3d, 4d and 5d transition metals of groups IV−XI. By exploring the dependence on the frequency and the polarization of light, we acquire insights into the origin and differences between spin and orbital flavors of IFE, as well as key features of their behavior. Our work provide a solid foundation for further advances in the field of optical magnetism and interaction of light and matter. # Results # Light-induced magnetism in transition metals We begin our discussion by exploring the IFE in a series of 3d, 4d and 5d transition metals of groups IV−XI. IFE is a non-linear optomagnetic effect in which magnetization δO ∝ E × E ∗ is induced as a second-order response to the electric field E of a laser pulse. We calculate the light-induced spin δS and orbital δL magnetic moments by means of the Keldysh formalism, see the section Methods [\[21,](#page-7-9) [39\]](#page-7-18). We focus on light energies hω̵ of 0.25 eV and 1.55 eV which have been routinely used before in our studies within the same formalism [\[29,](#page-7-8) [39,](#page-7-18) [60\]](#page-8-7). Additionally, we choose a lifetime broadening Γ of 25 meV that corresponds to room temperature in order to account for effects of disorder on the electronic states and to provide realistic estimations of the effect. In all calculations we choose the light to be circularly polarized in the xy or yz planes with a light intensity I of 10 GW/cm<sup>2</sup> , while for magnetic materials ferromagnetic magnetization is considered to be along the z-axis. In Fig. [1](#page-1-0) we present the calculated light-induced δL<sup>z</sup> [Fig. [1\(](#page-1-0)a-c)] and δS<sup>z</sup> [Fig. [1\(](#page-1-0)b-d)] moments of the considered transition metals at the Fermi energy. Light is circularly polarized in the xy plane, with energies hω̵ of 0.25 eV [Fig. [1\(](#page-1-0)ab)] and 1.55 eV [Fig. [1\(](#page-1-0)c-d)]. Regarding the 3d magnetic elements Fe, Co, Ni, we analyze both right and left handed circular polarizations of light. The exact values of δL<sup>z</sup> and δS<sup>z</sup> are listed in Tables [II](#page-14-0) and [III](#page-15-0) of the Supplemental Material (SM). Not shown are computed transverse x and y compo- ![<span id="page-2-1"></span>FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.](path) At first sight we notice a strong dependence of the magnitude and sign of both δL<sup>z</sup> and δS<sup>z</sup> on the light frequency. We point out the case of fcc Rh where an colossal δL<sup>z</sup> = −56 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> is predicted at hω̵ = 0.25 eV and which is drastically reduced at the higher frequency, while the spin component remains suppressed in both cases. For the case of magnetic hcp Co we notice a helicity-dependent change of sign for δL<sup>z</sup> at hω̵ = 0.25 eV which is not present for δSz. In general, δL<sup>z</sup> varies stronger with the light helicity than δSz. Moreover, δL<sup>z</sup> is one to two orders of magnitude larger than δS<sup>z</sup> for the non-magnetic elements, while they are of the same magnitude for the magnetic elemental materials. The last two observations are in agreement with the findings of Ref. [\[20\]](#page-7-5) and indicate how the spontaneous magnetization strongly influences the effect in ferromagnets by the time-reversal symmetry breaking. For comparison, in Table [I](#page-2-0) we list the values of the total light-induced moments, defined as the sum of δL<sup>z</sup> and δSz, for the transitional metals studied in [\[20\]](#page-7-5) with hω̵ = 1.55 eV. Our calculated values are of the same order of magnitude, with the exceptions of Au, and of Co for the case of left-handed polarization, which can be attributed to the difference in the computational methods. From Fig. [1\(](#page-1-0)c-d) we get a clear picture of how the induced moments scale with the strength of spin-orbit coupling (SOC). The effect is overall larger for the heavier 5d elements, both in the orbital δL<sup>z</sup> and spin δS<sup>z</sup> channels. However, at the lower frequency of hω̵ = 0.25 eV it is more difficult to draw such a conclusion since the corresponding photon energy falls into the range of SOC strength, which promotes the role of electronic transitions among spin-orbit split bands occurring within a narrow range around the Fermi energy and limited regions in k-space. In contrast, the use of a much larger frequency involves transitions among manifestly orbitally-distinct states which take place over larger portions <span id="page-2-0"></span>TABLE I. Total light-induced magnetic moments for transition metals previously studied in [\[20\]](#page-7-5) in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV. of the reciprocal space with a more uniform impact of the spin-orbit strength on the magnitude of the transition probabilities. From this discussion we have to exclude the ferromagnetic elements since the spontaneous magnetization induces bands splittings on the scale of exchange strength therefore making the effect much more complex. We study the dependence on the light polarization by performing additional calculations for light circularly polarized in the yz-plane, presenting the results for the z and x components of δL and δS respectively in Figs. [6](#page-10-0) and [7](#page-11-0) of the SM for hω̵ = 0.25 eV and hω̵ = 1.55 eV. Our results for δS<sup>z</sup> and δS<sup>x</sup> for the magnetic transition metals reproduce exactly the values of Ref. [\[21\]](#page-7-9), computed with the same method. We further present the values of δL<sup>z</sup> and δLx, where we find the orbital response along the magnetization, δLz, to be by a factor of two larger than the corresponding spin response, when averaged over all magnetic elements. Generally, for the z-component the effect at the magnetic elements is sim- ![<span id="page-3-0"></span>FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level E<sup>F</sup> (dashed gray line). The horizontal dotted red lines at E<sup>F</sup> ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.](path) ilar in magnitude with the case of polarization in the xy-plane shown in Fig. [1,](#page-1-0) and even in the light helicity, while it vanishes for the non-magnetic elements. On the other hand, for the x-component the effect at the magnetic elements becomes one-two orders of magnitude smaller and is odd in the light helicity, although for the non-magnetic elements the values are comparable to the case shown in Fig. [1.](#page-1-0) It is important to note that for light circularly-polarized in the plane containing ferromagnetic magnetization, the light-induced transverse to magnetization induced moments, even though being smaller than the longitudinal ones, are related to light-induced torques that are experimentally demonstrated to lead to helicity-dependent THz emission [\[21,](#page-7-9) [33\]](#page-7-12). In order to get a better understanding of the impact of the time-reversal symmetry breaking on IFE, we consider the cases of ferromagnetic, non-relativistic (i.e. computed without SOC) ferromagnetic, and antiferromagnetic fcc Ni. In Figs. [8](#page-12-0) and [9](#page-13-0) of the SM we present the band filling dependence of the Cartesian components of the light-induced orbital moments δL, for hω̵ = 0.25 eV and hω̵ = 1.55 eV, respectively. In the non-relativistic case only a δL<sup>i</sup> parallel to the light propagation axis is induced which is odd in the helicity, and no δS<sup>i</sup> is induced, exemplifying that the orbital response is the primary non-relativistic one, whereas the spin response is generated through SOC, as was also shown in [\[20\]](#page-7-5). Similarly, for the antiferromagnetic case, only a component parallel to the light propagation axis is induced which is odd in the helicity and remains unchanged under different polarization flavors. While this is the case also for the induced components δL<sup>x</sup> and δL<sup>y</sup> which are transverse to the magnetization in the ferromagnetic case, the situation drastically changes for the induced component δL<sup>z</sup> parallel to the magnetization, as discussed earlier. We note that a tiny, odd in the helicity, δL<sup>x</sup> or δL<sup>y</sup> is additionally induced when light is rotating in a plane containing ferromagnetic magnetization. On the other hand, the induced δL<sup>x</sup> and δLy, developing normal to the polarization plane, serve as a non-relativistic "background" which is independent of the magnetization, with features due to the crystal structure driving the effect over larger regions in energy. The additional band-splittings induced by SOC result in the IFE exhibiting more features with band filling in the relativistic scenario. Remarkably, while the relativistic antiferromagnetic and non-relativistic ferromagnetic cases in principle have similar to each other behavior in energy, a larger signal arises in the antiferromagnetic case by the virtue of flatter bands (see also the discussion for Hf and Pt below). Lastly, we note that a similar behavior has been observed for the in-plane spin IFE in PT-symmetric Mn2Au, however, in the latter case additional out-of-plane moments arise due to linearly polarized light as a result of broken by the magnetization inversion symmetry [\[29\]](#page-7-8). Next, we focus on the case of 5d transition metals where in Fig. [2\(](#page-2-1)a-b) we explore the relation of δL<sup>z</sup> and δSz, respectively, to the band filling, for light circularly polarized in the xy plane and the frequency of hω̵ = 1.55 eV. When going from group IV (hcp Hf) to XI (fcc Au) we observe a smooth ![<span id="page-4-0"></span>FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.](path) variation of δL<sup>z</sup> from positive to negative values, as well as nicely shaped plateaus, where δL<sup>z</sup> remains relatively robust in a wide energy region, for hcp Hf, bcc Ta, bcc W, fcc Ir, and fcc Pt. We note that such plateaus are often characteristic of orbital effects, as witnessed for example in orbital Hall insulators [\[61](#page-8-8)[–63\]](#page-8-9) and orbital Rashba systems [\[60\]](#page-8-7). On the contrary, δS<sup>z</sup> exhibits a very erratic behavior with strong variations for each material, at the same time being much smaller in magnitude than δLz. The above observation is a clear manifestation of how differently the orbital and spin degrees of freedom behave under light excitation, with the orbital angular momentum having its origin in intrinsic structural parameters as manifested in the crystal field splitting, whereas the spin angular momentum is more sensitive to finer details of the electronic structure mediated by SOC. Indeed, the light-induced spin and orbital moments exhibit similarly erratic behavior with band filling once the frequency of the light is drastically reduced to reach the range of spin-orbit interaction, see Fig. [10](#page-13-1) of the SM for hω̵ = 0.25 eV. # Anatomy of IFE in k-space Among the 5d transition metals, hcp Hf and fcc Pt exhibit the largest computed moments at the Fermi energy for hω̵ = 1.55 eV, with the corresponding values of δL<sup>z</sup> = 9.1 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −0.8 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Hf, and δL<sup>z</sup> = −7.4 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −1.1 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Pt. Therefore, we select these two materials and explore the behavior of their light-induced moments in reciprocal space. We present the band-resolved δL<sup>z</sup> and δS<sup>z</sup> for hcp Hf in Fig. [3\(](#page-3-0)a-b), as well as band-resolved δL<sup>z</sup> and δS<sup>z</sup> for fcc Pt in Fig. [3\(](#page-3-0)c-d). For the case of Hf, transitions along the bands near the A-point are the main source of δL<sup>z</sup> and δSz. Light-induced δL<sup>z</sup> consists of hotspot-like negative contributions and secondary, but extended over energy and k-space consistently positive contributions, which are hardly visible. Overall, however, the latter lead to a large positive orbital integrated response. On the other hand, δS<sup>z</sup> For the case of Pt, δL<sup>z</sup> and δS<sup>z</sup> arise from transitions close to X and L high-symmetry points, with both originating in roughly the same regions of (E, k)-space, but often having an opposite sign to each other. Note that the bands in Pt are much more dispersive in the considered energy window, which results in an effective reduction of the regions in (E, k)-space which contribute to the spin and orbital response alike. This is in contrast to Hf, where much flatter bands reside above and below the Fermi energy within the energy window of the laser pulse, providing significant integrated, albeit very small locally, contributions. Moreover, in fcc Pt the Fermi energy cuts through the band edges of the d-states, where the effect of spin-orbit interaction is the strongest, which explains the emergence of strong hotspot-like contributions with a clear correlation in the magnitude of spin and orbital response and the resultant similar behavior of δL<sup>z</sup> and δS<sup>z</sup> with band filling around the Fermi energy. We further scrutinize the reciprocal space distribution of δL<sup>z</sup> and δSz, shown for hcp Hf and fcc Pt in Fig. [3\(](#page-3-0)e-h). For both materials δL<sup>z</sup> distributions consist of large uniform areas of either positive or negative sign. On the contrary, δS<sup>z</sup> distributions are much finer and richer in details with more areas of opposite sign, consistent with the picture we drew above from the band-resolved analysis. A similar behavior of the light-induced magnetism in reciprocal space has been recently reported for rutile altermagnets [\[39\]](#page-7-18). We also observe that, as discussed above, while for Hf the contributions are well spread throughout the Brillouin zone, for Pt the spin and orbital contributions are located at the edges of the considered k<sup>z</sup> − k<sup>y</sup> plane. Overall, this fact indicates that the microscopic behavior of light-induced magnetism varies strongly among transition metals and crucially depends on the crystal structure and position of the Fermi level with respect to the states ![<span id="page-5-0"></span>FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ### Anisotropy of light-induced magnetism We first address the anisotropy of light-induced magnetism with respect to the magnetization by examining the response of magnetic elements under different flavors of circular polarization. In Fig. [4\(](#page-4-0)a-b) we present the computed δL<sup>z</sup> and δS<sup>z</sup> in relation to scattering lifetime Γ, for the cases of magnetic bcc Fe and hcp Co under excitation by light which is circularly polarized in the xy or yz planes at hω̵ = 1.55 eV. As we have already seen in Fig. [1](#page-1-0) for light polarized in the xyplane, the responses behave differently for right (solid lines) or left (dashed lines) polarization. This difference is more pronounced for δLz, for which we even observe a change of sign for Co. Surprisingly, the situation is drastically different for the case of yz-polarization. In this case, for bcc Fe the response is even in the helicity, while for hcp Co δL<sup>z</sup> is almost even and δS<sup>z</sup> is perfectly even in the helicity. Such different behavior with respect to the light helicity between the two magnetic elements can be traced back to the additional anisotropy originating in the crystal structure itself − an effect which we discuss below. Notably, we witness a highly nonlinear behavior with respect to Γ, with the response reaching colossal values in the clean limit. For example, δL<sup>z</sup> and δS<sup>z</sup> can reach as much as 40 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> and 70 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> in Co, respectively, given the scattering lifetime of 1 meV. Besides the potential tunability of the inverse Faraday effect by the degree of the disorder of the samples, one key message that we extract from our observations is that when comparing the values for laser-induced magnetic moments obtained with different methods, special care has to be taken since the implementations of disorder effects, even within a simple constant broadening model, may differ among various approaches. Finally, we analyze the anisotropy that the crystal structure induces in the light-induced magnetism. We select the case of hcp Re and present in Fig. [5\(](#page-5-0)a-b) the x and z components of δL and δS, respectively, in relation to Γ. Due to the inequivalence of x and z axes in the hcp structure, the δL<sup>z</sup> which originates in light circularly polarized in the xy-plane (light blue line) differs from the δL<sup>x</sup> for the case of the yz polarization (brown line). The situation is similar for the case of δS<sup>z</sup> and δSx. On the contrary, as presented in Fig. [11](#page-14-1) of the SM, in the case of fcc Pt we have a perfect match between δL<sup>z</sup> and δL<sup>x</sup> (or δS<sup>z</sup> and δSx) when changing the plane of circular polarization because the x and z axes are equivalent in the fcc structure. The situation is similar for the y component when light is circularly polarized in the xz-plane. As also expected from the symmetry of the crystal structure, we confirm that the light-induced moments are perfectly odd in the helicity for the nonmagnetic elements, which is not the case for the magnetic elements, see Figs. [5](#page-5-0) and [11](#page-14-1) of the SM [\[20\]](#page-7-5). ### Discussion The main goal of our work is to showcase the importance of the orbital component of IFE and its distinct behavior from the spin counterpart. While so far it was mainly the interaction of light with the spin magnetization that was taken into consideration for the interpretation of IFE-related effects, we speculate that the orbital IFE may provide a novel way to coherently induce magnetization and manipulate the magnetic order. For example, in a recent study, different types of optical torques that may arise in ferromagnetic layers were interpreted in terms of the light-induced orbital moment and its interaction with the magnetization through the spin-orbit interaction [\[64\]](#page-8-10). Since it is known that current-induced orbital accumulation and orbital torques exhibit a long-range behavior [\[49,](#page-8-0) [65](#page-8-11)[–67\]](#page-8-12) due to a characteristic small orbital decay, a question arises whether similar behavior can be exhibited by optical torques caused by the orbital IFE. The emergence of long-ranged orbital IFE would come as no surprise given the fact that several recent experiments reported that laser excitation can drive long-range ballistic orbital currents resulting in THz emission [\[68](#page-8-13)[–71\]](#page-8-14). a perturbation by the electric field of the original ground state Hamiltonian, it is common to utilize the angular momentum of light in order to understand the interaction with the magnetic order by the means of transfer of angular momentum. The spin of light through the helicity of circularly polarized pulses plays a crucial role in helicity-dependent all-optical switching scenarios. On the other hand, there is a strong recent interest in utilizing the orbital angular momentum of light via irradiation of matter with e.g. vortex beams or twisted light, in order to probe the magnetization [\[72\]](#page-8-15), generate photocurrents [\[73\]](#page-8-16), drive ultrafast demagnetization [\[74\]](#page-8-17), and induce IFE [\[40,](#page-7-24) [75\]](#page-8-18). Therefore, it is imperative to treat both spin and orbital degrees of freedom on equal footing when exploring the lightmatter interaction. This will not only trigger further advances in the field of ultrafast magnetism and THz spintronics, but also enable a transition to the novel field of attosecond spintronics [\[76,](#page-8-19) [77\]](#page-8-20). Method. In this work we calculate the first-principles electronic structures of 3d, 4d and 5d transition metals by using the full-potential linearized augmented plane wave FLEUR code [\[78\]](#page-8-21). We describe exchange and correlation effects by using the non-relativistic PBE [\[79\]](#page-8-22) functional, while relativistic effects are described by the second-variation scheme [\[80\]](#page-9-0). The parameters of our first-principles calculations, i.e. lattice constants, muffin-tin radii, plane-wave cutoffs, etc. are taken from Table I of Ref. [\[81\]](#page-9-1). Next, we construct maximally-localized Wannier functions (MLWFs) by employing the Wannier90 code [\[82\]](#page-9-2) and its interface with the FLEUR code [\[83\]](#page-9-3). Similarly to [\[81\]](#page-9-1), we choose s, p and d orbitals for the initial projections and disentagle 18 MLWFs out of 36 Bloch states within a frozen window of 5.0 eV above the Fermi energy for each atom in the unit cell. where O<sup>i</sup> is the i-th component of either the orbital angular momentum operator L<sup>i</sup> or of the spin operator S<sup>i</sup> . Moreover, a<sup>0</sup> = 4πϵ0h̵<sup>2</sup> /(mee 2 ) is the Bohr's radius, I = ϵ0cE<sup>2</sup> 0 /2 is the intensity of the pulse, ϵ<sup>0</sup> is the vacuum permittivity, m<sup>e</sup> is the electron mass, e is the elementary charge, h̵ is the reduced Planck constant, c is the light velocity, E<sup>H</sup> = e 2 /(4πϵ0a0) is the Hartree energy, and ϵ<sup>j</sup> is the j-th component of the polarization vector of the pulse. For example, we describe right/left-handedly polarized light in the xy-plane as ϵ = (1,±i, <sup>0</sup>)/<sup>√</sup> 2, respectively, and define its propagation vector to lie along the normal to the polarization plane. A detailed form of the tensor φijk can be seen in Eq.(14) of Ref. [\[21\]](#page-7-9). For the orbital response, the prefactor in Eq. [$1$](#page-6-2) must be multiplied by an additional factor of 2. A 128×128×128 interpolation k-mesh is sufficient to obtain well-converged results. In all calculations the lifetime broadening Γ was set at 25 meV, the light frequency hω̵ at 0.25 eV and 1.55 eV, the intensity of light at 10 GW/cm<sup>2</sup> , and we covered an energy region of [−2.5, 2.5] eV around the Fermi energy level E<sup>F</sup> . Acknowledgements. We thank Frank Freimuth and Maximilian Merte for discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR 173/3 − 268565370 (project A11) and by the K. and A. Wallenberg Foundation (Grants No. 2022.0079 and 2023.0336). We acknowledge support from the EIC Pathfinder OPEN grant 101129641 "OBELIX". We also gratefully acknowledge the Julich Supercomputing Cen- ¨ tre and RWTH Aachen University for providing computational resources under projects jiff40 and jara0062. Author Contributions. T. A. performed numerical calculations and analysed the results. T. A. and Y. M. wrote the manuscript. 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(a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-11-0"></span>FIG. 7. (a-d) Light-induced orbital δL<sup>x</sup> (a, c) and spin δS<sup>x</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-12-0"></span>FIG. 8. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-13-0"></span>FIG. 9. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ![<span id="page-13-1"></span>FIG. 10. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-14-1"></span>FIG. 11. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic fcc Pt. The induced magnetic moments along the x-axis and along the z-axis are presented. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) <span id="page-14-0"></span>TABLE II. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 0.25 eV. <span id="page-15-0"></span>TABLE III. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV.
Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (C<sup>p</sup>); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, C<sup>p</sup>=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52–54]) and two (C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53,54]) components, respectively.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd³ phase.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup>, AlII, AlIII, and AlIV.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd<sup>3</sup>. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between NdI and its surrounding Al atoms, along the bonds between various Al atoms around NdI , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e3 and f1-f3 are the same, respectively. For a better comparison, the lengths of the bonds are normalized.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 7. Local chemical environment (LCE) in α-Al11Nd³. (a1 a2) LCEs of Ndᴵ and Ndᴵᴵ. (b1-b4) LCEs of Alᴵ to Alᴵᴵ. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Ndⁱ. (a1−a3) The bonds between Ndⁱ and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Ndⁱ. In both a and b, from left to right, the bond length increases gradually.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage
Figure 3: 1D diffraction scans showing that the difference in monolayer-MoS<sup>2</sup> structure when placed directly onto SiO<sup>2</sup> (gray) versus when a few-layer hBN buffer is used between the monolayer and SiO<sup>2</sup> (red). Bulk MoS<sup>2</sup> and direct measurement of SiO<sup>2</sup> are included as references for known ordered/disordered scattering, respectively. Figure reproduced with permission from Radi´c et al. <sup>6</sup>	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage can leverage the spatial resolution and exclusive surface sensitivity of the technique to investigate inter-layer coupling strength in otherwise difficult to measure systems such as van Der Waals heterostructures where comparable techniques like LEED/M, PL or Raman struggle due to the inherent transmission of the probe through the sample. #### Thermal expansion coefficient Structural information, such as the lattice constant, can be accurately measured using 2D diffraction scans, found to be within 1%. <sup>1</sup> 2D diffraction scans offer increased accuracy over 1D scans, like those in figure 2, because more diffraction provide a statistical advantage. Initially, we measure the 2D diffraction scan on the surface of bulk MoS2, shown in figure S1, and determine the lattice constant to be 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A, matching literature values within experimental uncertainty. We extend the method to mechanically exfoliated monolayer-MoS2/hBN/SiO2, with diffraction pattern shown in figure 4, to showcase the exclusive surface sensitivity of the technique and determine that the lattice constant is <sup>3</sup>.14±0.07˚A, which is within experimental error ![Figure 4: A 2D diffraction pattern taken on monolayer-MoS2/hBN/SiO2, with the identified centres of the diffraction peaks shown as red dots. All of the lattice spacings between adjacent peaks were measured and averaged give a lattice constant 3.<sup>14</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) The thermal expansion coefficient (TEC) of 2D materials is crucial to understanding their behaviour for successful integration into optoelectronic devices, these materials also provide direct measurement of the anharmonic vibrational modes of low-dimensional systems, 7,8 however, difficult to measure due to their optical transparency. 9,10 Using helium atom microdiffraction one can measure the lattice constant, and therefore TEC, of microscopic optically transparent materials, whether supported by a substrate or free-standing. The diffraction pattern was measured along the principle ⟨10⟩ azimuth of MoS<sup>2</sup> at temperatures from 60 −450 ◦C (the data is presented in figure 4), as with work on HAS no change in the lattice constant was observed, allowing us to put an upper bound on the expansion coefficient of 16 × 10−<sup>6</sup> K−<sup>1</sup> . Our measurement is consistent with the upper bound determined as α < 14 × 10−<sup>6</sup> K−<sup>1</sup> reported by Anemone et al.,<sup>11</sup> and α = 7.6×10−<sup>6</sup> K−<sup>1</sup> measured by Zhang et al. using micro-Raman spectroscopy.<sup>10</sup> Our method is currently limited in angular resolution (radial dimension of figure 4) by the size of the limiting aperture that lies between the sample and detector. The current configuration uses a nominal 0.5 mm diameter hole for an in-plane angular resolution ∼ 7.9 ◦ , representing a compromise between measurement time and helium signal. By changing the limiting aperture to a rectangle with dimensions 0.075 × 0.5 mm we can integrate the signal in the axis perpendicular to any change in lattice parameter, the radial direction in figure 4, while improving in-plane angular resolution. We estimate that moving from the current circular aperture to the proposed rectangular aperture decreases helium signal by a factor of 5, but improves in-plane angular resolution by a factor of 6.7 from 7.9 ◦ → 1.2 ◦ which would be sufficient to resolve the TEC over the current temperature range. Details of the current instrument's spatial and angular resolutions, alongside their definitions, can be found in section 'Experimental Methods - Helium atom micro-diffraction instrument details'. ### Electron-phonon coupling Electron-phonon coupling is a major decoherence mechanism in semiconductors, often causing electron-phonon scattering, and eventually energy dissipation, in turn harming optoelectronic device performance. Optimisation of electron-phonon coupling is therefore critical in device design, both in cases where it must be minimised, or maximised. Electron-phonon coupling is typically measured using inelastic optical techniques, such as Raman or Brillouin scattering, and are primarily limited to optically transparent materials and access to electronic coupling to optical phonons only. It has been shown that helium atom scattering is an effective tool for determining the coupling strength between 2D materials and their substrates via Debye-Waller attenuation of helium scattering.12–15 However, previous atom scattering techniques have been limited to millimetre-scale spot sizes which made devicescale samples inaccessible. Here we introduce helium atom micro-diffraction as a technique well suited to the measurement of electronphonon coupling in a range of materials where its non-damaging, surface sensitive and microscopic neutral probe can be used to access electronic couplings to both optical and acoustic phonon modes in 2D materials. The temperature dependence of reflected and diffracted helium intensities can be modelled by Debye-Waller attenuation, 12,13 which describes the increasing motion of the surface atoms with temperature and causes an increase in inelastic scattering. Thus, the ordered scattering intensity is reduced. The attenuation is described by where 2W is the D-W factor. As the helium scattering occurs from the valence electron density rather than from the ionic cores themselves, one can link the D-W factor to electron phonon coupling, as described by Al Taleb et al. <sup>13</sup> D-W attenuation has been measured previously for macro-scale samples for both LiF and bulk MoS2, however without the spatial resolution enabled by SHeM, monolayer MoS<sup>2</sup> flakes could not be measured.<sup>13</sup> ![Figure 5: Log diffracted intensity as a function of temperature for monolayer MoS<sup>2</sup> on few-layer hBN on glass. In theory all diffraction orders should yield the same D-W, and therefore electron-phonon coupling, constants but the signal-to-noise ratios of each peak are a function of relative peak heights and instrument geometry, resulting in small differences.](path) From figure 5 we can extract the D-W factor as the straight-line gradient, and from that the electron-phonon coupling constant λ, contained in table 1 for monolayer and bulk MoS2. The exponents are compared relative to each other, and to literature values for the bulk case, and found to be in agreement with the expected behaviour with parallel momentum transfer (∆K). Table 1: Debye-Waller factors and electronphonon coupling constants extracted from the data presented in figure 5 using the model in equation 1 and theoretical relations outlined by Anemone et al.,<sup>13</sup> respectively. We report the electron-phonon coupling constant in bulk MoS<sup>2</sup> as λbulk ≈ 0.51 × 10−<sup>3</sup> K−<sup>1</sup> , showing good agreement to the literature values λ ≈ 0.41 × 10−<sup>3</sup> K−<sup>1</sup> , 0.49 × 10−<sup>3</sup> K−<sup>1</sup> measured using a typical helium atom scattering instrument with few-millimetre spot size, reported by Anemone et al. <sup>13</sup> We also find λML ≈ 0.40 × 10−<sup>3</sup> K−<sup>1</sup> for ML-MoS<sup>2</sup> on fewlayer hBN. We find the electron-phonon coupling constant is ∼ 20 % smaller in the monolayer compared to bulk MoS2. By leveraging the current best reported ≈ 350 nm spot size, the method can be extended to perform spatial mapping of the Debye-Waller factor, and therefore electron-phonon coupling constant, without sample preparation or damage in thin films and delicate materials, a class of materials typically difficult to characterise using standard optical and electron beam techniques. #### Vacancy-type defect density Precise control of defect density in semiconductors is instrumental for both current and future semiconductor device development. In particular, the optoelectronic properties of two-dimensional TMD semiconductors such as MoS2, can be tuned using single-atom defects. <sup>16</sup> Applications of these materials includes catalysis<sup>17</sup> and a plethora of devices, 18–21 all of which can affected by the material's defect density. In many applications a balance must be made between a sufficiently high number of defects for the material to acquire the desired properties while the material remaining sufficiently ordered to not degrade electronic performance, <sup>22</sup> or even entirely degrade the lattice structure. However, quantification of defect densities in 2D materials remains a significant experimental challenge, where typically used methods are XPS<sup>22</sup> and STEM, with conductive AFM (CAFM) being explored recently. <sup>23</sup> As all of these methods commonly require complicated sample preparation processes, there is a characterisation shortcoming that is only going to grow more acute as devices using 2D materials start being produced on an industrial scale, and therefore enter the commercial sphere. In this work we demonstrate how helium atom micro-diffraction can be used to characterise the vacancy-type defect density on the surface of few-layer materials using data reproduced from Radi´c et al. with permission. <sup>6</sup> Three mechanically exfoliated monolayer flakes of MoS<sup>2</sup> with increasing defect densities, ranging from the intrinsic 0.1 × 10<sup>14</sup> cm−<sup>2</sup> to 1.8 × 10<sup>14</sup> cm−<sup>2</sup> , and ≈ 15 µm lateral size were produced using high-temperature annealing under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere as outlined by Zhu et al.<sup>22</sup> Exact annealing parameters, and the resulting defect densities inferred using photoluminescence spectroscopy, which has been calibrated against stoichiometric beam-line XPS, are contained in Table 2. Figure 6 shows that diffracted helium intensity decreases approximately linearly as a function of defect density within the single-vacancy limit. Detailed analysis and discussion of the results in figure 6 can be found in the full manuscript.<sup>6</sup> ![Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS<sup>2</sup> flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>](path) tion can be employed as a lab-based method to quantify point-defect density without specific sample preparation or damage. The method is agnostic to sample chemistry, and thickness, because the mechanism through which it measures defect density is purely geometric, relying upon the degree of order of the sample surface, meaning that the method can be trivially extended to any system whose macroscopic properties are mediated by surface defects or dopants. Examples of current systems include hBN,<sup>24</sup> graphene,<sup>25</sup> doped systems such as diamond,<sup>26</sup> alongside other TMDs. In conjunction with its electrostatically neutral and low energy beam, microscopic helium atom diffraction presents itself as an ideal technique for the investigation of these few-layer materials. Further improvements in the instruments' lateral resolution resolution while performing diffraction measurements, down to the previously reported ∼ 300 nm spot sizes,<sup>2</sup> would allow for sub-micron scale mapping of defect density across samples. The technique is not currently capable of acting as a standalone, stoichimetric measure of defect density, like XPS, once an accurate helium-defect scattering crosssection is calculated this will also be possible. ## Conclusion The findings reported here have demonstrated the effectiveness of helium atom microdiffraction as a powerful, and completely noninvasive characterisation tool for 2D materials. The technique's ability to probe only the outermost atomic layers with micron spatial resolution makes it uniquely suited to addressing the challenges of analysing 2D materials, where established techniques often struggle. The results and discussion have highlighted several key applications of helium atom microdiffraction, including the measurement of surface cleanliness and contamination without altering the sample, a critical feature for ensuring the integrity of materials used in device fabrication. We also investigated the impact of substrate on monolayer properties, confirming that the use of a few-layer hBN buffer can preserve the structural integrity of monolayer MoS2, whereas direct contact with SiO2 results in significant disorder.? We have also explored the potential of helium atom micro-diffraction for determining the thermal expansion coefficient and electronphonon coupling in monolayer MoS2, and have demonstrated its capability for precise structural and thermal analysis. The technique's ability to quantify defect densities without sample preparation or damage also highlights it as an obvious tool for optimizing 2D materials in ever expanding areas of application, from optoelectronics to catalysis and more. Future enhancements to helium atom microdiffraction, such as achieving sub-50 nm spatial resolution, and incorporating out-of-plane scattering capabilities, will further expand its applicability, particularly for complex heterostructures and device-grade materials. In summary, helium atom micro-diffraction promises to provide a robust and adaptable platform for characterisation and development of 2D materials, paving the way for advancements in nanotechnology and material science. ## Experimental Methods ## Helium atom micro-diffraction instrument details In the presented work we used a Scanning Helium Microscope (SHeM) with 5 µm spatial and 7.9 ◦ in-plane angular resolution at the specular condition, representing a factor of 2 improvement in spatial resolution over the first iteration published by von Jeinsen et al. <sup>1</sup> Spatial resolution is defined as the full-width at halfmaximum of the beam spot on the sample at the designed working distance. In-plane angular resolution is given by the angle subtended by the limiting detector aperture and beam spot on the sample, with the correcting factor <sup>√</sup> 1 2 applied to account for the 45◦ instrument geometry. ## Sample details Monolayers of MoS<sup>2</sup> were produced by mechanical exfoliation of a bulk MoS<sup>2</sup> crystal, purchased from 2D Semiconductors Ltd., and were deposited onto few-layer thick hexgonal boron nitride (hBN) which in turn lies on a SiO<sup>2</sup> substrate (high-precision glass microscope slide). Each monolayer measured ≈ 15 µm laterally. The increased defect density monolayer MoS<sup>2</sup> samples, presented in section , were produced the same as previously described in this section with the addition of a thermal annealing process under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere for 0.5 h, following the method as described by Zhu et al.<sup>22</sup> Varying annealing temperature and time allows for control of defect density. Exact annealing parameters, with final defect densities characterised by stoichiometric XPS, are shown in Table 2. # Acknowledgements The work was supported by EPSRC grant EP/R008272/1, Innovate UK/Ionoptika Ltd. through Knowledge Transfer Partnership 10000925. The work was performed in part at CORDE, the Collaborative R&D Environment established to provide access to physics related facilities at the Cavendish Laboratory, University of Cambridge and EPSRC award EP/T00634X/1. SML acknowledges support from EPSRC grant EP/X525686/1. # References - 1. von Jeinsen, N. A.; Lambrick, S. M.; Bergin, M.; Radi´c, A.; Liu, B.; Seremet, D.; Jardine, A. P.; Ward, D. J. 2D Helium Atom Diffraction from a Microscopic Spot. Physical Review Letters 2023, 131, 236202, Publisher: American Physical Society. - 5. Man, M. K. L.; Deckoff-Jones, S.; Winchester, A.; Shi, G.; Gupta, G.; Mohite, A. D.; Kar, S.; Kioupakis, E.; Talapatra, S.; Dani, K. M. Protecting the properties of monolayer MoS2 on silicon based substrates with an atomically thin buffer. Scientific Reports 2016, 6, 20890, Number: 1 Publisher: Nature Publishing Group. - 6. Radi´c, A.; von Jeinsen, N.; Wang, K.; Zhu, Y.; Sami, I.; Perez, V.; Ward, D.; Jardine, A.; Chhowwalla, M.; Lambrick, S. Defect density quantification in monolayer MoS<sup>2</sup> using helium atom micro-diffraction. arXiv 2024, - 7. Late, D. J.; Shirodkar, S. N.; Waghmare, U. V.; Dravid, V. P.; Rao, C. N. R. Thermal Expansion, Anharmonicity and Temperature-Dependent Raman Spectra of Single- and Few-Layer MoSe2 and WSe2. ChemPhysChem 2014, 15, 1592–1598. - 10. Zhang, L.; Lu, Z.; Song, Y.; Zhao, L.; Bhatia, B.; Bagnall, K. R.; Wang, E. N. Thermal Expansion Coefficient of Monolayer Molybdenum Disulfide Using Micro-Raman Spectroscopy. Nano Letters 2019, 19, 4745–4751. - 11. Anemone, G.; Taleb, A. A.; Politano, A.; Kuo, C.-N.; Lue, C. S.; Miranda, R.; Far´ıas, D. Setting the limit for the lateral thermal expansion of layered crystals via helium atom scattering. Physical Chemistry Chemical Physics 2022, 24, 13229–13233, Publisher: The Royal Society of Chemistry. - 14. Anemone, G.; Taleb, A. A.; Benedek, G.; Castellanos-Gomez, A.; Far´ıas, D. Electron–Phonon Coupling Constant of 2H-MoS2(0001) from Helium-Atom Scattering. The Journal of Physical Chemistry C 2019, 123, 3682–3686, Publisher: American Chemical Society. - 15. Anemone, G.; Casado Aguilar, P.; Garnica, M.; Calleja, F.; Al Taleb, A.; Kuo, C.- N.; Lue, C. S.; Politano, A.; V´azquez de Parga, A. L.; Benedek, G.; Far´ıas, D.; Miranda, R. Electron–phonon coupling in superconducting 1T-PdTe2. npj 2D Materials and Applications 2021, 5, 1–7, Number: 1 Publisher: Nature Publishing Group. - 16. Regan, E. C.; Wang, D.; Paik, E. Y.; Zeng, Y.; Zhang, L.; Zhu, J.; MacDonald, A. H.; Deng, H.; Wang, F. Emerging exciton physics in transition metal dichalcogenide heterobilayers. 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Nature Materials 2024, # Helium atom micro-diffraction as a characterisation tool for 2D materials - supporting information # Bulk MoS<sup>2</sup> diffraction pattern ![Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure 5: Log diffracted intensity as a function of temperature for monolayer MoS<sup>2</sup> on few-layer hBN on glass. In theory all diffraction orders should yield the same D-W, and therefore electron-phonon coupling, constants but the signal-to-noise ratios of each peak are a function of relative peak heights and instrument geometry, resulting in small differences.	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage From figure 5 we can extract the D-W factor as the straight-line gradient, and from that the electron-phonon coupling constant λ, contained in table 1 for monolayer and bulk MoS2. The exponents are compared relative to each other, and to literature values for the bulk case, and found to be in agreement with the expected behaviour with parallel momentum transfer (∆K). Table 1: Debye-Waller factors and electronphonon coupling constants extracted from the data presented in figure 5 using the model in equation 1 and theoretical relations outlined by Anemone et al.,<sup>13</sup> respectively. We report the electron-phonon coupling constant in bulk MoS<sup>2</sup> as λbulk ≈ 0.51 × 10−<sup>3</sup> K−<sup>1</sup> , showing good agreement to the literature values λ ≈ 0.41 × 10−<sup>3</sup> K−<sup>1</sup> , 0.49 × 10−<sup>3</sup> K−<sup>1</sup> measured using a typical helium atom scattering instrument with few-millimetre spot size, reported by Anemone et al. <sup>13</sup> We also find λML ≈ 0.40 × 10−<sup>3</sup> K−<sup>1</sup> for ML-MoS<sup>2</sup> on fewlayer hBN. We find the electron-phonon coupling constant is ∼ 20 % smaller in the monolayer compared to bulk MoS2. By leveraging the current best reported ≈ 350 nm spot size, the method can be extended to perform spatial mapping of the Debye-Waller factor, and therefore electron-phonon coupling constant, without sample preparation or damage in thin films and delicate materials, a class of materials typically difficult to characterise using standard optical and electron beam techniques. #### Vacancy-type defect density Precise control of defect density in semiconductors is instrumental for both current and future semiconductor device development. In particular, the optoelectronic properties of two-dimensional TMD semiconductors such as MoS2, can be tuned using single-atom defects. <sup>16</sup> Applications of these materials includes catalysis<sup>17</sup> and a plethora of devices, 18–21 all of which can affected by the material's defect density. In many applications a balance must be made between a sufficiently high number of defects for the material to acquire the desired properties while the material remaining sufficiently ordered to not degrade electronic performance, <sup>22</sup> or even entirely degrade the lattice structure. However, quantification of defect densities in 2D materials remains a significant experimental challenge, where typically used methods are XPS<sup>22</sup> and STEM, with conductive AFM (CAFM) being explored recently. <sup>23</sup> As all of these methods commonly require complicated sample preparation processes, there is a characterisation shortcoming that is only going to grow more acute as devices using 2D materials start being produced on an industrial scale, and therefore enter the commercial sphere. In this work we demonstrate how helium atom micro-diffraction can be used to characterise the vacancy-type defect density on the surface of few-layer materials using data reproduced from Radi´c et al. with permission. <sup>6</sup> Three mechanically exfoliated monolayer flakes of MoS<sup>2</sup> with increasing defect densities, ranging from the intrinsic 0.1 × 10<sup>14</sup> cm−<sup>2</sup> to 1.8 × 10<sup>14</sup> cm−<sup>2</sup> , and ≈ 15 µm lateral size were produced using high-temperature annealing under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere as outlined by Zhu et al.<sup>22</sup> Exact annealing parameters, and the resulting defect densities inferred using photoluminescence spectroscopy, which has been calibrated against stoichiometric beam-line XPS, are contained in Table 2. Figure 6 shows that diffracted helium intensity decreases approximately linearly as a function of defect density within the single-vacancy limit. Detailed analysis and discussion of the results in figure 6 can be found in the full manuscript.<sup>6</sup> ![Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS<sup>2</sup> flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>](path) tion can be employed as a lab-based method to quantify point-defect density without specific sample preparation or damage. The method is agnostic to sample chemistry, and thickness, because the mechanism through which it measures defect density is purely geometric, relying upon the degree of order of the sample surface, meaning that the method can be trivially extended to any system whose macroscopic properties are mediated by surface defects or dopants. Examples of current systems include hBN,<sup>24</sup> graphene,<sup>25</sup> doped systems such as diamond,<sup>26</sup> alongside other TMDs. In conjunction with its electrostatically neutral and low energy beam, microscopic helium atom diffraction presents itself as an ideal technique for the investigation of these few-layer materials. Further improvements in the instruments' lateral resolution resolution while performing diffraction measurements, down to the previously reported ∼ 300 nm spot sizes,<sup>2</sup> would allow for sub-micron scale mapping of defect density across samples. The technique is not currently capable of acting as a standalone, stoichimetric measure of defect density, like XPS, once an accurate helium-defect scattering crosssection is calculated this will also be possible. ## Conclusion The findings reported here have demonstrated the effectiveness of helium atom microdiffraction as a powerful, and completely noninvasive characterisation tool for 2D materials. The technique's ability to probe only the outermost atomic layers with micron spatial resolution makes it uniquely suited to addressing the challenges of analysing 2D materials, where established techniques often struggle. The results and discussion have highlighted several key applications of helium atom microdiffraction, including the measurement of surface cleanliness and contamination without altering the sample, a critical feature for ensuring the integrity of materials used in device fabrication. We also investigated the impact of substrate on monolayer properties, confirming that the use of a few-layer hBN buffer can preserve the structural integrity of monolayer MoS2, whereas direct contact with SiO2 results in significant disorder.? We have also explored the potential of helium atom micro-diffraction for determining the thermal expansion coefficient and electronphonon coupling in monolayer MoS2, and have demonstrated its capability for precise structural and thermal analysis. The technique's ability to quantify defect densities without sample preparation or damage also highlights it as an obvious tool for optimizing 2D materials in ever expanding areas of application, from optoelectronics to catalysis and more. Future enhancements to helium atom microdiffraction, such as achieving sub-50 nm spatial resolution, and incorporating out-of-plane scattering capabilities, will further expand its applicability, particularly for complex heterostructures and device-grade materials. In summary, helium atom micro-diffraction promises to provide a robust and adaptable platform for characterisation and development of 2D materials, paving the way for advancements in nanotechnology and material science. ## Experimental Methods ## Helium atom micro-diffraction instrument details In the presented work we used a Scanning Helium Microscope (SHeM) with 5 µm spatial and 7.9 ◦ in-plane angular resolution at the specular condition, representing a factor of 2 improvement in spatial resolution over the first iteration published by von Jeinsen et al. <sup>1</sup> Spatial resolution is defined as the full-width at halfmaximum of the beam spot on the sample at the designed working distance. In-plane angular resolution is given by the angle subtended by the limiting detector aperture and beam spot on the sample, with the correcting factor <sup>√</sup> 1 2 applied to account for the 45◦ instrument geometry. ## Sample details Monolayers of MoS<sup>2</sup> were produced by mechanical exfoliation of a bulk MoS<sup>2</sup> crystal, purchased from 2D Semiconductors Ltd., and were deposited onto few-layer thick hexgonal boron nitride (hBN) which in turn lies on a SiO<sup>2</sup> substrate (high-precision glass microscope slide). Each monolayer measured ≈ 15 µm laterally. The increased defect density monolayer MoS<sup>2</sup> samples, presented in section , were produced the same as previously described in this section with the addition of a thermal annealing process under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere for 0.5 h, following the method as described by Zhu et al.<sup>22</sup> Varying annealing temperature and time allows for control of defect density. Exact annealing parameters, with final defect densities characterised by stoichiometric XPS, are shown in Table 2. # Acknowledgements The work was supported by EPSRC grant EP/R008272/1, Innovate UK/Ionoptika Ltd. through Knowledge Transfer Partnership 10000925. The work was performed in part at CORDE, the Collaborative R&D Environment established to provide access to physics related facilities at the Cavendish Laboratory, University of Cambridge and EPSRC award EP/T00634X/1. SML acknowledges support from EPSRC grant EP/X525686/1. # References - 1. von Jeinsen, N. A.; Lambrick, S. M.; Bergin, M.; Radi´c, A.; Liu, B.; Seremet, D.; Jardine, A. P.; Ward, D. J. 2D Helium Atom Diffraction from a Microscopic Spot. Physical Review Letters 2023, 131, 236202, Publisher: American Physical Society. - 5. Man, M. K. L.; Deckoff-Jones, S.; Winchester, A.; Shi, G.; Gupta, G.; Mohite, A. D.; Kar, S.; Kioupakis, E.; Talapatra, S.; Dani, K. M. Protecting the properties of monolayer MoS2 on silicon based substrates with an atomically thin buffer. Scientific Reports 2016, 6, 20890, Number: 1 Publisher: Nature Publishing Group. - 6. Radi´c, A.; von Jeinsen, N.; Wang, K.; Zhu, Y.; Sami, I.; Perez, V.; Ward, D.; Jardine, A.; Chhowwalla, M.; Lambrick, S. Defect density quantification in monolayer MoS<sup>2</sup> using helium atom micro-diffraction. arXiv 2024, - 7. Late, D. J.; Shirodkar, S. N.; Waghmare, U. V.; Dravid, V. P.; Rao, C. N. R. Thermal Expansion, Anharmonicity and Temperature-Dependent Raman Spectra of Single- and Few-Layer MoSe2 and WSe2. ChemPhysChem 2014, 15, 1592–1598. - 10. Zhang, L.; Lu, Z.; Song, Y.; Zhao, L.; Bhatia, B.; Bagnall, K. R.; Wang, E. N. Thermal Expansion Coefficient of Monolayer Molybdenum Disulfide Using Micro-Raman Spectroscopy. Nano Letters 2019, 19, 4745–4751. - 11. Anemone, G.; Taleb, A. A.; Politano, A.; Kuo, C.-N.; Lue, C. S.; Miranda, R.; Far´ıas, D. Setting the limit for the lateral thermal expansion of layered crystals via helium atom scattering. Physical Chemistry Chemical Physics 2022, 24, 13229–13233, Publisher: The Royal Society of Chemistry. - 14. Anemone, G.; Taleb, A. A.; Benedek, G.; Castellanos-Gomez, A.; Far´ıas, D. Electron–Phonon Coupling Constant of 2H-MoS2(0001) from Helium-Atom Scattering. The Journal of Physical Chemistry C 2019, 123, 3682–3686, Publisher: American Chemical Society. - 15. Anemone, G.; Casado Aguilar, P.; Garnica, M.; Calleja, F.; Al Taleb, A.; Kuo, C.- N.; Lue, C. S.; Politano, A.; V´azquez de Parga, A. L.; Benedek, G.; Far´ıas, D.; Miranda, R. Electron–phonon coupling in superconducting 1T-PdTe2. npj 2D Materials and Applications 2021, 5, 1–7, Number: 1 Publisher: Nature Publishing Group. - 16. Regan, E. C.; Wang, D.; Paik, E. Y.; Zeng, Y.; Zhang, L.; Zhu, J.; MacDonald, A. H.; Deng, H.; Wang, F. Emerging exciton physics in transition metal dichalcogenide heterobilayers. Nature Reviews Materials 2022, 7, 778–795. - 17. Yang, J.; Wang, Y.; Lagos, M. J.; Manichev, V.; Fullon, R.; Song, X.; Voiry, D.; Chakraborty, S.; Zhang, W.; Batson, P. E.; Feldman, L.; Gustafsson, T.; Chhowalla, M. Single Atomic Vacancy Catalysis. ACS Nano 2019, 13, 9958–9964, Publisher: American Chemical Society. - 18. Mitterreiter, E.; Schuler, B.; Micevic, A.; Hernang´omez-P´erez, D.; Barthelmi, K.; Cochrane, K. A.; Kiemle, J.; Sigger, F.; Klein, J.; Wong, E.; Barnard, E. S.; Watanabe, K.; Taniguchi, T.; Lorke, M.; Jahnke, F.; Finley, J. J.; Schwartzberg, A. M.; Qiu, D. Y.; Refaely-Abramson, S.; Holleitner, A. W. et al. The role of chalcogen vacancies for - 19. Barthelmi, K.; Klein, J.; H¨otger, A.; Sigl, L.; Sigger, F.; Mitterreiter, E.; Rey, S.; Gyger, S.; Lorke, M.; Florian, M.; Jahnke, F.; Taniguchi, T.; Watanabe, K.; Zwiller, V.; J¨ons, K. D.; Wurstbauer, U.; Kastl, C.; Weber-Bargioni, A.; Finley, J. J.; M¨uller, K. et al. Atomistic defects as singlephoton emitters in atomically thin MoS2. Applied Physics Letters 2020, 117 . - 22. Zhu, Y.; Lim, J.; Zhang, Z.; Wang, Y.; Sarkar, S.; Ramsden, H.; Li, Y.; Yan, H.; Phuyal, D.; Gauriot, N.; Rao, A.; Hoye, R. L. Z.; Eda, G.; Chhowalla, M. Room-Temperature Photoluminescence Mediated by Sulfur Vacancies in 2D Molybdenum Disulfide. ACS Nano 2023, 17, 13545– 13553, Publisher: American Chemical Society. - 23. Xu, K.; Holbrook, M.; Holtzman, L. N.; Pasupathy, A. N.; Barmak, K.; Hone, J. C.; Rosenberger, M. R. Validating the Use of Conductive Atomic Force Microscopy for Defect Quantification in 2D Materials. ACS Nano 2023, 17, 24743–24752, PMID: 38095969. - 24. Stern, H. L.; M. Gilardoni, C.; Gu, Q.; Eizagirre Barker, S.; Powell, O. F. J.; Deng, X.; Fraser, S. A.; Follet, L.; Li, C.; Ramsay, A. J.; Tan, H. H.; Aharonovich, I.; Atat¨ure, M. A quantum coherent spin in hexagonal boron nitride at ambient conditions. Nature Materials 2024, # Helium atom micro-diffraction as a characterisation tool for 2D materials - supporting information # Bulk MoS<sup>2</sup> diffraction pattern ![Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure 4: A 2D diffraction pattern taken on monolayer-MoS2/hBN/SiO2, with the identified centres of the diffraction peaks shown as red dots. All of the lattice spacings between adjacent peaks were measured and averaged give a lattice constant 3.14±0.07˚A.	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage ![Figure 1: Reflection mode optical (top-left) and real-space SHeM (bottom) images of bulk MoS2, hBN/SiO<sup>2</sup> and monolayer MoS2/hBN/SiO<sup>2</sup> (Shaded intersection). A 2D micro diffraction measurement (top-right) was taken to identify diffraction conditions for the real space imaging. An obvious change in contrast can be seen as the instrument is configured for](path) and species that one expects during standard device fabrication. Once under high-vacuum (2×10−<sup>8</sup> mbar) in the SHeM, a real-space image is taken of the as-prepared sample (Figure 2a) which demonstrates it is difficult to differentiate between MoS<sup>2</sup> and hBN, with the surrounding SiO<sup>2</sup> appearing less intense. Upon heating to 240 ◦C for 8 hours, another real-space image (figure 2b) reveals a stark difference in contrast between MoS2, hBN, and SiO2. The MoS<sup>2</sup> and hBN display significantly more intense scattering than the surrounding SiO<sup>2</sup> which can be quantitatively compared in Figure 2c. Having demonstrated the sensitivity of helium atom micro-diffraction to adsorbates, one can trivially extend the method to quantitatively map adsorbate densities across real-space images. By cleaning and measuring the sample with the presented procedure, one can identify the positions of diffraction peaks in reciprocal space (∆K). This is followed by acquiring a real-space image at a diffraction peak's maximum scattering condition in reciprocal space, achieved by translation of the sample in the zaxis. One can now dose the sample environment with a given contaminant and measure a realspace image to determine if there is a change adsorbate coverage. Through dosing, adsorbation behaviour of a given sample to varying species can also be investigated by monitoring a specific diffraction peak while dosing. ### Monolayer-substrate interactions It is well documented that the choice of substrate placed under a few-layer material has a significant effect on a range of material properties. One can measure the effect of the substrate on optoelectronic or structural properties of the sample via methods such as photoluminescence spectroscopy (PL), Raman, or low-energy electron microscopy/diffraction (LEEM/D), respectively.<sup>5</sup> To demonstrate the exclusive surface sensitivity of helium atom micro-diffraction, monolayer-MoS<sup>2</sup> was placed directly onto both an SiO<sup>2</sup> substrate and few-layer hBN which is in turn on SiO2, sample geometry is shown in figure 1. It has been demonstrated that certain atomically thin buffer layers, such as hBN, ![Figure 2: Real-space images and diffraction scans of monolayer-MoS2/hBN/SiO<sup>2</sup> (red), fewlayer hBN/SiO<sup>2</sup> (black) and SiO<sup>2</sup> (blue) before and after heating (left to right) to 240 ◦C for 8 hours to remove physisorbed species. For monolayer-MoS<sup>2</sup> and hBN we see ordered scattering emerging upon cleaning in contrast to the cosine-like scattering prior. SiO<sup>2</sup> remains cosine-like post-heating. Diffraction scans were acquired at the indicated points in Panels a,b. Real-space images were taken at a single ∆K, marked with a vertical line in Panel C.](path) LaAlO<sup>3</sup> and SrTiO3, can protect the optoelectronic properties of monolayers mounted on them from strongly interacting substrates like SiO2. <sup>5</sup> Figure 3 shows that the helium scattering from monolayer MoS<sup>2</sup> becomes almost entirely disordered when mounted directly on SiO<sup>2</sup> (gray), with faint signs of structure remaining at the expected ∆K diffraction peak positions. In contrast, placing the monolayer onto a few-layer hBN buffer protects its structure and produces diffraction that matches bulk MoS<sup>2</sup> to within experimental bounds, suggesting that their surface morphologies are the same. ![Figure 3: 1D diffraction scans showing that the difference in monolayer-MoS<sup>2</sup> structure when placed directly onto SiO<sup>2</sup> (gray) versus when a few-layer hBN buffer is used between the monolayer and SiO<sup>2</sup> (red). Bulk MoS<sup>2</sup> and direct measurement of SiO<sup>2</sup> are included as references for known ordered/disordered scattering, respectively. Figure reproduced with permission from Radi´c et al. <sup>6</sup>](path) can leverage the spatial resolution and exclusive surface sensitivity of the technique to investigate inter-layer coupling strength in otherwise difficult to measure systems such as van Der Waals heterostructures where comparable techniques like LEED/M, PL or Raman struggle due to the inherent transmission of the probe through the sample. #### Thermal expansion coefficient Structural information, such as the lattice constant, can be accurately measured using 2D diffraction scans, found to be within 1%. <sup>1</sup> 2D diffraction scans offer increased accuracy over 1D scans, like those in figure 2, because more diffraction provide a statistical advantage. Initially, we measure the 2D diffraction scan on the surface of bulk MoS2, shown in figure S1, and determine the lattice constant to be 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A, matching literature values within experimental uncertainty. We extend the method to mechanically exfoliated monolayer-MoS2/hBN/SiO2, with diffraction pattern shown in figure 4, to showcase the exclusive surface sensitivity of the technique and determine that the lattice constant is <sup>3</sup>.14±0.07˚A, which is within experimental error ![Figure 4: A 2D diffraction pattern taken on monolayer-MoS2/hBN/SiO2, with the identified centres of the diffraction peaks shown as red dots. All of the lattice spacings between adjacent peaks were measured and averaged give a lattice constant 3.<sup>14</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) The thermal expansion coefficient (TEC) of 2D materials is crucial to understanding their behaviour for successful integration into optoelectronic devices, these materials also provide direct measurement of the anharmonic vibrational modes of low-dimensional systems, 7,8 however, difficult to measure due to their optical transparency. 9,10 Using helium atom microdiffraction one can measure the lattice constant, and therefore TEC, of microscopic optically transparent materials, whether supported by a substrate or free-standing. The diffraction pattern was measured along the principle ⟨10⟩ azimuth of MoS<sup>2</sup> at temperatures from 60 −450 ◦C (the data is presented in figure 4), as with work on HAS no change in the lattice constant was observed, allowing us to put an upper bound on the expansion coefficient of 16 × 10−<sup>6</sup> K−<sup>1</sup> . Our measurement is consistent with the upper bound determined as α < 14 × 10−<sup>6</sup> K−<sup>1</sup> reported by Anemone et al.,<sup>11</sup> and α = 7.6×10−<sup>6</sup> K−<sup>1</sup> measured by Zhang et al. using micro-Raman spectroscopy.<sup>10</sup> Our method is currently limited in angular resolution (radial dimension of figure 4) by the size of the limiting aperture that lies between the sample and detector. The current configuration uses a nominal 0.5 mm diameter hole for an in-plane angular resolution ∼ 7.9 ◦ , representing a compromise between measurement time and helium signal. By changing the limiting aperture to a rectangle with dimensions 0.075 × 0.5 mm we can integrate the signal in the axis perpendicular to any change in lattice parameter, the radial direction in figure 4, while improving in-plane angular resolution. We estimate that moving from the current circular aperture to the proposed rectangular aperture decreases helium signal by a factor of 5, but improves in-plane angular resolution by a factor of 6.7 from 7.9 ◦ → 1.2 ◦ which would be sufficient to resolve the TEC over the current temperature range. Details of the current instrument's spatial and angular resolutions, alongside their definitions, can be found in section 'Experimental Methods - Helium atom micro-diffraction instrument details'. ### Electron-phonon coupling Electron-phonon coupling is a major decoherence mechanism in semiconductors, often causing electron-phonon scattering, and eventually energy dissipation, in turn harming optoelectronic device performance. Optimisation of electron-phonon coupling is therefore critical in device design, both in cases where it must be minimised, or maximised. Electron-phonon coupling is typically measured using inelastic optical techniques, such as Raman or Brillouin scattering, and are primarily limited to optically transparent materials and access to electronic coupling to optical phonons only. It has been shown that helium atom scattering is an effective tool for determining the coupling strength between 2D materials and their substrates via Debye-Waller attenuation of helium scattering.12–15 However, previous atom scattering techniques have been limited to millimetre-scale spot sizes which made devicescale samples inaccessible. Here we introduce helium atom micro-diffraction as a technique well suited to the measurement of electronphonon coupling in a range of materials where its non-damaging, surface sensitive and microscopic neutral probe can be used to access electronic couplings to both optical and acoustic phonon modes in 2D materials. The temperature dependence of reflected and diffracted helium intensities can be modelled by Debye-Waller attenuation, 12,13 which describes the increasing motion of the surface atoms with temperature and causes an increase in inelastic scattering. Thus, the ordered scattering intensity is reduced. The attenuation is described by where 2W is the D-W factor. As the helium scattering occurs from the valence electron density rather than from the ionic cores themselves, one can link the D-W factor to electron phonon coupling, as described by Al Taleb et al. <sup>13</sup> D-W attenuation has been measured previously for macro-scale samples for both LiF and bulk MoS2, however without the spatial resolution enabled by SHeM, monolayer MoS<sup>2</sup> flakes could not be measured.<sup>13</sup> ![Figure 5: Log diffracted intensity as a function of temperature for monolayer MoS<sup>2</sup> on few-layer hBN on glass. In theory all diffraction orders should yield the same D-W, and therefore electron-phonon coupling, constants but the signal-to-noise ratios of each peak are a function of relative peak heights and instrument geometry, resulting in small differences.](path) From figure 5 we can extract the D-W factor as the straight-line gradient, and from that the electron-phonon coupling constant λ, contained in table 1 for monolayer and bulk MoS2. The exponents are compared relative to each other, and to literature values for the bulk case, and found to be in agreement with the expected behaviour with parallel momentum transfer (∆K). Table 1: Debye-Waller factors and electronphonon coupling constants extracted from the data presented in figure 5 using the model in equation 1 and theoretical relations outlined by Anemone et al.,<sup>13</sup> respectively. We report the electron-phonon coupling constant in bulk MoS<sup>2</sup> as λbulk ≈ 0.51 × 10−<sup>3</sup> K−<sup>1</sup> , showing good agreement to the literature values λ ≈ 0.41 × 10−<sup>3</sup> K−<sup>1</sup> , 0.49 × 10−<sup>3</sup> K−<sup>1</sup> measured using a typical helium atom scattering instrument with few-millimetre spot size, reported by Anemone et al. <sup>13</sup> We also find λML ≈ 0.40 × 10−<sup>3</sup> K−<sup>1</sup> for ML-MoS<sup>2</sup> on fewlayer hBN. We find the electron-phonon coupling constant is ∼ 20 % smaller in the monolayer compared to bulk MoS2. By leveraging the current best reported ≈ 350 nm spot size, the method can be extended to perform spatial mapping of the Debye-Waller factor, and therefore electron-phonon coupling constant, without sample preparation or damage in thin films and delicate materials, a class of materials typically difficult to characterise using standard optical and electron beam techniques. #### Vacancy-type defect density Precise control of defect density in semiconductors is instrumental for both current and future semiconductor device development. In particular, the optoelectronic properties of two-dimensional TMD semiconductors such as MoS2, can be tuned using single-atom defects. <sup>16</sup> Applications of these materials includes catalysis<sup>17</sup> and a plethora of devices, 18–21 all of which can affected by the material's defect density. In many applications a balance must be made between a sufficiently high number of defects for the material to acquire the desired properties while the material remaining sufficiently ordered to not degrade electronic performance, <sup>22</sup> or even entirely degrade the lattice structure. However, quantification of defect densities in 2D materials remains a significant experimental challenge, where typically used methods are XPS<sup>22</sup> and STEM, with conductive AFM (CAFM) being explored recently. <sup>23</sup> As all of these methods commonly require complicated sample preparation processes, there is a characterisation shortcoming that is only going to grow more acute as devices using 2D materials start being produced on an industrial scale, and therefore enter the commercial sphere. In this work we demonstrate how helium atom micro-diffraction can be used to characterise the vacancy-type defect density on the surface of few-layer materials using data reproduced from Radi´c et al. with permission. <sup>6</sup> Three mechanically exfoliated monolayer flakes of MoS<sup>2</sup> with increasing defect densities, ranging from the intrinsic 0.1 × 10<sup>14</sup> cm−<sup>2</sup> to 1.8 × 10<sup>14</sup> cm−<sup>2</sup> , and ≈ 15 µm lateral size were produced using high-temperature annealing under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere as outlined by Zhu et al.<sup>22</sup> Exact annealing parameters, and the resulting defect densities inferred using photoluminescence spectroscopy, which has been calibrated against stoichiometric beam-line XPS, are contained in Table 2. Figure 6 shows that diffracted helium intensity decreases approximately linearly as a function of defect density within the single-vacancy limit. Detailed analysis and discussion of the results in figure 6 can be found in the full manuscript.<sup>6</sup> ![Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS<sup>2</sup> flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>](path) tion can be employed as a lab-based method to quantify point-defect density without specific sample preparation or damage. The method is agnostic to sample chemistry, and thickness, because the mechanism through which it measures defect density is purely geometric, relying upon the degree of order of the sample surface, meaning that the method can be trivially extended to any system whose macroscopic properties are mediated by surface defects or dopants. Examples of current systems include hBN,<sup>24</sup> graphene,<sup>25</sup> doped systems such as diamond,<sup>26</sup> alongside other TMDs. In conjunction with its electrostatically neutral and low energy beam, microscopic helium atom diffraction presents itself as an ideal technique for the investigation of these few-layer materials. Further improvements in the instruments' lateral resolution resolution while performing diffraction measurements, down to the previously reported ∼ 300 nm spot sizes,<sup>2</sup> would allow for sub-micron scale mapping of defect density across samples. The technique is not currently capable of acting as a standalone, stoichimetric measure of defect density, like XPS, once an accurate helium-defect scattering crosssection is calculated this will also be possible. ## Conclusion The findings reported here have demonstrated the effectiveness of helium atom microdiffraction as a powerful, and completely noninvasive characterisation tool for 2D materials. The technique's ability to probe only the outermost atomic layers with micron spatial resolution makes it uniquely suited to addressing the challenges of analysing 2D materials, where established techniques often struggle. The results and discussion have highlighted several key applications of helium atom microdiffraction, including the measurement of surface cleanliness and contamination without altering the sample, a critical feature for ensuring the integrity of materials used in device fabrication. We also investigated the impact of substrate on monolayer properties, confirming that the use of a few-layer hBN buffer can preserve the structural integrity of monolayer MoS2, whereas direct contact with SiO2 results in significant disorder.? We have also explored the potential of helium atom micro-diffraction for determining the thermal expansion coefficient and electronphonon coupling in monolayer MoS2, and have demonstrated its capability for precise structural and thermal analysis. The technique's ability to quantify defect densities without sample preparation or damage also highlights it as an obvious tool for optimizing 2D materials in ever expanding areas of application, from optoelectronics to catalysis and more. Future enhancements to helium atom microdiffraction, such as achieving sub-50 nm spatial resolution, and incorporating out-of-plane scattering capabilities, will further expand its applicability, particularly for complex heterostructures and device-grade materials. In summary, helium atom micro-diffraction promises to provide a robust and adaptable platform for characterisation and development of 2D materials, paving the way for advancements in nanotechnology and material science. ## Experimental Methods ## Helium atom micro-diffraction instrument details In the presented work we used a Scanning Helium Microscope (SHeM) with 5 µm spatial and 7.9 ◦ in-plane angular resolution at the specular condition, representing a factor of 2 improvement in spatial resolution over the first iteration published by von Jeinsen et al. <sup>1</sup> Spatial resolution is defined as the full-width at halfmaximum of the beam spot on the sample at the designed working distance. In-plane angular resolution is given by the angle subtended by the limiting detector aperture and beam spot on the sample, with the correcting factor <sup>√</sup> 1 2 applied to account for the 45◦ instrument geometry. ## Sample details Monolayers of MoS<sup>2</sup> were produced by mechanical exfoliation of a bulk MoS<sup>2</sup> crystal, purchased from 2D Semiconductors Ltd., and were deposited onto few-layer thick hexgonal boron nitride (hBN) which in turn lies on a SiO<sup>2</sup> substrate (high-precision glass microscope slide). Each monolayer measured ≈ 15 µm laterally. The increased defect density monolayer MoS<sup>2</sup> samples, presented in section , were produced the same as previously described in this section with the addition of a thermal annealing process under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere for 0.5 h, following the method as described by Zhu et al.<sup>22</sup> Varying annealing temperature and time allows for control of defect density. Exact annealing parameters, with final defect densities characterised by stoichiometric XPS, are shown in Table 2. # Acknowledgements The work was supported by EPSRC grant EP/R008272/1, Innovate UK/Ionoptika Ltd. through Knowledge Transfer Partnership 10000925. The work was performed in part at CORDE, the Collaborative R&D Environment established to provide access to physics related facilities at the Cavendish Laboratory, University of Cambridge and EPSRC award EP/T00634X/1. SML acknowledges support from EPSRC grant EP/X525686/1. # References - 1. von Jeinsen, N. A.; Lambrick, S. M.; Bergin, M.; Radi´c, A.; Liu, B.; Seremet, D.; Jardine, A. P.; Ward, D. J. 2D Helium Atom Diffraction from a Microscopic Spot. Physical Review Letters 2023, 131, 236202, Publisher: American Physical Society. - 5. Man, M. K. L.; Deckoff-Jones, S.; Winchester, A.; Shi, G.; Gupta, G.; Mohite, A. D.; Kar, S.; Kioupakis, E.; Talapatra, S.; Dani, K. M. 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Nature Materials 2024, # Helium atom micro-diffraction as a characterisation tool for 2D materials - supporting information # Bulk MoS<sup>2</sup> diffraction pattern ![Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure 1: Reflection mode optical (top-left) and real-space SHeM (bottom) images of bulk MoS2, hBN/SiO2 and monolayer MoS2/hBN/SiO2 (Shaded intersection). A 2D micro diffraction measurement (top-right) was taken to identify diffraction conditions for the real space imaging. An obvious change in contrast can be seen as the instrument is configured for	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage ![Figure 1: Reflection mode optical (top-left) and real-space SHeM (bottom) images of bulk MoS2, hBN/SiO<sup>2</sup> and monolayer MoS2/hBN/SiO<sup>2</sup> (Shaded intersection). A 2D micro diffraction measurement (top-right) was taken to identify diffraction conditions for the real space imaging. An obvious change in contrast can be seen as the instrument is configured for](path) and species that one expects during standard device fabrication. Once under high-vacuum (2×10−<sup>8</sup> mbar) in the SHeM, a real-space image is taken of the as-prepared sample (Figure 2a) which demonstrates it is difficult to differentiate between MoS<sup>2</sup> and hBN, with the surrounding SiO<sup>2</sup> appearing less intense. Upon heating to 240 ◦C for 8 hours, another real-space image (figure 2b) reveals a stark difference in contrast between MoS2, hBN, and SiO2. The MoS<sup>2</sup> and hBN display significantly more intense scattering than the surrounding SiO<sup>2</sup> which can be quantitatively compared in Figure 2c. Having demonstrated the sensitivity of helium atom micro-diffraction to adsorbates, one can trivially extend the method to quantitatively map adsorbate densities across real-space images. By cleaning and measuring the sample with the presented procedure, one can identify the positions of diffraction peaks in reciprocal space (∆K). This is followed by acquiring a real-space image at a diffraction peak's maximum scattering condition in reciprocal space, achieved by translation of the sample in the zaxis. One can now dose the sample environment with a given contaminant and measure a realspace image to determine if there is a change adsorbate coverage. Through dosing, adsorbation behaviour of a given sample to varying species can also be investigated by monitoring a specific diffraction peak while dosing. ### Monolayer-substrate interactions It is well documented that the choice of substrate placed under a few-layer material has a significant effect on a range of material properties. One can measure the effect of the substrate on optoelectronic or structural properties of the sample via methods such as photoluminescence spectroscopy (PL), Raman, or low-energy electron microscopy/diffraction (LEEM/D), respectively.<sup>5</sup> To demonstrate the exclusive surface sensitivity of helium atom micro-diffraction, monolayer-MoS<sup>2</sup> was placed directly onto both an SiO<sup>2</sup> substrate and few-layer hBN which is in turn on SiO2, sample geometry is shown in figure 1. It has been demonstrated that certain atomically thin buffer layers, such as hBN, ![Figure 2: Real-space images and diffraction scans of monolayer-MoS2/hBN/SiO<sup>2</sup> (red), fewlayer hBN/SiO<sup>2</sup> (black) and SiO<sup>2</sup> (blue) before and after heating (left to right) to 240 ◦C for 8 hours to remove physisorbed species. For monolayer-MoS<sup>2</sup> and hBN we see ordered scattering emerging upon cleaning in contrast to the cosine-like scattering prior. SiO<sup>2</sup> remains cosine-like post-heating. Diffraction scans were acquired at the indicated points in Panels a,b. Real-space images were taken at a single ∆K, marked with a vertical line in Panel C.](path) LaAlO<sup>3</sup> and SrTiO3, can protect the optoelectronic properties of monolayers mounted on them from strongly interacting substrates like SiO2. <sup>5</sup> Figure 3 shows that the helium scattering from monolayer MoS<sup>2</sup> becomes almost entirely disordered when mounted directly on SiO<sup>2</sup> (gray), with faint signs of structure remaining at the expected ∆K diffraction peak positions. In contrast, placing the monolayer onto a few-layer hBN buffer protects its structure and produces diffraction that matches bulk MoS<sup>2</sup> to within experimental bounds, suggesting that their surface morphologies are the same. ![Figure 3: 1D diffraction scans showing that the difference in monolayer-MoS<sup>2</sup> structure when placed directly onto SiO<sup>2</sup> (gray) versus when a few-layer hBN buffer is used between the monolayer and SiO<sup>2</sup> (red). Bulk MoS<sup>2</sup> and direct measurement of SiO<sup>2</sup> are included as references for known ordered/disordered scattering, respectively. Figure reproduced with permission from Radi´c et al. <sup>6</sup>](path) can leverage the spatial resolution and exclusive surface sensitivity of the technique to investigate inter-layer coupling strength in otherwise difficult to measure systems such as van Der Waals heterostructures where comparable techniques like LEED/M, PL or Raman struggle due to the inherent transmission of the probe through the sample. #### Thermal expansion coefficient Structural information, such as the lattice constant, can be accurately measured using 2D diffraction scans, found to be within 1%. <sup>1</sup> 2D diffraction scans offer increased accuracy over 1D scans, like those in figure 2, because more diffraction provide a statistical advantage. Initially, we measure the 2D diffraction scan on the surface of bulk MoS2, shown in figure S1, and determine the lattice constant to be 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A, matching literature values within experimental uncertainty. We extend the method to mechanically exfoliated monolayer-MoS2/hBN/SiO2, with diffraction pattern shown in figure 4, to showcase the exclusive surface sensitivity of the technique and determine that the lattice constant is <sup>3</sup>.14±0.07˚A, which is within experimental error ![Figure 4: A 2D diffraction pattern taken on monolayer-MoS2/hBN/SiO2, with the identified centres of the diffraction peaks shown as red dots. All of the lattice spacings between adjacent peaks were measured and averaged give a lattice constant 3.<sup>14</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) The thermal expansion coefficient (TEC) of 2D materials is crucial to understanding their behaviour for successful integration into optoelectronic devices, these materials also provide direct measurement of the anharmonic vibrational modes of low-dimensional systems, 7,8 however, difficult to measure due to their optical transparency. 9,10 Using helium atom microdiffraction one can measure the lattice constant, and therefore TEC, of microscopic optically transparent materials, whether supported by a substrate or free-standing. The diffraction pattern was measured along the principle ⟨10⟩ azimuth of MoS<sup>2</sup> at temperatures from 60 −450 ◦C (the data is presented in figure 4), as with work on HAS no change in the lattice constant was observed, allowing us to put an upper bound on the expansion coefficient of 16 × 10−<sup>6</sup> K−<sup>1</sup> . Our measurement is consistent with the upper bound determined as α < 14 × 10−<sup>6</sup> K−<sup>1</sup> reported by Anemone et al.,<sup>11</sup> and α = 7.6×10−<sup>6</sup> K−<sup>1</sup> measured by Zhang et al. using micro-Raman spectroscopy.<sup>10</sup> Our method is currently limited in angular resolution (radial dimension of figure 4) by the size of the limiting aperture that lies between the sample and detector. The current configuration uses a nominal 0.5 mm diameter hole for an in-plane angular resolution ∼ 7.9 ◦ , representing a compromise between measurement time and helium signal. By changing the limiting aperture to a rectangle with dimensions 0.075 × 0.5 mm we can integrate the signal in the axis perpendicular to any change in lattice parameter, the radial direction in figure 4, while improving in-plane angular resolution. We estimate that moving from the current circular aperture to the proposed rectangular aperture decreases helium signal by a factor of 5, but improves in-plane angular resolution by a factor of 6.7 from 7.9 ◦ → 1.2 ◦ which would be sufficient to resolve the TEC over the current temperature range. Details of the current instrument's spatial and angular resolutions, alongside their definitions, can be found in section 'Experimental Methods - Helium atom micro-diffraction instrument details'. ### Electron-phonon coupling Electron-phonon coupling is a major decoherence mechanism in semiconductors, often causing electron-phonon scattering, and eventually energy dissipation, in turn harming optoelectronic device performance. Optimisation of electron-phonon coupling is therefore critical in device design, both in cases where it must be minimised, or maximised. Electron-phonon coupling is typically measured using inelastic optical techniques, such as Raman or Brillouin scattering, and are primarily limited to optically transparent materials and access to electronic coupling to optical phonons only. It has been shown that helium atom scattering is an effective tool for determining the coupling strength between 2D materials and their substrates via Debye-Waller attenuation of helium scattering.12–15 However, previous atom scattering techniques have been limited to millimetre-scale spot sizes which made devicescale samples inaccessible. Here we introduce helium atom micro-diffraction as a technique well suited to the measurement of electronphonon coupling in a range of materials where its non-damaging, surface sensitive and microscopic neutral probe can be used to access electronic couplings to both optical and acoustic phonon modes in 2D materials. The temperature dependence of reflected and diffracted helium intensities can be modelled by Debye-Waller attenuation, 12,13 which describes the increasing motion of the surface atoms with temperature and causes an increase in inelastic scattering. Thus, the ordered scattering intensity is reduced. The attenuation is described by where 2W is the D-W factor. As the helium scattering occurs from the valence electron density rather than from the ionic cores themselves, one can link the D-W factor to electron phonon coupling, as described by Al Taleb et al. <sup>13</sup> D-W attenuation has been measured previously for macro-scale samples for both LiF and bulk MoS2, however without the spatial resolution enabled by SHeM, monolayer MoS<sup>2</sup> flakes could not be measured.<sup>13</sup> ![Figure 5: Log diffracted intensity as a function of temperature for monolayer MoS<sup>2</sup> on few-layer hBN on glass. In theory all diffraction orders should yield the same D-W, and therefore electron-phonon coupling, constants but the signal-to-noise ratios of each peak are a function of relative peak heights and instrument geometry, resulting in small differences.](path) From figure 5 we can extract the D-W factor as the straight-line gradient, and from that the electron-phonon coupling constant λ, contained in table 1 for monolayer and bulk MoS2. The exponents are compared relative to each other, and to literature values for the bulk case, and found to be in agreement with the expected behaviour with parallel momentum transfer (∆K). Table 1: Debye-Waller factors and electronphonon coupling constants extracted from the data presented in figure 5 using the model in equation 1 and theoretical relations outlined by Anemone et al.,<sup>13</sup> respectively. We report the electron-phonon coupling constant in bulk MoS<sup>2</sup> as λbulk ≈ 0.51 × 10−<sup>3</sup> K−<sup>1</sup> , showing good agreement to the literature values λ ≈ 0.41 × 10−<sup>3</sup> K−<sup>1</sup> , 0.49 × 10−<sup>3</sup> K−<sup>1</sup> measured using a typical helium atom scattering instrument with few-millimetre spot size, reported by Anemone et al. <sup>13</sup> We also find λML ≈ 0.40 × 10−<sup>3</sup> K−<sup>1</sup> for ML-MoS<sup>2</sup> on fewlayer hBN. We find the electron-phonon coupling constant is ∼ 20 % smaller in the monolayer compared to bulk MoS2. By leveraging the current best reported ≈ 350 nm spot size, the method can be extended to perform spatial mapping of the Debye-Waller factor, and therefore electron-phonon coupling constant, without sample preparation or damage in thin films and delicate materials, a class of materials typically difficult to characterise using standard optical and electron beam techniques. #### Vacancy-type defect density Precise control of defect density in semiconductors is instrumental for both current and future semiconductor device development. In particular, the optoelectronic properties of two-dimensional TMD semiconductors such as MoS2, can be tuned using single-atom defects. <sup>16</sup> Applications of these materials includes catalysis<sup>17</sup> and a plethora of devices, 18–21 all of which can affected by the material's defect density. In many applications a balance must be made between a sufficiently high number of defects for the material to acquire the desired properties while the material remaining sufficiently ordered to not degrade electronic performance, <sup>22</sup> or even entirely degrade the lattice structure. However, quantification of defect densities in 2D materials remains a significant experimental challenge, where typically used methods are XPS<sup>22</sup> and STEM, with conductive AFM (CAFM) being explored recently. <sup>23</sup> As all of these methods commonly require complicated sample preparation processes, there is a characterisation shortcoming that is only going to grow more acute as devices using 2D materials start being produced on an industrial scale, and therefore enter the commercial sphere. In this work we demonstrate how helium atom micro-diffraction can be used to characterise the vacancy-type defect density on the surface of few-layer materials using data reproduced from Radi´c et al. with permission. <sup>6</sup> Three mechanically exfoliated monolayer flakes of MoS<sup>2</sup> with increasing defect densities, ranging from the intrinsic 0.1 × 10<sup>14</sup> cm−<sup>2</sup> to 1.8 × 10<sup>14</sup> cm−<sup>2</sup> , and ≈ 15 µm lateral size were produced using high-temperature annealing under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere as outlined by Zhu et al.<sup>22</sup> Exact annealing parameters, and the resulting defect densities inferred using photoluminescence spectroscopy, which has been calibrated against stoichiometric beam-line XPS, are contained in Table 2. Figure 6 shows that diffracted helium intensity decreases approximately linearly as a function of defect density within the single-vacancy limit. Detailed analysis and discussion of the results in figure 6 can be found in the full manuscript.<sup>6</sup> ![Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS<sup>2</sup> flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>](path) tion can be employed as a lab-based method to quantify point-defect density without specific sample preparation or damage. The method is agnostic to sample chemistry, and thickness, because the mechanism through which it measures defect density is purely geometric, relying upon the degree of order of the sample surface, meaning that the method can be trivially extended to any system whose macroscopic properties are mediated by surface defects or dopants. Examples of current systems include hBN,<sup>24</sup> graphene,<sup>25</sup> doped systems such as diamond,<sup>26</sup> alongside other TMDs. In conjunction with its electrostatically neutral and low energy beam, microscopic helium atom diffraction presents itself as an ideal technique for the investigation of these few-layer materials. Further improvements in the instruments' lateral resolution resolution while performing diffraction measurements, down to the previously reported ∼ 300 nm spot sizes,<sup>2</sup> would allow for sub-micron scale mapping of defect density across samples. The technique is not currently capable of acting as a standalone, stoichimetric measure of defect density, like XPS, once an accurate helium-defect scattering crosssection is calculated this will also be possible. ## Conclusion The findings reported here have demonstrated the effectiveness of helium atom microdiffraction as a powerful, and completely noninvasive characterisation tool for 2D materials. The technique's ability to probe only the outermost atomic layers with micron spatial resolution makes it uniquely suited to addressing the challenges of analysing 2D materials, where established techniques often struggle. The results and discussion have highlighted several key applications of helium atom microdiffraction, including the measurement of surface cleanliness and contamination without altering the sample, a critical feature for ensuring the integrity of materials used in device fabrication. We also investigated the impact of substrate on monolayer properties, confirming that the use of a few-layer hBN buffer can preserve the structural integrity of monolayer MoS2, whereas direct contact with SiO2 results in significant disorder.? We have also explored the potential of helium atom micro-diffraction for determining the thermal expansion coefficient and electronphonon coupling in monolayer MoS2, and have demonstrated its capability for precise structural and thermal analysis. The technique's ability to quantify defect densities without sample preparation or damage also highlights it as an obvious tool for optimizing 2D materials in ever expanding areas of application, from optoelectronics to catalysis and more. Future enhancements to helium atom microdiffraction, such as achieving sub-50 nm spatial resolution, and incorporating out-of-plane scattering capabilities, will further expand its applicability, particularly for complex heterostructures and device-grade materials. In summary, helium atom micro-diffraction promises to provide a robust and adaptable platform for characterisation and development of 2D materials, paving the way for advancements in nanotechnology and material science. ## Experimental Methods ## Helium atom micro-diffraction instrument details In the presented work we used a Scanning Helium Microscope (SHeM) with 5 µm spatial and 7.9 ◦ in-plane angular resolution at the specular condition, representing a factor of 2 improvement in spatial resolution over the first iteration published by von Jeinsen et al. <sup>1</sup> Spatial resolution is defined as the full-width at halfmaximum of the beam spot on the sample at the designed working distance. In-plane angular resolution is given by the angle subtended by the limiting detector aperture and beam spot on the sample, with the correcting factor <sup>√</sup> 1 2 applied to account for the 45◦ instrument geometry. ## Sample details Monolayers of MoS<sup>2</sup> were produced by mechanical exfoliation of a bulk MoS<sup>2</sup> crystal, purchased from 2D Semiconductors Ltd., and were deposited onto few-layer thick hexgonal boron nitride (hBN) which in turn lies on a SiO<sup>2</sup> substrate (high-precision glass microscope slide). Each monolayer measured ≈ 15 µm laterally. The increased defect density monolayer MoS<sup>2</sup> samples, presented in section , were produced the same as previously described in this section with the addition of a thermal annealing process under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere for 0.5 h, following the method as described by Zhu et al.<sup>22</sup> Varying annealing temperature and time allows for control of defect density. Exact annealing parameters, with final defect densities characterised by stoichiometric XPS, are shown in Table 2. # Acknowledgements The work was supported by EPSRC grant EP/R008272/1, Innovate UK/Ionoptika Ltd. through Knowledge Transfer Partnership 10000925. The work was performed in part at CORDE, the Collaborative R&D Environment established to provide access to physics related facilities at the Cavendish Laboratory, University of Cambridge and EPSRC award EP/T00634X/1. SML acknowledges support from EPSRC grant EP/X525686/1. # References - 1. von Jeinsen, N. A.; Lambrick, S. M.; Bergin, M.; Radi´c, A.; Liu, B.; Seremet, D.; Jardine, A. P.; Ward, D. J. 2D Helium Atom Diffraction from a Microscopic Spot. Physical Review Letters 2023, 131, 236202, Publisher: American Physical Society. - 5. Man, M. K. L.; Deckoff-Jones, S.; Winchester, A.; Shi, G.; Gupta, G.; Mohite, A. D.; Kar, S.; Kioupakis, E.; Talapatra, S.; Dani, K. M. 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Nature Materials 2024, # Helium atom micro-diffraction as a characterisation tool for 2D materials - supporting information # Bulk MoS<sup>2</sup> diffraction pattern ![Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS² flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage ![Figure 1: Reflection mode optical (top-left) and real-space SHeM (bottom) images of bulk MoS2, hBN/SiO<sup>2</sup> and monolayer MoS2/hBN/SiO<sup>2</sup> (Shaded intersection). A 2D micro diffraction measurement (top-right) was taken to identify diffraction conditions for the real space imaging. An obvious change in contrast can be seen as the instrument is configured for](path) and species that one expects during standard device fabrication. Once under high-vacuum (2×10−<sup>8</sup> mbar) in the SHeM, a real-space image is taken of the as-prepared sample (Figure 2a) which demonstrates it is difficult to differentiate between MoS<sup>2</sup> and hBN, with the surrounding SiO<sup>2</sup> appearing less intense. Upon heating to 240 ◦C for 8 hours, another real-space image (figure 2b) reveals a stark difference in contrast between MoS2, hBN, and SiO2. The MoS<sup>2</sup> and hBN display significantly more intense scattering than the surrounding SiO<sup>2</sup> which can be quantitatively compared in Figure 2c. Having demonstrated the sensitivity of helium atom micro-diffraction to adsorbates, one can trivially extend the method to quantitatively map adsorbate densities across real-space images. By cleaning and measuring the sample with the presented procedure, one can identify the positions of diffraction peaks in reciprocal space (∆K). This is followed by acquiring a real-space image at a diffraction peak's maximum scattering condition in reciprocal space, achieved by translation of the sample in the zaxis. One can now dose the sample environment with a given contaminant and measure a realspace image to determine if there is a change adsorbate coverage. Through dosing, adsorbation behaviour of a given sample to varying species can also be investigated by monitoring a specific diffraction peak while dosing. ### Monolayer-substrate interactions It is well documented that the choice of substrate placed under a few-layer material has a significant effect on a range of material properties. One can measure the effect of the substrate on optoelectronic or structural properties of the sample via methods such as photoluminescence spectroscopy (PL), Raman, or low-energy electron microscopy/diffraction (LEEM/D), respectively.<sup>5</sup> To demonstrate the exclusive surface sensitivity of helium atom micro-diffraction, monolayer-MoS<sup>2</sup> was placed directly onto both an SiO<sup>2</sup> substrate and few-layer hBN which is in turn on SiO2, sample geometry is shown in figure 1. It has been demonstrated that certain atomically thin buffer layers, such as hBN, ![Figure 2: Real-space images and diffraction scans of monolayer-MoS2/hBN/SiO<sup>2</sup> (red), fewlayer hBN/SiO<sup>2</sup> (black) and SiO<sup>2</sup> (blue) before and after heating (left to right) to 240 ◦C for 8 hours to remove physisorbed species. For monolayer-MoS<sup>2</sup> and hBN we see ordered scattering emerging upon cleaning in contrast to the cosine-like scattering prior. SiO<sup>2</sup> remains cosine-like post-heating. Diffraction scans were acquired at the indicated points in Panels a,b. Real-space images were taken at a single ∆K, marked with a vertical line in Panel C.](path) LaAlO<sup>3</sup> and SrTiO3, can protect the optoelectronic properties of monolayers mounted on them from strongly interacting substrates like SiO2. <sup>5</sup> Figure 3 shows that the helium scattering from monolayer MoS<sup>2</sup> becomes almost entirely disordered when mounted directly on SiO<sup>2</sup> (gray), with faint signs of structure remaining at the expected ∆K diffraction peak positions. In contrast, placing the monolayer onto a few-layer hBN buffer protects its structure and produces diffraction that matches bulk MoS<sup>2</sup> to within experimental bounds, suggesting that their surface morphologies are the same. ![Figure 3: 1D diffraction scans showing that the difference in monolayer-MoS<sup>2</sup> structure when placed directly onto SiO<sup>2</sup> (gray) versus when a few-layer hBN buffer is used between the monolayer and SiO<sup>2</sup> (red). Bulk MoS<sup>2</sup> and direct measurement of SiO<sup>2</sup> are included as references for known ordered/disordered scattering, respectively. Figure reproduced with permission from Radi´c et al. <sup>6</sup>](path) can leverage the spatial resolution and exclusive surface sensitivity of the technique to investigate inter-layer coupling strength in otherwise difficult to measure systems such as van Der Waals heterostructures where comparable techniques like LEED/M, PL or Raman struggle due to the inherent transmission of the probe through the sample. #### Thermal expansion coefficient Structural information, such as the lattice constant, can be accurately measured using 2D diffraction scans, found to be within 1%. <sup>1</sup> 2D diffraction scans offer increased accuracy over 1D scans, like those in figure 2, because more diffraction provide a statistical advantage. Initially, we measure the 2D diffraction scan on the surface of bulk MoS2, shown in figure S1, and determine the lattice constant to be 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A, matching literature values within experimental uncertainty. We extend the method to mechanically exfoliated monolayer-MoS2/hBN/SiO2, with diffraction pattern shown in figure 4, to showcase the exclusive surface sensitivity of the technique and determine that the lattice constant is <sup>3</sup>.14±0.07˚A, which is within experimental error ![Figure 4: A 2D diffraction pattern taken on monolayer-MoS2/hBN/SiO2, with the identified centres of the diffraction peaks shown as red dots. All of the lattice spacings between adjacent peaks were measured and averaged give a lattice constant 3.<sup>14</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) The thermal expansion coefficient (TEC) of 2D materials is crucial to understanding their behaviour for successful integration into optoelectronic devices, these materials also provide direct measurement of the anharmonic vibrational modes of low-dimensional systems, 7,8 however, difficult to measure due to their optical transparency. 9,10 Using helium atom microdiffraction one can measure the lattice constant, and therefore TEC, of microscopic optically transparent materials, whether supported by a substrate or free-standing. The diffraction pattern was measured along the principle ⟨10⟩ azimuth of MoS<sup>2</sup> at temperatures from 60 −450 ◦C (the data is presented in figure 4), as with work on HAS no change in the lattice constant was observed, allowing us to put an upper bound on the expansion coefficient of 16 × 10−<sup>6</sup> K−<sup>1</sup> . Our measurement is consistent with the upper bound determined as α < 14 × 10−<sup>6</sup> K−<sup>1</sup> reported by Anemone et al.,<sup>11</sup> and α = 7.6×10−<sup>6</sup> K−<sup>1</sup> measured by Zhang et al. using micro-Raman spectroscopy.<sup>10</sup> Our method is currently limited in angular resolution (radial dimension of figure 4) by the size of the limiting aperture that lies between the sample and detector. The current configuration uses a nominal 0.5 mm diameter hole for an in-plane angular resolution ∼ 7.9 ◦ , representing a compromise between measurement time and helium signal. By changing the limiting aperture to a rectangle with dimensions 0.075 × 0.5 mm we can integrate the signal in the axis perpendicular to any change in lattice parameter, the radial direction in figure 4, while improving in-plane angular resolution. We estimate that moving from the current circular aperture to the proposed rectangular aperture decreases helium signal by a factor of 5, but improves in-plane angular resolution by a factor of 6.7 from 7.9 ◦ → 1.2 ◦ which would be sufficient to resolve the TEC over the current temperature range. Details of the current instrument's spatial and angular resolutions, alongside their definitions, can be found in section 'Experimental Methods - Helium atom micro-diffraction instrument details'. ### Electron-phonon coupling Electron-phonon coupling is a major decoherence mechanism in semiconductors, often causing electron-phonon scattering, and eventually energy dissipation, in turn harming optoelectronic device performance. Optimisation of electron-phonon coupling is therefore critical in device design, both in cases where it must be minimised, or maximised. Electron-phonon coupling is typically measured using inelastic optical techniques, such as Raman or Brillouin scattering, and are primarily limited to optically transparent materials and access to electronic coupling to optical phonons only. It has been shown that helium atom scattering is an effective tool for determining the coupling strength between 2D materials and their substrates via Debye-Waller attenuation of helium scattering.12–15 However, previous atom scattering techniques have been limited to millimetre-scale spot sizes which made devicescale samples inaccessible. Here we introduce helium atom micro-diffraction as a technique well suited to the measurement of electronphonon coupling in a range of materials where its non-damaging, surface sensitive and microscopic neutral probe can be used to access electronic couplings to both optical and acoustic phonon modes in 2D materials. The temperature dependence of reflected and diffracted helium intensities can be modelled by Debye-Waller attenuation, 12,13 which describes the increasing motion of the surface atoms with temperature and causes an increase in inelastic scattering. Thus, the ordered scattering intensity is reduced. The attenuation is described by where 2W is the D-W factor. As the helium scattering occurs from the valence electron density rather than from the ionic cores themselves, one can link the D-W factor to electron phonon coupling, as described by Al Taleb et al. <sup>13</sup> D-W attenuation has been measured previously for macro-scale samples for both LiF and bulk MoS2, however without the spatial resolution enabled by SHeM, monolayer MoS<sup>2</sup> flakes could not be measured.<sup>13</sup> ![Figure 5: Log diffracted intensity as a function of temperature for monolayer MoS<sup>2</sup> on few-layer hBN on glass. In theory all diffraction orders should yield the same D-W, and therefore electron-phonon coupling, constants but the signal-to-noise ratios of each peak are a function of relative peak heights and instrument geometry, resulting in small differences.](path) From figure 5 we can extract the D-W factor as the straight-line gradient, and from that the electron-phonon coupling constant λ, contained in table 1 for monolayer and bulk MoS2. The exponents are compared relative to each other, and to literature values for the bulk case, and found to be in agreement with the expected behaviour with parallel momentum transfer (∆K). Table 1: Debye-Waller factors and electronphonon coupling constants extracted from the data presented in figure 5 using the model in equation 1 and theoretical relations outlined by Anemone et al.,<sup>13</sup> respectively. We report the electron-phonon coupling constant in bulk MoS<sup>2</sup> as λbulk ≈ 0.51 × 10−<sup>3</sup> K−<sup>1</sup> , showing good agreement to the literature values λ ≈ 0.41 × 10−<sup>3</sup> K−<sup>1</sup> , 0.49 × 10−<sup>3</sup> K−<sup>1</sup> measured using a typical helium atom scattering instrument with few-millimetre spot size, reported by Anemone et al. <sup>13</sup> We also find λML ≈ 0.40 × 10−<sup>3</sup> K−<sup>1</sup> for ML-MoS<sup>2</sup> on fewlayer hBN. We find the electron-phonon coupling constant is ∼ 20 % smaller in the monolayer compared to bulk MoS2. By leveraging the current best reported ≈ 350 nm spot size, the method can be extended to perform spatial mapping of the Debye-Waller factor, and therefore electron-phonon coupling constant, without sample preparation or damage in thin films and delicate materials, a class of materials typically difficult to characterise using standard optical and electron beam techniques. #### Vacancy-type defect density Precise control of defect density in semiconductors is instrumental for both current and future semiconductor device development. In particular, the optoelectronic properties of two-dimensional TMD semiconductors such as MoS2, can be tuned using single-atom defects. <sup>16</sup> Applications of these materials includes catalysis<sup>17</sup> and a plethora of devices, 18–21 all of which can affected by the material's defect density. In many applications a balance must be made between a sufficiently high number of defects for the material to acquire the desired properties while the material remaining sufficiently ordered to not degrade electronic performance, <sup>22</sup> or even entirely degrade the lattice structure. However, quantification of defect densities in 2D materials remains a significant experimental challenge, where typically used methods are XPS<sup>22</sup> and STEM, with conductive AFM (CAFM) being explored recently. <sup>23</sup> As all of these methods commonly require complicated sample preparation processes, there is a characterisation shortcoming that is only going to grow more acute as devices using 2D materials start being produced on an industrial scale, and therefore enter the commercial sphere. In this work we demonstrate how helium atom micro-diffraction can be used to characterise the vacancy-type defect density on the surface of few-layer materials using data reproduced from Radi´c et al. with permission. <sup>6</sup> Three mechanically exfoliated monolayer flakes of MoS<sup>2</sup> with increasing defect densities, ranging from the intrinsic 0.1 × 10<sup>14</sup> cm−<sup>2</sup> to 1.8 × 10<sup>14</sup> cm−<sup>2</sup> , and ≈ 15 µm lateral size were produced using high-temperature annealing under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere as outlined by Zhu et al.<sup>22</sup> Exact annealing parameters, and the resulting defect densities inferred using photoluminescence spectroscopy, which has been calibrated against stoichiometric beam-line XPS, are contained in Table 2. Figure 6 shows that diffracted helium intensity decreases approximately linearly as a function of defect density within the single-vacancy limit. Detailed analysis and discussion of the results in figure 6 can be found in the full manuscript.<sup>6</sup> ![Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS<sup>2</sup> flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>](path) tion can be employed as a lab-based method to quantify point-defect density without specific sample preparation or damage. The method is agnostic to sample chemistry, and thickness, because the mechanism through which it measures defect density is purely geometric, relying upon the degree of order of the sample surface, meaning that the method can be trivially extended to any system whose macroscopic properties are mediated by surface defects or dopants. Examples of current systems include hBN,<sup>24</sup> graphene,<sup>25</sup> doped systems such as diamond,<sup>26</sup> alongside other TMDs. In conjunction with its electrostatically neutral and low energy beam, microscopic helium atom diffraction presents itself as an ideal technique for the investigation of these few-layer materials. Further improvements in the instruments' lateral resolution resolution while performing diffraction measurements, down to the previously reported ∼ 300 nm spot sizes,<sup>2</sup> would allow for sub-micron scale mapping of defect density across samples. The technique is not currently capable of acting as a standalone, stoichimetric measure of defect density, like XPS, once an accurate helium-defect scattering crosssection is calculated this will also be possible. ## Conclusion The findings reported here have demonstrated the effectiveness of helium atom microdiffraction as a powerful, and completely noninvasive characterisation tool for 2D materials. The technique's ability to probe only the outermost atomic layers with micron spatial resolution makes it uniquely suited to addressing the challenges of analysing 2D materials, where established techniques often struggle. The results and discussion have highlighted several key applications of helium atom microdiffraction, including the measurement of surface cleanliness and contamination without altering the sample, a critical feature for ensuring the integrity of materials used in device fabrication. We also investigated the impact of substrate on monolayer properties, confirming that the use of a few-layer hBN buffer can preserve the structural integrity of monolayer MoS2, whereas direct contact with SiO2 results in significant disorder.? We have also explored the potential of helium atom micro-diffraction for determining the thermal expansion coefficient and electronphonon coupling in monolayer MoS2, and have demonstrated its capability for precise structural and thermal analysis. The technique's ability to quantify defect densities without sample preparation or damage also highlights it as an obvious tool for optimizing 2D materials in ever expanding areas of application, from optoelectronics to catalysis and more. Future enhancements to helium atom microdiffraction, such as achieving sub-50 nm spatial resolution, and incorporating out-of-plane scattering capabilities, will further expand its applicability, particularly for complex heterostructures and device-grade materials. In summary, helium atom micro-diffraction promises to provide a robust and adaptable platform for characterisation and development of 2D materials, paving the way for advancements in nanotechnology and material science. ## Experimental Methods ## Helium atom micro-diffraction instrument details In the presented work we used a Scanning Helium Microscope (SHeM) with 5 µm spatial and 7.9 ◦ in-plane angular resolution at the specular condition, representing a factor of 2 improvement in spatial resolution over the first iteration published by von Jeinsen et al. <sup>1</sup> Spatial resolution is defined as the full-width at halfmaximum of the beam spot on the sample at the designed working distance. In-plane angular resolution is given by the angle subtended by the limiting detector aperture and beam spot on the sample, with the correcting factor <sup>√</sup> 1 2 applied to account for the 45◦ instrument geometry. ## Sample details Monolayers of MoS<sup>2</sup> were produced by mechanical exfoliation of a bulk MoS<sup>2</sup> crystal, purchased from 2D Semiconductors Ltd., and were deposited onto few-layer thick hexgonal boron nitride (hBN) which in turn lies on a SiO<sup>2</sup> substrate (high-precision glass microscope slide). Each monolayer measured ≈ 15 µm laterally. The increased defect density monolayer MoS<sup>2</sup> samples, presented in section , were produced the same as previously described in this section with the addition of a thermal annealing process under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere for 0.5 h, following the method as described by Zhu et al.<sup>22</sup> Varying annealing temperature and time allows for control of defect density. Exact annealing parameters, with final defect densities characterised by stoichiometric XPS, are shown in Table 2. # Acknowledgements The work was supported by EPSRC grant EP/R008272/1, Innovate UK/Ionoptika Ltd. through Knowledge Transfer Partnership 10000925. The work was performed in part at CORDE, the Collaborative R&D Environment established to provide access to physics related facilities at the Cavendish Laboratory, University of Cambridge and EPSRC award EP/T00634X/1. SML acknowledges support from EPSRC grant EP/X525686/1. # References - 1. von Jeinsen, N. A.; Lambrick, S. M.; Bergin, M.; Radi´c, A.; Liu, B.; Seremet, D.; Jardine, A. P.; Ward, D. J. 2D Helium Atom Diffraction from a Microscopic Spot. Physical Review Letters 2023, 131, 236202, Publisher: American Physical Society. - 5. Man, M. K. L.; Deckoff-Jones, S.; Winchester, A.; Shi, G.; Gupta, G.; Mohite, A. D.; Kar, S.; Kioupakis, E.; Talapatra, S.; Dani, K. M. 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Nature Materials 2024, # Helium atom micro-diffraction as a characterisation tool for 2D materials - supporting information # Bulk MoS<sup>2</sup> diffraction pattern ![Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure 3: 1D diffraction scans showing that the difference in monolayer-MoS² structure when placed directly onto SiO² (gray) versus when a few-layer hBN buffer is used between the monolayer and SiO² (red). Bulk MoS² and direct measurement of SiO² are included as references for known ordered/disordered scattering, respectively. Figure reproduced with permission from Radi´c et al. <sup>6</sup>	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage ![Figure 1: Reflection mode optical (top-left) and real-space SHeM (bottom) images of bulk MoS2, hBN/SiO<sup>2</sup> and monolayer MoS2/hBN/SiO<sup>2</sup> (Shaded intersection). A 2D micro diffraction measurement (top-right) was taken to identify diffraction conditions for the real space imaging. An obvious change in contrast can be seen as the instrument is configured for](path) and species that one expects during standard device fabrication. Once under high-vacuum (2×10−<sup>8</sup> mbar) in the SHeM, a real-space image is taken of the as-prepared sample (Figure 2a) which demonstrates it is difficult to differentiate between MoS<sup>2</sup> and hBN, with the surrounding SiO<sup>2</sup> appearing less intense. Upon heating to 240 ◦C for 8 hours, another real-space image (figure 2b) reveals a stark difference in contrast between MoS2, hBN, and SiO2. The MoS<sup>2</sup> and hBN display significantly more intense scattering than the surrounding SiO<sup>2</sup> which can be quantitatively compared in Figure 2c. Having demonstrated the sensitivity of helium atom micro-diffraction to adsorbates, one can trivially extend the method to quantitatively map adsorbate densities across real-space images. By cleaning and measuring the sample with the presented procedure, one can identify the positions of diffraction peaks in reciprocal space (∆K). This is followed by acquiring a real-space image at a diffraction peak's maximum scattering condition in reciprocal space, achieved by translation of the sample in the zaxis. One can now dose the sample environment with a given contaminant and measure a realspace image to determine if there is a change adsorbate coverage. Through dosing, adsorbation behaviour of a given sample to varying species can also be investigated by monitoring a specific diffraction peak while dosing. ### Monolayer-substrate interactions It is well documented that the choice of substrate placed under a few-layer material has a significant effect on a range of material properties. One can measure the effect of the substrate on optoelectronic or structural properties of the sample via methods such as photoluminescence spectroscopy (PL), Raman, or low-energy electron microscopy/diffraction (LEEM/D), respectively.<sup>5</sup> To demonstrate the exclusive surface sensitivity of helium atom micro-diffraction, monolayer-MoS<sup>2</sup> was placed directly onto both an SiO<sup>2</sup> substrate and few-layer hBN which is in turn on SiO2, sample geometry is shown in figure 1. It has been demonstrated that certain atomically thin buffer layers, such as hBN, ![Figure 2: Real-space images and diffraction scans of monolayer-MoS2/hBN/SiO<sup>2</sup> (red), fewlayer hBN/SiO<sup>2</sup> (black) and SiO<sup>2</sup> (blue) before and after heating (left to right) to 240 ◦C for 8 hours to remove physisorbed species. For monolayer-MoS<sup>2</sup> and hBN we see ordered scattering emerging upon cleaning in contrast to the cosine-like scattering prior. SiO<sup>2</sup> remains cosine-like post-heating. Diffraction scans were acquired at the indicated points in Panels a,b. Real-space images were taken at a single ∆K, marked with a vertical line in Panel C.](path) LaAlO<sup>3</sup> and SrTiO3, can protect the optoelectronic properties of monolayers mounted on them from strongly interacting substrates like SiO2. <sup>5</sup> Figure 3 shows that the helium scattering from monolayer MoS<sup>2</sup> becomes almost entirely disordered when mounted directly on SiO<sup>2</sup> (gray), with faint signs of structure remaining at the expected ∆K diffraction peak positions. In contrast, placing the monolayer onto a few-layer hBN buffer protects its structure and produces diffraction that matches bulk MoS<sup>2</sup> to within experimental bounds, suggesting that their surface morphologies are the same. ![Figure 3: 1D diffraction scans showing that the difference in monolayer-MoS<sup>2</sup> structure when placed directly onto SiO<sup>2</sup> (gray) versus when a few-layer hBN buffer is used between the monolayer and SiO<sup>2</sup> (red). Bulk MoS<sup>2</sup> and direct measurement of SiO<sup>2</sup> are included as references for known ordered/disordered scattering, respectively. Figure reproduced with permission from Radi´c et al. <sup>6</sup>](path) can leverage the spatial resolution and exclusive surface sensitivity of the technique to investigate inter-layer coupling strength in otherwise difficult to measure systems such as van Der Waals heterostructures where comparable techniques like LEED/M, PL or Raman struggle due to the inherent transmission of the probe through the sample. #### Thermal expansion coefficient Structural information, such as the lattice constant, can be accurately measured using 2D diffraction scans, found to be within 1%. <sup>1</sup> 2D diffraction scans offer increased accuracy over 1D scans, like those in figure 2, because more diffraction provide a statistical advantage. Initially, we measure the 2D diffraction scan on the surface of bulk MoS2, shown in figure S1, and determine the lattice constant to be 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A, matching literature values within experimental uncertainty. We extend the method to mechanically exfoliated monolayer-MoS2/hBN/SiO2, with diffraction pattern shown in figure 4, to showcase the exclusive surface sensitivity of the technique and determine that the lattice constant is <sup>3</sup>.14±0.07˚A, which is within experimental error ![Figure 4: A 2D diffraction pattern taken on monolayer-MoS2/hBN/SiO2, with the identified centres of the diffraction peaks shown as red dots. All of the lattice spacings between adjacent peaks were measured and averaged give a lattice constant 3.<sup>14</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) The thermal expansion coefficient (TEC) of 2D materials is crucial to understanding their behaviour for successful integration into optoelectronic devices, these materials also provide direct measurement of the anharmonic vibrational modes of low-dimensional systems, 7,8 however, difficult to measure due to their optical transparency. 9,10 Using helium atom microdiffraction one can measure the lattice constant, and therefore TEC, of microscopic optically transparent materials, whether supported by a substrate or free-standing. The diffraction pattern was measured along the principle ⟨10⟩ azimuth of MoS<sup>2</sup> at temperatures from 60 −450 ◦C (the data is presented in figure 4), as with work on HAS no change in the lattice constant was observed, allowing us to put an upper bound on the expansion coefficient of 16 × 10−<sup>6</sup> K−<sup>1</sup> . Our measurement is consistent with the upper bound determined as α < 14 × 10−<sup>6</sup> K−<sup>1</sup> reported by Anemone et al.,<sup>11</sup> and α = 7.6×10−<sup>6</sup> K−<sup>1</sup> measured by Zhang et al. using micro-Raman spectroscopy.<sup>10</sup> Our method is currently limited in angular resolution (radial dimension of figure 4) by the size of the limiting aperture that lies between the sample and detector. The current configuration uses a nominal 0.5 mm diameter hole for an in-plane angular resolution ∼ 7.9 ◦ , representing a compromise between measurement time and helium signal. By changing the limiting aperture to a rectangle with dimensions 0.075 × 0.5 mm we can integrate the signal in the axis perpendicular to any change in lattice parameter, the radial direction in figure 4, while improving in-plane angular resolution. We estimate that moving from the current circular aperture to the proposed rectangular aperture decreases helium signal by a factor of 5, but improves in-plane angular resolution by a factor of 6.7 from 7.9 ◦ → 1.2 ◦ which would be sufficient to resolve the TEC over the current temperature range. Details of the current instrument's spatial and angular resolutions, alongside their definitions, can be found in section 'Experimental Methods - Helium atom micro-diffraction instrument details'. ### Electron-phonon coupling Electron-phonon coupling is a major decoherence mechanism in semiconductors, often causing electron-phonon scattering, and eventually energy dissipation, in turn harming optoelectronic device performance. Optimisation of electron-phonon coupling is therefore critical in device design, both in cases where it must be minimised, or maximised. Electron-phonon coupling is typically measured using inelastic optical techniques, such as Raman or Brillouin scattering, and are primarily limited to optically transparent materials and access to electronic coupling to optical phonons only. It has been shown that helium atom scattering is an effective tool for determining the coupling strength between 2D materials and their substrates via Debye-Waller attenuation of helium scattering.12–15 However, previous atom scattering techniques have been limited to millimetre-scale spot sizes which made devicescale samples inaccessible. Here we introduce helium atom micro-diffraction as a technique well suited to the measurement of electronphonon coupling in a range of materials where its non-damaging, surface sensitive and microscopic neutral probe can be used to access electronic couplings to both optical and acoustic phonon modes in 2D materials. The temperature dependence of reflected and diffracted helium intensities can be modelled by Debye-Waller attenuation, 12,13 which describes the increasing motion of the surface atoms with temperature and causes an increase in inelastic scattering. Thus, the ordered scattering intensity is reduced. The attenuation is described by where 2W is the D-W factor. As the helium scattering occurs from the valence electron density rather than from the ionic cores themselves, one can link the D-W factor to electron phonon coupling, as described by Al Taleb et al. <sup>13</sup> D-W attenuation has been measured previously for macro-scale samples for both LiF and bulk MoS2, however without the spatial resolution enabled by SHeM, monolayer MoS<sup>2</sup> flakes could not be measured.<sup>13</sup> ![Figure 5: Log diffracted intensity as a function of temperature for monolayer MoS<sup>2</sup> on few-layer hBN on glass. In theory all diffraction orders should yield the same D-W, and therefore electron-phonon coupling, constants but the signal-to-noise ratios of each peak are a function of relative peak heights and instrument geometry, resulting in small differences.](path) From figure 5 we can extract the D-W factor as the straight-line gradient, and from that the electron-phonon coupling constant λ, contained in table 1 for monolayer and bulk MoS2. The exponents are compared relative to each other, and to literature values for the bulk case, and found to be in agreement with the expected behaviour with parallel momentum transfer (∆K). Table 1: Debye-Waller factors and electronphonon coupling constants extracted from the data presented in figure 5 using the model in equation 1 and theoretical relations outlined by Anemone et al.,<sup>13</sup> respectively. We report the electron-phonon coupling constant in bulk MoS<sup>2</sup> as λbulk ≈ 0.51 × 10−<sup>3</sup> K−<sup>1</sup> , showing good agreement to the literature values λ ≈ 0.41 × 10−<sup>3</sup> K−<sup>1</sup> , 0.49 × 10−<sup>3</sup> K−<sup>1</sup> measured using a typical helium atom scattering instrument with few-millimetre spot size, reported by Anemone et al. <sup>13</sup> We also find λML ≈ 0.40 × 10−<sup>3</sup> K−<sup>1</sup> for ML-MoS<sup>2</sup> on fewlayer hBN. We find the electron-phonon coupling constant is ∼ 20 % smaller in the monolayer compared to bulk MoS2. By leveraging the current best reported ≈ 350 nm spot size, the method can be extended to perform spatial mapping of the Debye-Waller factor, and therefore electron-phonon coupling constant, without sample preparation or damage in thin films and delicate materials, a class of materials typically difficult to characterise using standard optical and electron beam techniques. #### Vacancy-type defect density Precise control of defect density in semiconductors is instrumental for both current and future semiconductor device development. In particular, the optoelectronic properties of two-dimensional TMD semiconductors such as MoS2, can be tuned using single-atom defects. <sup>16</sup> Applications of these materials includes catalysis<sup>17</sup> and a plethora of devices, 18–21 all of which can affected by the material's defect density. In many applications a balance must be made between a sufficiently high number of defects for the material to acquire the desired properties while the material remaining sufficiently ordered to not degrade electronic performance, <sup>22</sup> or even entirely degrade the lattice structure. However, quantification of defect densities in 2D materials remains a significant experimental challenge, where typically used methods are XPS<sup>22</sup> and STEM, with conductive AFM (CAFM) being explored recently. <sup>23</sup> As all of these methods commonly require complicated sample preparation processes, there is a characterisation shortcoming that is only going to grow more acute as devices using 2D materials start being produced on an industrial scale, and therefore enter the commercial sphere. In this work we demonstrate how helium atom micro-diffraction can be used to characterise the vacancy-type defect density on the surface of few-layer materials using data reproduced from Radi´c et al. with permission. <sup>6</sup> Three mechanically exfoliated monolayer flakes of MoS<sup>2</sup> with increasing defect densities, ranging from the intrinsic 0.1 × 10<sup>14</sup> cm−<sup>2</sup> to 1.8 × 10<sup>14</sup> cm−<sup>2</sup> , and ≈ 15 µm lateral size were produced using high-temperature annealing under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere as outlined by Zhu et al.<sup>22</sup> Exact annealing parameters, and the resulting defect densities inferred using photoluminescence spectroscopy, which has been calibrated against stoichiometric beam-line XPS, are contained in Table 2. Figure 6 shows that diffracted helium intensity decreases approximately linearly as a function of defect density within the single-vacancy limit. Detailed analysis and discussion of the results in figure 6 can be found in the full manuscript.<sup>6</sup> ![Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS<sup>2</sup> flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>](path) tion can be employed as a lab-based method to quantify point-defect density without specific sample preparation or damage. The method is agnostic to sample chemistry, and thickness, because the mechanism through which it measures defect density is purely geometric, relying upon the degree of order of the sample surface, meaning that the method can be trivially extended to any system whose macroscopic properties are mediated by surface defects or dopants. Examples of current systems include hBN,<sup>24</sup> graphene,<sup>25</sup> doped systems such as diamond,<sup>26</sup> alongside other TMDs. In conjunction with its electrostatically neutral and low energy beam, microscopic helium atom diffraction presents itself as an ideal technique for the investigation of these few-layer materials. Further improvements in the instruments' lateral resolution resolution while performing diffraction measurements, down to the previously reported ∼ 300 nm spot sizes,<sup>2</sup> would allow for sub-micron scale mapping of defect density across samples. The technique is not currently capable of acting as a standalone, stoichimetric measure of defect density, like XPS, once an accurate helium-defect scattering crosssection is calculated this will also be possible. ## Conclusion The findings reported here have demonstrated the effectiveness of helium atom microdiffraction as a powerful, and completely noninvasive characterisation tool for 2D materials. The technique's ability to probe only the outermost atomic layers with micron spatial resolution makes it uniquely suited to addressing the challenges of analysing 2D materials, where established techniques often struggle. The results and discussion have highlighted several key applications of helium atom microdiffraction, including the measurement of surface cleanliness and contamination without altering the sample, a critical feature for ensuring the integrity of materials used in device fabrication. We also investigated the impact of substrate on monolayer properties, confirming that the use of a few-layer hBN buffer can preserve the structural integrity of monolayer MoS2, whereas direct contact with SiO2 results in significant disorder.? We have also explored the potential of helium atom micro-diffraction for determining the thermal expansion coefficient and electronphonon coupling in monolayer MoS2, and have demonstrated its capability for precise structural and thermal analysis. The technique's ability to quantify defect densities without sample preparation or damage also highlights it as an obvious tool for optimizing 2D materials in ever expanding areas of application, from optoelectronics to catalysis and more. Future enhancements to helium atom microdiffraction, such as achieving sub-50 nm spatial resolution, and incorporating out-of-plane scattering capabilities, will further expand its applicability, particularly for complex heterostructures and device-grade materials. In summary, helium atom micro-diffraction promises to provide a robust and adaptable platform for characterisation and development of 2D materials, paving the way for advancements in nanotechnology and material science. ## Experimental Methods ## Helium atom micro-diffraction instrument details In the presented work we used a Scanning Helium Microscope (SHeM) with 5 µm spatial and 7.9 ◦ in-plane angular resolution at the specular condition, representing a factor of 2 improvement in spatial resolution over the first iteration published by von Jeinsen et al. <sup>1</sup> Spatial resolution is defined as the full-width at halfmaximum of the beam spot on the sample at the designed working distance. In-plane angular resolution is given by the angle subtended by the limiting detector aperture and beam spot on the sample, with the correcting factor <sup>√</sup> 1 2 applied to account for the 45◦ instrument geometry. ## Sample details Monolayers of MoS<sup>2</sup> were produced by mechanical exfoliation of a bulk MoS<sup>2</sup> crystal, purchased from 2D Semiconductors Ltd., and were deposited onto few-layer thick hexgonal boron nitride (hBN) which in turn lies on a SiO<sup>2</sup> substrate (high-precision glass microscope slide). Each monolayer measured ≈ 15 µm laterally. The increased defect density monolayer MoS<sup>2</sup> samples, presented in section , were produced the same as previously described in this section with the addition of a thermal annealing process under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere for 0.5 h, following the method as described by Zhu et al.<sup>22</sup> Varying annealing temperature and time allows for control of defect density. Exact annealing parameters, with final defect densities characterised by stoichiometric XPS, are shown in Table 2. # Acknowledgements The work was supported by EPSRC grant EP/R008272/1, Innovate UK/Ionoptika Ltd. through Knowledge Transfer Partnership 10000925. The work was performed in part at CORDE, the Collaborative R&D Environment established to provide access to physics related facilities at the Cavendish Laboratory, University of Cambridge and EPSRC award EP/T00634X/1. SML acknowledges support from EPSRC grant EP/X525686/1. # References - 1. von Jeinsen, N. A.; Lambrick, S. M.; Bergin, M.; Radi´c, A.; Liu, B.; Seremet, D.; Jardine, A. P.; Ward, D. J. 2D Helium Atom Diffraction from a Microscopic Spot. Physical Review Letters 2023, 131, 236202, Publisher: American Physical Society. - 5. Man, M. K. L.; Deckoff-Jones, S.; Winchester, A.; Shi, G.; Gupta, G.; Mohite, A. D.; Kar, S.; Kioupakis, E.; Talapatra, S.; Dani, K. M. 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Nature Materials 2024, # Helium atom micro-diffraction as a characterisation tool for 2D materials - supporting information # Bulk MoS<sup>2</sup> diffraction pattern ![Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure 1*: XRD pattern of dip coated samples of precursor 4	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. Morphology of dip coated samples was investigated with field emission scanning electron microscopy (FESEM). For rough one-layer samples, the layer thickness is that thin that the topography of substrate (FTO) can mostly be seen (see figure 6). Also, the film is not constant but showing cracks and bare FTO substrate at some places (see red circle figure 6 a and b). The samples show two different types of morphology. There is a worm-like the structure of particles (orange rectangles figure 6a and 6c) as well as round shaped particles (green circles/ rectangle figure c). They can be either in separate spaces like figure 28 shows or in between the worm-like structure one can find round shaped particles (see figure 6 c). Further investigation of the fraction of single round shaped and single worm like shaped particles could give information about their relevance to the PEC performance. Particle sizes of round shaped particles are from around 70nm of large particles to 20nm for small ones. Samples that show higher photocurrent density also show round shaped particles that display a bigger size, from 90nm up to 250nm (see figure 7). ![Figure 6: a. SEM Image of hematite film made of iron oleate precursor the green circles and orange rectangles indicate round shaped and worm-like shaped particles, respectively, b. Bare FTO substrate, c. SEM picture is showing worm like shaped (orange rectangle) and round shaped (green box) particles separately.](path) ![Figure7: Round shaped particles with a bigger particle size that can be found with samples of higher PEC performance](path) A second layer deposited on the rough samples leads to a higher thickness of the film. Therefore, the topography of the substrate cannot be seen anymore. Morphology on the surface does not change, still, there are round shaped and also worm-like particles. For spin-coated samples, the morphology is the same as for dip coated samples, consisting of worm-like and also round formed particles, single and even mixed. 3.2.3. Crystallographic properties of hematite thin films prepared from iron salt stearic acid precursor The XRD - patterns reveal the presence of hematite, no other phase could be identified. Compared to the XRD - trends of samples of iron oleate precursor, the intensity of the hematite phase is higher here. The [110] peak shows higher intensity than the [104] peak, which is the average peak of highest intensity for hematite (see figure 8). Therefore, there should be a preferred orientation of crystals along this crystallographic direction. It has also been described elsewhere for hematite films [5]. The calculated crystallite size for one and two-layer samples is not different, around 24 nm to 26 nm (see table 3). ![Figure 8: XRD patterns for dip coated samples of stearic acid precursor](path) #### 3.2.4. Morphology of hematite thin films prepared from iron salt stearic acid precursor Morphology of the samples investigated with SEM shows a very thin layer of worm-like shaped particles on FTO substrate for one-layer samples. There are no round shaped particles, the topography of FTO substrate can be seen (see figure 9a). For two-layer samples, the film thickness increases as the substrate topography are not visible anymore (see figure 9b). Still, there is the only worm like shaped particles that seem to emerge from the bulk. They show an aspect ratio between two and three. ![Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor](path) Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure 3: XRR patterns for samples of precursor 3 (a) and one (b) and two (c) layer samples of precursor 4, the red arrows in the insets indicate the increase of roughness of samples with higher decay.	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. #### 3.2 Crystallographic and morphological properties of nanoparticulate hematite films #### 3.2.1. Crystallographic properties of hematite thin films prepared from iron salt oleic acid precursor Characteristic rough samples of the weight dependent study and two-layer samples of precursor 3 and precursor four were investigated with x-ray diffraction to determine the crystallographic structure and crystallite size. The XRD patterns also show the tin oxide peaks of the substrate beside the peaks of hematite. No other phase could be identified (see figures 4 and 5). Samples with a higher weight of organic film show slightly increased peak intensity and also for two-layer samples the peak intensity is getting bigger. The crystallite sizes for rough one-layer dip coated samples of precursor three do not differ (see table 1). But for two-layer samples, the size increases slightly. On the contrary one - layer dip coated samples of precursor four show differences in crystallite sizes from 25 nm to 34 nm. By deposition of a second layer, the crystallite size also increases. The photocurrent density of samples of precursor 4 is higher with a bigger crystallite size. Furthermore, a one-layer sample with the same crystallite size of a two-layer sample displays the same current density (see table 2). The increase of crystallite size for two-layer samples can be explained with crystallite growth of the first layer due to additional heat treatment. ![Figure 4: XRD pattern of dip coated samples of precursor 3](path) ![Figure 1: XRD pattern of dip coated samples of precursor 4](path) Morphology of dip coated samples was investigated with field emission scanning electron microscopy (FESEM). For rough one-layer samples, the layer thickness is that thin that the topography of substrate (FTO) can mostly be seen (see figure 6). Also, the film is not constant but showing cracks and bare FTO substrate at some places (see red circle figure 6 a and b). The samples show two different types of morphology. There is a worm-like the structure of particles (orange rectangles figure 6a and 6c) as well as round shaped particles (green circles/ rectangle figure c). They can be either in separate spaces like figure 28 shows or in between the worm-like structure one can find round shaped particles (see figure 6 c). Further investigation of the fraction of single round shaped and single worm like shaped particles could give information about their relevance to the PEC performance. Particle sizes of round shaped particles are from around 70nm of large particles to 20nm for small ones. Samples that show higher photocurrent density also show round shaped particles that display a bigger size, from 90nm up to 250nm (see figure 7). ![Figure 6: a. SEM Image of hematite film made of iron oleate precursor the green circles and orange rectangles indicate round shaped and worm-like shaped particles, respectively, b. Bare FTO substrate, c. SEM picture is showing worm like shaped (orange rectangle) and round shaped (green box) particles separately.](path) ![Figure7: Round shaped particles with a bigger particle size that can be found with samples of higher PEC performance](path) A second layer deposited on the rough samples leads to a higher thickness of the film. Therefore, the topography of the substrate cannot be seen anymore. Morphology on the surface does not change, still, there are round shaped and also worm-like particles. For spin-coated samples, the morphology is the same as for dip coated samples, consisting of worm-like and also round formed particles, single and even mixed. 3.2.3. Crystallographic properties of hematite thin films prepared from iron salt stearic acid precursor The XRD - patterns reveal the presence of hematite, no other phase could be identified. Compared to the XRD - trends of samples of iron oleate precursor, the intensity of the hematite phase is higher here. The [110] peak shows higher intensity than the [104] peak, which is the average peak of highest intensity for hematite (see figure 8). Therefore, there should be a preferred orientation of crystals along this crystallographic direction. It has also been described elsewhere for hematite films [5]. The calculated crystallite size for one and two-layer samples is not different, around 24 nm to 26 nm (see table 3). ![Figure 8: XRD patterns for dip coated samples of stearic acid precursor](path) #### 3.2.4. Morphology of hematite thin films prepared from iron salt stearic acid precursor Morphology of the samples investigated with SEM shows a very thin layer of worm-like shaped particles on FTO substrate for one-layer samples. There are no round shaped particles, the topography of FTO substrate can be seen (see figure 9a). For two-layer samples, the film thickness increases as the substrate topography are not visible anymore (see figure 9b). Still, there is the only worm like shaped particles that seem to emerge from the bulk. They show an aspect ratio between two and three. ![Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor](path) Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure 8: XRD patterns for dip coated samples of stearic acid precursor	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. ![Figure1: a) XRR pattern of a smooth hematite film on FTO substrate, b): XRR pattern of a tungsten oxide film on Titania substrate](path) Therefore, the aim is to make a qualitative discussion of the measurements as a higher roughness of the surface increases the decay for angles bigger than the critical angle [1]. Unfortunately, the analysis is found to be sensitive to the adjustment of the slot between the sample and beam knife. Also, different ranges of the roughness of the samples complicate the right alignment. With the bigger size of the slot, the starting intensity of normalized curves is smaller; a shoulder comes up where there is already a decline in intensity for measurements with a lower slot. After this shoulder, the decay starts. However, the slope difference is small for a too big slot and the right size, whereas a too small slot would indicate less roughness because of less decay (see figure 2). For angles 2θ bigger than 1° there is an intensity difference for the measured samples that also should be due to different adjustments of the slot (see figure3). ![Figure 2: Different XRR decays of one sample due to various changes in the aperture between the sample and wedge](path) When comparing the measured curves (see figure 3) the difference in decline could be the difference of roughness of samples, still influenced by the adjustment of the slot. Lines can describe the different decays with the same slope that is taken in the range 0.6°≤2θ≤0.7°. The decays are qualitatively displayed as a function of the measured value Rrms of each sample (see inset figure 3). For each set of samples, a trend of higher decay with bigger Rrms can be estimated; it is indicated with a red arrow in the insets. As scattering is very high, it is still to question if the differences in decay represent the roughness of the samples. Slight variations in the adjustment of the slot and differences in alignment of the sample before a measurement could also be the reason for small measurement differences. Furthermore, it is doubtful if the measurement method is sensitive to such high roughnesses as the samples display. Investigation of samples with differently known roughness's on flat substrates should be used to validate the XRR results. ![Figure 3: XRR patterns for samples of precursor 3 (a) and one (b) and two (c) layer samples of precursor 4, the red arrows in the insets indicate the increase of roughness of samples with higher decay.](path) #### 3.2 Crystallographic and morphological properties of nanoparticulate hematite films #### 3.2.1. Crystallographic properties of hematite thin films prepared from iron salt oleic acid precursor Characteristic rough samples of the weight dependent study and two-layer samples of precursor 3 and precursor four were investigated with x-ray diffraction to determine the crystallographic structure and crystallite size. The XRD patterns also show the tin oxide peaks of the substrate beside the peaks of hematite. No other phase could be identified (see figures 4 and 5). Samples with a higher weight of organic film show slightly increased peak intensity and also for two-layer samples the peak intensity is getting bigger. The crystallite sizes for rough one-layer dip coated samples of precursor three do not differ (see table 1). But for two-layer samples, the size increases slightly. On the contrary one - layer dip coated samples of precursor four show differences in crystallite sizes from 25 nm to 34 nm. By deposition of a second layer, the crystallite size also increases. The photocurrent density of samples of precursor 4 is higher with a bigger crystallite size. Furthermore, a one-layer sample with the same crystallite size of a two-layer sample displays the same current density (see table 2). The increase of crystallite size for two-layer samples can be explained with crystallite growth of the first layer due to additional heat treatment. ![Figure 4: XRD pattern of dip coated samples of precursor 3](path) ![Figure 1: XRD pattern of dip coated samples of precursor 4](path) Morphology of dip coated samples was investigated with field emission scanning electron microscopy (FESEM). For rough one-layer samples, the layer thickness is that thin that the topography of substrate (FTO) can mostly be seen (see figure 6). Also, the film is not constant but showing cracks and bare FTO substrate at some places (see red circle figure 6 a and b). The samples show two different types of morphology. There is a worm-like the structure of particles (orange rectangles figure 6a and 6c) as well as round shaped particles (green circles/ rectangle figure c). They can be either in separate spaces like figure 28 shows or in between the worm-like structure one can find round shaped particles (see figure 6 c). Further investigation of the fraction of single round shaped and single worm like shaped particles could give information about their relevance to the PEC performance. Particle sizes of round shaped particles are from around 70nm of large particles to 20nm for small ones. Samples that show higher photocurrent density also show round shaped particles that display a bigger size, from 90nm up to 250nm (see figure 7). ![Figure 6: a. SEM Image of hematite film made of iron oleate precursor the green circles and orange rectangles indicate round shaped and worm-like shaped particles, respectively, b. Bare FTO substrate, c. SEM picture is showing worm like shaped (orange rectangle) and round shaped (green box) particles separately.](path) ![Figure7: Round shaped particles with a bigger particle size that can be found with samples of higher PEC performance](path) A second layer deposited on the rough samples leads to a higher thickness of the film. Therefore, the topography of the substrate cannot be seen anymore. Morphology on the surface does not change, still, there are round shaped and also worm-like particles. For spin-coated samples, the morphology is the same as for dip coated samples, consisting of worm-like and also round formed particles, single and even mixed. 3.2.3. Crystallographic properties of hematite thin films prepared from iron salt stearic acid precursor The XRD - patterns reveal the presence of hematite, no other phase could be identified. Compared to the XRD - trends of samples of iron oleate precursor, the intensity of the hematite phase is higher here. The [110] peak shows higher intensity than the [104] peak, which is the average peak of highest intensity for hematite (see figure 8). Therefore, there should be a preferred orientation of crystals along this crystallographic direction. It has also been described elsewhere for hematite films [5]. The calculated crystallite size for one and two-layer samples is not different, around 24 nm to 26 nm (see table 3). ![Figure 8: XRD patterns for dip coated samples of stearic acid precursor](path) #### 3.2.4. Morphology of hematite thin films prepared from iron salt stearic acid precursor Morphology of the samples investigated with SEM shows a very thin layer of worm-like shaped particles on FTO substrate for one-layer samples. There are no round shaped particles, the topography of FTO substrate can be seen (see figure 9a). For two-layer samples, the film thickness increases as the substrate topography are not visible anymore (see figure 9b). Still, there is the only worm like shaped particles that seem to emerge from the bulk. They show an aspect ratio between two and three. ![Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor](path) Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure7: Round shaped particles with a bigger particle size that can be found with samples of higher PEC performance	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. A second layer deposited on the rough samples leads to a higher thickness of the film. Therefore, the topography of the substrate cannot be seen anymore. Morphology on the surface does not change, still, there are round shaped and also worm-like particles. For spin-coated samples, the morphology is the same as for dip coated samples, consisting of worm-like and also round formed particles, single and even mixed. 3.2.3. Crystallographic properties of hematite thin films prepared from iron salt stearic acid precursor The XRD - patterns reveal the presence of hematite, no other phase could be identified. Compared to the XRD - trends of samples of iron oleate precursor, the intensity of the hematite phase is higher here. The [110] peak shows higher intensity than the [104] peak, which is the average peak of highest intensity for hematite (see figure 8). Therefore, there should be a preferred orientation of crystals along this crystallographic direction. It has also been described elsewhere for hematite films [5]. The calculated crystallite size for one and two-layer samples is not different, around 24 nm to 26 nm (see table 3). ![Figure 8: XRD patterns for dip coated samples of stearic acid precursor](path) #### 3.2.4. Morphology of hematite thin films prepared from iron salt stearic acid precursor Morphology of the samples investigated with SEM shows a very thin layer of worm-like shaped particles on FTO substrate for one-layer samples. There are no round shaped particles, the topography of FTO substrate can be seen (see figure 9a). For two-layer samples, the film thickness increases as the substrate topography are not visible anymore (see figure 9b). Still, there is the only worm like shaped particles that seem to emerge from the bulk. They show an aspect ratio between two and three. ![Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor](path) Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure 4*: XRD pattern of dip coated samples of precursor 3	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. ![Figure1: a) XRR pattern of a smooth hematite film on FTO substrate, b): XRR pattern of a tungsten oxide film on Titania substrate](path) Therefore, the aim is to make a qualitative discussion of the measurements as a higher roughness of the surface increases the decay for angles bigger than the critical angle [1]. Unfortunately, the analysis is found to be sensitive to the adjustment of the slot between the sample and beam knife. Also, different ranges of the roughness of the samples complicate the right alignment. With the bigger size of the slot, the starting intensity of normalized curves is smaller; a shoulder comes up where there is already a decline in intensity for measurements with a lower slot. After this shoulder, the decay starts. However, the slope difference is small for a too big slot and the right size, whereas a too small slot would indicate less roughness because of less decay (see figure 2). For angles 2θ bigger than 1° there is an intensity difference for the measured samples that also should be due to different adjustments of the slot (see figure3). ![Figure 2: Different XRR decays of one sample due to various changes in the aperture between the sample and wedge](path) When comparing the measured curves (see figure 3) the difference in decline could be the difference of roughness of samples, still influenced by the adjustment of the slot. Lines can describe the different decays with the same slope that is taken in the range 0.6°≤2θ≤0.7°. The decays are qualitatively displayed as a function of the measured value Rrms of each sample (see inset figure 3). For each set of samples, a trend of higher decay with bigger Rrms can be estimated; it is indicated with a red arrow in the insets. As scattering is very high, it is still to question if the differences in decay represent the roughness of the samples. Slight variations in the adjustment of the slot and differences in alignment of the sample before a measurement could also be the reason for small measurement differences. Furthermore, it is doubtful if the measurement method is sensitive to such high roughnesses as the samples display. Investigation of samples with differently known roughness's on flat substrates should be used to validate the XRR results. ![Figure 3: XRR patterns for samples of precursor 3 (a) and one (b) and two (c) layer samples of precursor 4, the red arrows in the insets indicate the increase of roughness of samples with higher decay.](path) #### 3.2 Crystallographic and morphological properties of nanoparticulate hematite films #### 3.2.1. Crystallographic properties of hematite thin films prepared from iron salt oleic acid precursor Characteristic rough samples of the weight dependent study and two-layer samples of precursor 3 and precursor four were investigated with x-ray diffraction to determine the crystallographic structure and crystallite size. The XRD patterns also show the tin oxide peaks of the substrate beside the peaks of hematite. No other phase could be identified (see figures 4 and 5). Samples with a higher weight of organic film show slightly increased peak intensity and also for two-layer samples the peak intensity is getting bigger. The crystallite sizes for rough one-layer dip coated samples of precursor three do not differ (see table 1). But for two-layer samples, the size increases slightly. On the contrary one - layer dip coated samples of precursor four show differences in crystallite sizes from 25 nm to 34 nm. By deposition of a second layer, the crystallite size also increases. The photocurrent density of samples of precursor 4 is higher with a bigger crystallite size. Furthermore, a one-layer sample with the same crystallite size of a two-layer sample displays the same current density (see table 2). The increase of crystallite size for two-layer samples can be explained with crystallite growth of the first layer due to additional heat treatment. ![Figure 4: XRD pattern of dip coated samples of precursor 3](path) ![Figure 1: XRD pattern of dip coated samples of precursor 4](path) Morphology of dip coated samples was investigated with field emission scanning electron microscopy (FESEM). For rough one-layer samples, the layer thickness is that thin that the topography of substrate (FTO) can mostly be seen (see figure 6). Also, the film is not constant but showing cracks and bare FTO substrate at some places (see red circle figure 6 a and b). The samples show two different types of morphology. There is a worm-like the structure of particles (orange rectangles figure 6a and 6c) as well as round shaped particles (green circles/ rectangle figure c). They can be either in separate spaces like figure 28 shows or in between the worm-like structure one can find round shaped particles (see figure 6 c). Further investigation of the fraction of single round shaped and single worm like shaped particles could give information about their relevance to the PEC performance. Particle sizes of round shaped particles are from around 70nm of large particles to 20nm for small ones. Samples that show higher photocurrent density also show round shaped particles that display a bigger size, from 90nm up to 250nm (see figure 7). ![Figure 6: a. SEM Image of hematite film made of iron oleate precursor the green circles and orange rectangles indicate round shaped and worm-like shaped particles, respectively, b. Bare FTO substrate, c. SEM picture is showing worm like shaped (orange rectangle) and round shaped (green box) particles separately.](path) ![Figure7: Round shaped particles with a bigger particle size that can be found with samples of higher PEC performance](path) A second layer deposited on the rough samples leads to a higher thickness of the film. Therefore, the topography of the substrate cannot be seen anymore. Morphology on the surface does not change, still, there are round shaped and also worm-like particles. For spin-coated samples, the morphology is the same as for dip coated samples, consisting of worm-like and also round formed particles, single and even mixed. 3.2.3. Crystallographic properties of hematite thin films prepared from iron salt stearic acid precursor The XRD - patterns reveal the presence of hematite, no other phase could be identified. Compared to the XRD - trends of samples of iron oleate precursor, the intensity of the hematite phase is higher here. The [110] peak shows higher intensity than the [104] peak, which is the average peak of highest intensity for hematite (see figure 8). Therefore, there should be a preferred orientation of crystals along this crystallographic direction. It has also been described elsewhere for hematite films [5]. The calculated crystallite size for one and two-layer samples is not different, around 24 nm to 26 nm (see table 3). ![Figure 8: XRD patterns for dip coated samples of stearic acid precursor](path) #### 3.2.4. Morphology of hematite thin films prepared from iron salt stearic acid precursor Morphology of the samples investigated with SEM shows a very thin layer of worm-like shaped particles on FTO substrate for one-layer samples. There are no round shaped particles, the topography of FTO substrate can be seen (see figure 9a). For two-layer samples, the film thickness increases as the substrate topography are not visible anymore (see figure 9b). Still, there is the only worm like shaped particles that seem to emerge from the bulk. They show an aspect ratio between two and three. ![Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor](path) Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure10: XRD patterns for dip coated samples of lauric acid precursor	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure 6*: a. SEM Image of hematite film made of iron oleate precursor the green circles and orange rectangles indicate round shaped and worm-like shaped particles, respectively, b. Bare FTO substrate, c. SEM picture is showing worm like shaped (orange rectangle) and round shaped (green box) particles separately.	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. ![Figure7: Round shaped particles with a bigger particle size that can be found with samples of higher PEC performance](path) A second layer deposited on the rough samples leads to a higher thickness of the film. Therefore, the topography of the substrate cannot be seen anymore. Morphology on the surface does not change, still, there are round shaped and also worm-like particles. For spin-coated samples, the morphology is the same as for dip coated samples, consisting of worm-like and also round formed particles, single and even mixed. 3.2.3. Crystallographic properties of hematite thin films prepared from iron salt stearic acid precursor The XRD - patterns reveal the presence of hematite, no other phase could be identified. Compared to the XRD - trends of samples of iron oleate precursor, the intensity of the hematite phase is higher here. The [110] peak shows higher intensity than the [104] peak, which is the average peak of highest intensity for hematite (see figure 8). Therefore, there should be a preferred orientation of crystals along this crystallographic direction. It has also been described elsewhere for hematite films [5]. The calculated crystallite size for one and two-layer samples is not different, around 24 nm to 26 nm (see table 3). ![Figure 8: XRD patterns for dip coated samples of stearic acid precursor](path) #### 3.2.4. Morphology of hematite thin films prepared from iron salt stearic acid precursor Morphology of the samples investigated with SEM shows a very thin layer of worm-like shaped particles on FTO substrate for one-layer samples. There are no round shaped particles, the topography of FTO substrate can be seen (see figure 9a). For two-layer samples, the film thickness increases as the substrate topography are not visible anymore (see figure 9b). Still, there is the only worm like shaped particles that seem to emerge from the bulk. They show an aspect ratio between two and three. ![Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor](path) Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. ![Fig. 1 Supercell lattice model of (a)BTO and (b)BZCT (light blue for barium atoms, orange spheres for titanium atoms, purple spheres for cobalt atoms, magenta spheres for zinc atoms, yellow spheres for oxygen atoms)](path) It is evident that doping results in a reduction in the lattice parameter of BZCT, an increase in the c lattice parameter, a slight rise in the c/a ratio, and an enhancement of the tetragonal nature of the lattice. Concurrently, the β angle, which is associated with the Zn atoms, undergoes a slight decline, and the lattice symmetry is diminished. The lattice aberration induced by (Zn, Co) doping is particularly pronounced. This is attributed to the difference in ion size between the dopant elements, Zn and Co, and the substituted ions. Additionally, the cell volume of the doped system is smaller than that of the pure BTO, indicating that the dopantelements induce lattice shrinkage. Following atomic relaxation, the optimized structure exhibits a smaller lattice volume, enhanced tetragonality, and relatively weakened symmetry, which is a contributing factor to the enhanced intrinsic polarization observed in BZCT ferroelectrics. ## 3.2. Electronic properties density of states(DOS), can reflect a multitude of physical properties and bonding characteristics. The study of the electronic energy band structure provides useful information for realizing paraelectricity and ferroelectricity, including semiconductor behavior. In order to gain a deeper understanding of the underlying mechanism of lattice distortion, we selected the 100-plane and the 110-plane, which contain dopant elements in the form of Zn and Co atoms, for a charge density analysis. This is illustrated in Fig. 2, where the upward direction corresponds to the c-axis. As illustrated in Fig. 2, the Ti and Co ions in BZCT exhibit relaxation along the c-axis direction, deviating from the Ti-O facets in comparison to the pure BTO structure. Additionally, as illustrated in Fig. 2(c), there is an overlapping region of electron densities between the Zn ions and the adjacent O ions, indicating a robust interaction between Zn and O. This observation implies the presence of covalent bonding character in the Zn-O linkage. These results in the adjacent O2 atoms of the Zn ions undergoing reverse relaxation along the C-axis, which causes a significant zigzagging of the Ti-O plane in this layer of the O2 ions and an increase in the distortion of the oxygen octahedron. Consequently, this causes an increase in the displacement of Ti atoms from the center of the oxygen octahedron, which accounts for the observed enhancement in the intrinsic polarization of the doped system. Furthermore, Fig. 2(b) illustrates that the electronic charge densities of Ti ions and Co ions overlap with the oxygen ions, forming covalent bonds. This phenomenon is analogous to the bonding observed between Zn ions and corresponding O2 ions, as depicted in Fig. 2(c). Furthermore, the nature of the bonding between the Ti ions and the O2 ions is markedly different, with no overlap of electronic layers between Ti-O2. It can thusbe postulated that the promotion of oxygen octahedral distortion by the Ti-O planar fluctuation of BZCT,caused by Co-O chemical bonding and the formation of covalent bonds between Zn-O, represents the primary factors responsible for the enhancement of ferroelectricity and the improvement of other properties of BZCT. ![Fig. 2 (100)plane charge density plots of(a)BTO,(b)BZCT, and (c) (110)plane charge density plots of BZCT.](path) To study the electronic structure of BTO perovskites and their doping modification changes, the electronic energy band structures of pure BTO and BZCT in the Brillouin zone along the high symmetry direction were calculated. The Fermi energy level was set to zero, which is shown as a gray dashed line in the figure. From Fig. 3(a), it can be seen that the CBM is located at the G-point due to the dominance of the Ti-3d state, while the VBM is located at the G-point at FL(0 eV), which is guided by the O-2p-state. The valence band tops (VBM) and conduction band bottoms (CBM) of the BTOs are located at highly symmetric G-points, indicating that the BTOs are direct bandgap semiconductors with an energy band value of 1.7918 eV, which is in close agreement with the reported values of 1.723 eV[36] and 1.778 eV[37]. Fig. 3(b) shows the electrified energy band structure of BZCT, where it is observed that the valence band top (VBM), predominantly constituted by the Ti-3d state, is situated at the X point, whereas the conduction band bottom (CBM), primarily comprising the O-2p state, is located at the Y point. This suggests that ![Fig. 3 Band structure of (a)BTO,(b)BZCT without Hubbard U and band structure of (c)BTO,(d)BZCT withHubbard U](path) The previous experimental study indicated that the band gap of BTO is approximately 3.2 eV[38]. This discrepancy can be attributed to the fact that the generalized gradient approximation (GGA) methodology employed in the calculations of p-d repulsion for cations and anions, as well as the estimation of band gaps, often results in an underestimation of the latter[39]. The observed trend of decreasing band gaps for BTO is consistent with the typicalunderestimation of density functional theory observed for another perovskite material, SrTiO3[40]. To rectify this discrepancy, we employed the Hubbard+U algorithm to compute the revised electronic energy band structure of pure BTO and BZCT. The outcomes of this calculation are illustrated in Fig. 3(c) and Fig. 3(d). Following the correction by the Hubbard+U algorithm, the electronic energy band gap of pure BTO is 3.21 eV, which is in agreement with other experimental findings [38] ,[41]. The U-added algorithm of BZCT demonstrates that its energy bandwidth bandgap is 2.34 eV, a value that is smaller than that ofpure BTO. The electrons and holes can be excited by lower electron energy, and due to its status as a direct bandgap semiconductor, the material is conducive to carrier migration. Consequently, the BZCT canbe applied to optoelectronic materials. As this paper is concerned with the comparative alterations in properties prior to and following doping, along with the underlying mechanisms, the data will be used without the inclusion of U in the subsequent investigation. ![Fig. 4 The total and partial density of states of(a) BTO-TDOS, (b) BZCT-TDOS, (c) BZCT-PDOS.](path) The total density of states (TDOS) and partial wave density of states (PDOS) for BTO and BCTO were calculated and are presented in Fig. 4. In the case of pure BTO, the energy range for the density of states (DOS) was selected to be between -6 eV and 6 eV. In the valence band (VB), the O-2p state is the primary contributor. In the conduction band (CB) region, the primary contributions are made by the Ti-3d and O-2p states. The hybridization of Ti-3d and O-2p states in the valence band (VB) and conduction band (CB) regions is a key factor contributing to the ferroelectricity observed in pure BTO[42]. Furthermore, the introduction of Zn and Co elements into the BTO cell results in the emergence of a new peak situated in close proximity to the Fermi energy level, specifically at 0 eV. The DOS plots of Fig. 4(b)(c) demonstrate that the primary contributions to this new state are Co-3d and O-2p states. The emergence of this peak results in the Fermi energy level of BCZT being situated in close proximity to the valence band. Concurrently, Co introduces a new electronic state at the base of the conduction band of BTO, thereby reducing the band gap of the energy band of BZCT. The main contributors to the valence band and the conduction band of BCTO remain the Ti-3d and O-2p states, respectively. The impact of Zn doping is primarily manifested in the peak at -6eV in the Zn-3d valence band. In conclusion, the introduction of (Zn,Co) co-doping results in the emergence of new impurity energy levels within the material, leading to a downward shift in the conduction band and a shift in the Fermi energy levels towards the valence band. This phenomenon contributes to a reduction in the energy band gap of the entire system. Furthermore, the strong hybridization between Ti-3d and O-2p orbitals, as well as between Co-4d and O-2p orbitals, suggests that this is the factor responsible for the enhancement of ferroelectricity in the BZCT materials[42]. ## 3.3 Electrical properties Fig. 5 shows the energy difference and polarization function of pure BTO and BSZT, respectively. The energy polarization curve isitted by the phenomenological Landau–Ginzburg–Devonshire theory, and the equation is as follows. where ΔGis the energy difference between the ferroelectric phase and the paraelectric phase, and α, β, γ are coefficient constants. The potential curves for both BTO and BCZT are well-fitted by the image-only Landau equation, respectively. The two minima in the double-trap potential curves correspond to the two stable polarisation states, the P+ state and the P- state,and, for each of these minima,the depth of the trap with respect to P=0 corresponds to the effective barrier for the reversal of the polarization. The P=0 state represents the paraelectric phase, which has no Ti displacement. For both cases of pure BTO and BZCT, the P+ and P- states are identical. It is generally accepted that the magnitude of the double-trap depth is proportional to the magnitude of the ferroelectric polarisation[43]. Therefore, analyzing the variation of the ferroelectric double-well potential depth will help to predict the ferroelectric phase transition and the response to the external electric field. As can be seen from Table 2, the double-well potential depth of BTO is 0.397 meV. The double-potential depth of BZCT is 0.766 meV, which is larger than that of BTO, and we predictthat it is more difficult for BZCT to achieve polarisation switching. This is due to the larger ionic shift and stronger hybridization of Co-3d with O-2p orbitals and Ti-3d with O-2p orbitals. The spontaneous polarisation intensity of BZCT is 32.41 μC/cm<sup>2</sup> , which is higher than that of pure BTO, which is 26.86 μC/cm<sup>2</sup> . The changes in the spontaneous polarisation and the depth of the traps suggest that the spontaneous ferroelectricity of BCZT is higher than that of pure BTO ferroelectric is enhanced. ![Fig. 5 Double-well potentials of (a) pure BTO and (b) BZCT. The red dots are the total energy calculated by DFT and theblue line is from the phenomenological Landau model.](path) In the realm of energy storage devices, superior performance is primarily determined by two critical metrics: enhanced energy storage density and optimal energy storage efficiency. The underpinnings of these attributes are typically reflected in the material's high spontaneous polarization and significant breakdown strength, according to the existing studies. Fig. 6 illustrates the transport properties of four distinct BTO systems, for which we have ascertained the conductivities. These calculations were grounded in the semi-empirical framework of the Boltzmann transport theory. where ƒμ(T,ε) is the Fermi-Dirac distribution function, μ is the chemical potential, and T is the temperature. When a constant relaxation time isknown, it can be shown that the conductivity at RTA and RBA can be obtained for a certain chemical potential and temperature by calculating the energy band structure. Only their conductivities have been discussed qualitatively, so their relaxation time constants have not been estimated here. ![Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT](path) between the pure BTO and doped BTO systems. A comparative analysis of the conductivity and relaxation time between the individual systems is illustrated in figure, where the horizontal coordinates are the maximum and minimum values of the chemical potential with respect to the Fermi energy level, We can see that the (Zn,Co) co-doped BZCT shows lower conductivity than the other three systems in a more stable state, and fewer carriers are transferred in the BZCT; and when the electrons in the system are in a higher energy state, i.e. the BZCT is in an excited state or there is an external energy input, such as sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, promoting carrier migration, which has good potential applications in the field of photovoltaics, When the electrons in the system are in a higher energy state, i.e., the BZCT is in an excited state or there is an external energy input, e.g., sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, which promotes carrier migration, which has good potential applications in the field of photovoltaics. This promotes carrier migration, which has good potential applications in the field of photovoltaics and is also consistentwith the previous results of electronic energy band analysis. Conductivity describes the conductivity of the material, from the overall point of view, the conductivity of BZCT is weaker than the pure BTO system, indicating that the breakdown strength of BZCT is higher than that of pure BTO system, BZCT has a higher spontaneous polarization and higher breakdown strength than that of pure BTO[45], and BZCT is expected to be used in the field of energy storage. ## 3.4 Optical properties In recent years, the effective utilization of solar energy has attracted the attention of many researchers [46, 47]. Ferroelectric photovoltaic materials have been demonstrated to exhibit an excellent photovoltaic effect, with photogenerated voltage not limited by the forbidden bandwidth (band gap) ofthe material itself. Furthermore, the photogenerated current can be regulated by the built-in electric field[48]. Accordingly, we have investigated the impact of modified ferroelectric characteristics on the optical attributes of the materials and explored the prospective applications of perovskite materials in photorefractive, optoelectronic, and solar cell(photovoltaic) domains. ![Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.](path) In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. 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Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. Wang, H. Lu, H. Huyan, S. Lindemann, S. Ryu, H. Lee, T.H. Kim, In-plane quasi-single-domain BaTiO<sup>3</sup> via interfacial symmetry engineering, Nature Communications (2021). [8] Hoffmann, Michael, J., Glaum, Julia, Genenko, Yuri, A., Albe, Karsten, Mechanisms of aging and fatigue in ferroelectrics, Materials Science & Engineering B Solid State Materials for Advanced Technology (2015). [9] H. Shen, K. Xia, P. Wang, R. Tan, The electronic, structural, ferroelectric and optical properties of strontium and zirconium co-doped BaTiO3: First-principles calculations, Solid State Communications (2022). [10] D.S. Fu, S. Hao, J.L. Li, L.S. 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Fig. 3 Band structure of (a)BTO,(b)BZCT without Hubbard U and band structure of (c)BTO,(d)BZCT withHubbard U	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. The previous experimental study indicated that the band gap of BTO is approximately 3.2 eV[38]. This discrepancy can be attributed to the fact that the generalized gradient approximation (GGA) methodology employed in the calculations of p-d repulsion for cations and anions, as well as the estimation of band gaps, often results in an underestimation of the latter[39]. The observed trend of decreasing band gaps for BTO is consistent with the typicalunderestimation of density functional theory observed for another perovskite material, SrTiO3[40]. To rectify this discrepancy, we employed the Hubbard+U algorithm to compute the revised electronic energy band structure of pure BTO and BZCT. The outcomes of this calculation are illustrated in Fig. 3(c) and Fig. 3(d). Following the correction by the Hubbard+U algorithm, the electronic energy band gap of pure BTO is 3.21 eV, which is in agreement with other experimental findings [38] ,[41]. The U-added algorithm of BZCT demonstrates that its energy bandwidth bandgap is 2.34 eV, a value that is smaller than that ofpure BTO. The electrons and holes can be excited by lower electron energy, and due to its status as a direct bandgap semiconductor, the material is conducive to carrier migration. Consequently, the BZCT canbe applied to optoelectronic materials. As this paper is concerned with the comparative alterations in properties prior to and following doping, along with the underlying mechanisms, the data will be used without the inclusion of U in the subsequent investigation. ![Fig. 4 The total and partial density of states of(a) BTO-TDOS, (b) BZCT-TDOS, (c) BZCT-PDOS.](path) The total density of states (TDOS) and partial wave density of states (PDOS) for BTO and BCTO were calculated and are presented in Fig. 4. In the case of pure BTO, the energy range for the density of states (DOS) was selected to be between -6 eV and 6 eV. In the valence band (VB), the O-2p state is the primary contributor. In the conduction band (CB) region, the primary contributions are made by the Ti-3d and O-2p states. The hybridization of Ti-3d and O-2p states in the valence band (VB) and conduction band (CB) regions is a key factor contributing to the ferroelectricity observed in pure BTO[42]. Furthermore, the introduction of Zn and Co elements into the BTO cell results in the emergence of a new peak situated in close proximity to the Fermi energy level, specifically at 0 eV. The DOS plots of Fig. 4(b)(c) demonstrate that the primary contributions to this new state are Co-3d and O-2p states. The emergence of this peak results in the Fermi energy level of BCZT being situated in close proximity to the valence band. Concurrently, Co introduces a new electronic state at the base of the conduction band of BTO, thereby reducing the band gap of the energy band of BZCT. The main contributors to the valence band and the conduction band of BCTO remain the Ti-3d and O-2p states, respectively. The impact of Zn doping is primarily manifested in the peak at -6eV in the Zn-3d valence band. In conclusion, the introduction of (Zn,Co) co-doping results in the emergence of new impurity energy levels within the material, leading to a downward shift in the conduction band and a shift in the Fermi energy levels towards the valence band. This phenomenon contributes to a reduction in the energy band gap of the entire system. Furthermore, the strong hybridization between Ti-3d and O-2p orbitals, as well as between Co-4d and O-2p orbitals, suggests that this is the factor responsible for the enhancement of ferroelectricity in the BZCT materials[42]. ## 3.3 Electrical properties Fig. 5 shows the energy difference and polarization function of pure BTO and BSZT, respectively. The energy polarization curve isitted by the phenomenological Landau–Ginzburg–Devonshire theory, and the equation is as follows. where ΔGis the energy difference between the ferroelectric phase and the paraelectric phase, and α, β, γ are coefficient constants. The potential curves for both BTO and BCZT are well-fitted by the image-only Landau equation, respectively. The two minima in the double-trap potential curves correspond to the two stable polarisation states, the P+ state and the P- state,and, for each of these minima,the depth of the trap with respect to P=0 corresponds to the effective barrier for the reversal of the polarization. The P=0 state represents the paraelectric phase, which has no Ti displacement. For both cases of pure BTO and BZCT, the P+ and P- states are identical. It is generally accepted that the magnitude of the double-trap depth is proportional to the magnitude of the ferroelectric polarisation[43]. Therefore, analyzing the variation of the ferroelectric double-well potential depth will help to predict the ferroelectric phase transition and the response to the external electric field. As can be seen from Table 2, the double-well potential depth of BTO is 0.397 meV. The double-potential depth of BZCT is 0.766 meV, which is larger than that of BTO, and we predictthat it is more difficult for BZCT to achieve polarisation switching. This is due to the larger ionic shift and stronger hybridization of Co-3d with O-2p orbitals and Ti-3d with O-2p orbitals. The spontaneous polarisation intensity of BZCT is 32.41 μC/cm<sup>2</sup> , which is higher than that of pure BTO, which is 26.86 μC/cm<sup>2</sup> . The changes in the spontaneous polarisation and the depth of the traps suggest that the spontaneous ferroelectricity of BCZT is higher than that of pure BTO ferroelectric is enhanced. ![Fig. 5 Double-well potentials of (a) pure BTO and (b) BZCT. The red dots are the total energy calculated by DFT and theblue line is from the phenomenological Landau model.](path) In the realm of energy storage devices, superior performance is primarily determined by two critical metrics: enhanced energy storage density and optimal energy storage efficiency. The underpinnings of these attributes are typically reflected in the material's high spontaneous polarization and significant breakdown strength, according to the existing studies. Fig. 6 illustrates the transport properties of four distinct BTO systems, for which we have ascertained the conductivities. These calculations were grounded in the semi-empirical framework of the Boltzmann transport theory. where ƒμ(T,ε) is the Fermi-Dirac distribution function, μ is the chemical potential, and T is the temperature. When a constant relaxation time isknown, it can be shown that the conductivity at RTA and RBA can be obtained for a certain chemical potential and temperature by calculating the energy band structure. Only their conductivities have been discussed qualitatively, so their relaxation time constants have not been estimated here. ![Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT](path) between the pure BTO and doped BTO systems. A comparative analysis of the conductivity and relaxation time between the individual systems is illustrated in figure, where the horizontal coordinates are the maximum and minimum values of the chemical potential with respect to the Fermi energy level, We can see that the (Zn,Co) co-doped BZCT shows lower conductivity than the other three systems in a more stable state, and fewer carriers are transferred in the BZCT; and when the electrons in the system are in a higher energy state, i.e. the BZCT is in an excited state or there is an external energy input, such as sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, promoting carrier migration, which has good potential applications in the field of photovoltaics, When the electrons in the system are in a higher energy state, i.e., the BZCT is in an excited state or there is an external energy input, e.g., sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, which promotes carrier migration, which has good potential applications in the field of photovoltaics. This promotes carrier migration, which has good potential applications in the field of photovoltaics and is also consistentwith the previous results of electronic energy band analysis. Conductivity describes the conductivity of the material, from the overall point of view, the conductivity of BZCT is weaker than the pure BTO system, indicating that the breakdown strength of BZCT is higher than that of pure BTO system, BZCT has a higher spontaneous polarization and higher breakdown strength than that of pure BTO[45], and BZCT is expected to be used in the field of energy storage. ## 3.4 Optical properties In recent years, the effective utilization of solar energy has attracted the attention of many researchers [46, 47]. Ferroelectric photovoltaic materials have been demonstrated to exhibit an excellent photovoltaic effect, with photogenerated voltage not limited by the forbidden bandwidth (band gap) ofthe material itself. Furthermore, the photogenerated current can be regulated by the built-in electric field[48]. Accordingly, we have investigated the impact of modified ferroelectric characteristics on the optical attributes of the materials and explored the prospective applications of perovskite materials in photorefractive, optoelectronic, and solar cell(photovoltaic) domains. ![Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.](path) In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. 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Fig. 4 The total and partial density of states of(a) BTO-TDOS, (b) BZCT-TDOS, (c) BZCT-PDOS.	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. The total density of states (TDOS) and partial wave density of states (PDOS) for BTO and BCTO were calculated and are presented in Fig. 4. In the case of pure BTO, the energy range for the density of states (DOS) was selected to be between -6 eV and 6 eV. In the valence band (VB), the O-2p state is the primary contributor. In the conduction band (CB) region, the primary contributions are made by the Ti-3d and O-2p states. The hybridization of Ti-3d and O-2p states in the valence band (VB) and conduction band (CB) regions is a key factor contributing to the ferroelectricity observed in pure BTO[42]. Furthermore, the introduction of Zn and Co elements into the BTO cell results in the emergence of a new peak situated in close proximity to the Fermi energy level, specifically at 0 eV. The DOS plots of Fig. 4(b)(c) demonstrate that the primary contributions to this new state are Co-3d and O-2p states. The emergence of this peak results in the Fermi energy level of BCZT being situated in close proximity to the valence band. Concurrently, Co introduces a new electronic state at the base of the conduction band of BTO, thereby reducing the band gap of the energy band of BZCT. The main contributors to the valence band and the conduction band of BCTO remain the Ti-3d and O-2p states, respectively. The impact of Zn doping is primarily manifested in the peak at -6eV in the Zn-3d valence band. In conclusion, the introduction of (Zn,Co) co-doping results in the emergence of new impurity energy levels within the material, leading to a downward shift in the conduction band and a shift in the Fermi energy levels towards the valence band. This phenomenon contributes to a reduction in the energy band gap of the entire system. Furthermore, the strong hybridization between Ti-3d and O-2p orbitals, as well as between Co-4d and O-2p orbitals, suggests that this is the factor responsible for the enhancement of ferroelectricity in the BZCT materials[42]. ## 3.3 Electrical properties Fig. 5 shows the energy difference and polarization function of pure BTO and BSZT, respectively. The energy polarization curve isitted by the phenomenological Landau–Ginzburg–Devonshire theory, and the equation is as follows. where ΔGis the energy difference between the ferroelectric phase and the paraelectric phase, and α, β, γ are coefficient constants. The potential curves for both BTO and BCZT are well-fitted by the image-only Landau equation, respectively. The two minima in the double-trap potential curves correspond to the two stable polarisation states, the P+ state and the P- state,and, for each of these minima,the depth of the trap with respect to P=0 corresponds to the effective barrier for the reversal of the polarization. The P=0 state represents the paraelectric phase, which has no Ti displacement. For both cases of pure BTO and BZCT, the P+ and P- states are identical. It is generally accepted that the magnitude of the double-trap depth is proportional to the magnitude of the ferroelectric polarisation[43]. Therefore, analyzing the variation of the ferroelectric double-well potential depth will help to predict the ferroelectric phase transition and the response to the external electric field. As can be seen from Table 2, the double-well potential depth of BTO is 0.397 meV. The double-potential depth of BZCT is 0.766 meV, which is larger than that of BTO, and we predictthat it is more difficult for BZCT to achieve polarisation switching. This is due to the larger ionic shift and stronger hybridization of Co-3d with O-2p orbitals and Ti-3d with O-2p orbitals. The spontaneous polarisation intensity of BZCT is 32.41 μC/cm<sup>2</sup> , which is higher than that of pure BTO, which is 26.86 μC/cm<sup>2</sup> . The changes in the spontaneous polarisation and the depth of the traps suggest that the spontaneous ferroelectricity of BCZT is higher than that of pure BTO ferroelectric is enhanced. ![Fig. 5 Double-well potentials of (a) pure BTO and (b) BZCT. The red dots are the total energy calculated by DFT and theblue line is from the phenomenological Landau model.](path) In the realm of energy storage devices, superior performance is primarily determined by two critical metrics: enhanced energy storage density and optimal energy storage efficiency. The underpinnings of these attributes are typically reflected in the material's high spontaneous polarization and significant breakdown strength, according to the existing studies. Fig. 6 illustrates the transport properties of four distinct BTO systems, for which we have ascertained the conductivities. These calculations were grounded in the semi-empirical framework of the Boltzmann transport theory. where ƒμ(T,ε) is the Fermi-Dirac distribution function, μ is the chemical potential, and T is the temperature. When a constant relaxation time isknown, it can be shown that the conductivity at RTA and RBA can be obtained for a certain chemical potential and temperature by calculating the energy band structure. Only their conductivities have been discussed qualitatively, so their relaxation time constants have not been estimated here. ![Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT](path) between the pure BTO and doped BTO systems. A comparative analysis of the conductivity and relaxation time between the individual systems is illustrated in figure, where the horizontal coordinates are the maximum and minimum values of the chemical potential with respect to the Fermi energy level, We can see that the (Zn,Co) co-doped BZCT shows lower conductivity than the other three systems in a more stable state, and fewer carriers are transferred in the BZCT; and when the electrons in the system are in a higher energy state, i.e. the BZCT is in an excited state or there is an external energy input, such as sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, promoting carrier migration, which has good potential applications in the field of photovoltaics, When the electrons in the system are in a higher energy state, i.e., the BZCT is in an excited state or there is an external energy input, e.g., sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, which promotes carrier migration, which has good potential applications in the field of photovoltaics. This promotes carrier migration, which has good potential applications in the field of photovoltaics and is also consistentwith the previous results of electronic energy band analysis. Conductivity describes the conductivity of the material, from the overall point of view, the conductivity of BZCT is weaker than the pure BTO system, indicating that the breakdown strength of BZCT is higher than that of pure BTO system, BZCT has a higher spontaneous polarization and higher breakdown strength than that of pure BTO[45], and BZCT is expected to be used in the field of energy storage. ## 3.4 Optical properties In recent years, the effective utilization of solar energy has attracted the attention of many researchers [46, 47]. Ferroelectric photovoltaic materials have been demonstrated to exhibit an excellent photovoltaic effect, with photogenerated voltage not limited by the forbidden bandwidth (band gap) ofthe material itself. Furthermore, the photogenerated current can be regulated by the built-in electric field[48]. Accordingly, we have investigated the impact of modified ferroelectric characteristics on the optical attributes of the materials and explored the prospective applications of perovskite materials in photorefractive, optoelectronic, and solar cell(photovoltaic) domains. ![Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.](path) In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. 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Fig. 5 Double-well potentials of (a) pure BTO and (b) BZCT. The red dots are the total energy calculated by DFT and theblue line is from the phenomenological Landau model.	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. In the realm of energy storage devices, superior performance is primarily determined by two critical metrics: enhanced energy storage density and optimal energy storage efficiency. The underpinnings of these attributes are typically reflected in the material's high spontaneous polarization and significant breakdown strength, according to the existing studies. Fig. 6 illustrates the transport properties of four distinct BTO systems, for which we have ascertained the conductivities. These calculations were grounded in the semi-empirical framework of the Boltzmann transport theory. where ƒμ(T,ε) is the Fermi-Dirac distribution function, μ is the chemical potential, and T is the temperature. When a constant relaxation time isknown, it can be shown that the conductivity at RTA and RBA can be obtained for a certain chemical potential and temperature by calculating the energy band structure. Only their conductivities have been discussed qualitatively, so their relaxation time constants have not been estimated here. ![Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT](path) between the pure BTO and doped BTO systems. A comparative analysis of the conductivity and relaxation time between the individual systems is illustrated in figure, where the horizontal coordinates are the maximum and minimum values of the chemical potential with respect to the Fermi energy level, We can see that the (Zn,Co) co-doped BZCT shows lower conductivity than the other three systems in a more stable state, and fewer carriers are transferred in the BZCT; and when the electrons in the system are in a higher energy state, i.e. the BZCT is in an excited state or there is an external energy input, such as sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, promoting carrier migration, which has good potential applications in the field of photovoltaics, When the electrons in the system are in a higher energy state, i.e., the BZCT is in an excited state or there is an external energy input, e.g., sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, which promotes carrier migration, which has good potential applications in the field of photovoltaics. This promotes carrier migration, which has good potential applications in the field of photovoltaics and is also consistentwith the previous results of electronic energy band analysis. Conductivity describes the conductivity of the material, from the overall point of view, the conductivity of BZCT is weaker than the pure BTO system, indicating that the breakdown strength of BZCT is higher than that of pure BTO system, BZCT has a higher spontaneous polarization and higher breakdown strength than that of pure BTO[45], and BZCT is expected to be used in the field of energy storage. ## 3.4 Optical properties In recent years, the effective utilization of solar energy has attracted the attention of many researchers [46, 47]. Ferroelectric photovoltaic materials have been demonstrated to exhibit an excellent photovoltaic effect, with photogenerated voltage not limited by the forbidden bandwidth (band gap) ofthe material itself. Furthermore, the photogenerated current can be regulated by the built-in electric field[48]. Accordingly, we have investigated the impact of modified ferroelectric characteristics on the optical attributes of the materials and explored the prospective applications of perovskite materials in photorefractive, optoelectronic, and solar cell(photovoltaic) domains. ![Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.](path) In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. 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Fig. 1 Supercell lattice model of (a)BTO and (b)BZCT (light blue for barium atoms, orange spheres for titanium atoms, purple spheres for cobalt atoms, magenta spheres for zinc atoms, yellow spheres for oxygen atoms)	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. It is evident that doping results in a reduction in the lattice parameter of BZCT, an increase in the c lattice parameter, a slight rise in the c/a ratio, and an enhancement of the tetragonal nature of the lattice. Concurrently, the β angle, which is associated with the Zn atoms, undergoes a slight decline, and the lattice symmetry is diminished. The lattice aberration induced by (Zn, Co) doping is particularly pronounced. This is attributed to the difference in ion size between the dopant elements, Zn and Co, and the substituted ions. Additionally, the cell volume of the doped system is smaller than that of the pure BTO, indicating that the dopantelements induce lattice shrinkage. Following atomic relaxation, the optimized structure exhibits a smaller lattice volume, enhanced tetragonality, and relatively weakened symmetry, which is a contributing factor to the enhanced intrinsic polarization observed in BZCT ferroelectrics. ## 3.2. Electronic properties density of states(DOS), can reflect a multitude of physical properties and bonding characteristics. The study of the electronic energy band structure provides useful information for realizing paraelectricity and ferroelectricity, including semiconductor behavior. In order to gain a deeper understanding of the underlying mechanism of lattice distortion, we selected the 100-plane and the 110-plane, which contain dopant elements in the form of Zn and Co atoms, for a charge density analysis. This is illustrated in Fig. 2, where the upward direction corresponds to the c-axis. As illustrated in Fig. 2, the Ti and Co ions in BZCT exhibit relaxation along the c-axis direction, deviating from the Ti-O facets in comparison to the pure BTO structure. Additionally, as illustrated in Fig. 2(c), there is an overlapping region of electron densities between the Zn ions and the adjacent O ions, indicating a robust interaction between Zn and O. This observation implies the presence of covalent bonding character in the Zn-O linkage. These results in the adjacent O2 atoms of the Zn ions undergoing reverse relaxation along the C-axis, which causes a significant zigzagging of the Ti-O plane in this layer of the O2 ions and an increase in the distortion of the oxygen octahedron. Consequently, this causes an increase in the displacement of Ti atoms from the center of the oxygen octahedron, which accounts for the observed enhancement in the intrinsic polarization of the doped system. Furthermore, Fig. 2(b) illustrates that the electronic charge densities of Ti ions and Co ions overlap with the oxygen ions, forming covalent bonds. This phenomenon is analogous to the bonding observed between Zn ions and corresponding O2 ions, as depicted in Fig. 2(c). Furthermore, the nature of the bonding between the Ti ions and the O2 ions is markedly different, with no overlap of electronic layers between Ti-O2. It can thusbe postulated that the promotion of oxygen octahedral distortion by the Ti-O planar fluctuation of BZCT,caused by Co-O chemical bonding and the formation of covalent bonds between Zn-O, represents the primary factors responsible for the enhancement of ferroelectricity and the improvement of other properties of BZCT. ![Fig. 2 (100)plane charge density plots of(a)BTO,(b)BZCT, and (c) (110)plane charge density plots of BZCT.](path) To study the electronic structure of BTO perovskites and their doping modification changes, the electronic energy band structures of pure BTO and BZCT in the Brillouin zone along the high symmetry direction were calculated. The Fermi energy level was set to zero, which is shown as a gray dashed line in the figure. From Fig. 3(a), it can be seen that the CBM is located at the G-point due to the dominance of the Ti-3d state, while the VBM is located at the G-point at FL(0 eV), which is guided by the O-2p-state. The valence band tops (VBM) and conduction band bottoms (CBM) of the BTOs are located at highly symmetric G-points, indicating that the BTOs are direct bandgap semiconductors with an energy band value of 1.7918 eV, which is in close agreement with the reported values of 1.723 eV[36] and 1.778 eV[37]. Fig. 3(b) shows the electrified energy band structure of BZCT, where it is observed that the valence band top (VBM), predominantly constituted by the Ti-3d state, is situated at the X point, whereas the conduction band bottom (CBM), primarily comprising the O-2p state, is located at the Y point. This suggests that ![Fig. 3 Band structure of (a)BTO,(b)BZCT without Hubbard U and band structure of (c)BTO,(d)BZCT withHubbard U](path) The previous experimental study indicated that the band gap of BTO is approximately 3.2 eV[38]. This discrepancy can be attributed to the fact that the generalized gradient approximation (GGA) methodology employed in the calculations of p-d repulsion for cations and anions, as well as the estimation of band gaps, often results in an underestimation of the latter[39]. The observed trend of decreasing band gaps for BTO is consistent with the typicalunderestimation of density functional theory observed for another perovskite material, SrTiO3[40]. To rectify this discrepancy, we employed the Hubbard+U algorithm to compute the revised electronic energy band structure of pure BTO and BZCT. The outcomes of this calculation are illustrated in Fig. 3(c) and Fig. 3(d). Following the correction by the Hubbard+U algorithm, the electronic energy band gap of pure BTO is 3.21 eV, which is in agreement with other experimental findings [38] ,[41]. The U-added algorithm of BZCT demonstrates that its energy bandwidth bandgap is 2.34 eV, a value that is smaller than that ofpure BTO. The electrons and holes can be excited by lower electron energy, and due to its status as a direct bandgap semiconductor, the material is conducive to carrier migration. Consequently, the BZCT canbe applied to optoelectronic materials. As this paper is concerned with the comparative alterations in properties prior to and following doping, along with the underlying mechanisms, the data will be used without the inclusion of U in the subsequent investigation. ![Fig. 4 The total and partial density of states of(a) BTO-TDOS, (b) BZCT-TDOS, (c) BZCT-PDOS.](path) The total density of states (TDOS) and partial wave density of states (PDOS) for BTO and BCTO were calculated and are presented in Fig. 4. In the case of pure BTO, the energy range for the density of states (DOS) was selected to be between -6 eV and 6 eV. In the valence band (VB), the O-2p state is the primary contributor. In the conduction band (CB) region, the primary contributions are made by the Ti-3d and O-2p states. The hybridization of Ti-3d and O-2p states in the valence band (VB) and conduction band (CB) regions is a key factor contributing to the ferroelectricity observed in pure BTO[42]. Furthermore, the introduction of Zn and Co elements into the BTO cell results in the emergence of a new peak situated in close proximity to the Fermi energy level, specifically at 0 eV. The DOS plots of Fig. 4(b)(c) demonstrate that the primary contributions to this new state are Co-3d and O-2p states. The emergence of this peak results in the Fermi energy level of BCZT being situated in close proximity to the valence band. Concurrently, Co introduces a new electronic state at the base of the conduction band of BTO, thereby reducing the band gap of the energy band of BZCT. The main contributors to the valence band and the conduction band of BCTO remain the Ti-3d and O-2p states, respectively. The impact of Zn doping is primarily manifested in the peak at -6eV in the Zn-3d valence band. In conclusion, the introduction of (Zn,Co) co-doping results in the emergence of new impurity energy levels within the material, leading to a downward shift in the conduction band and a shift in the Fermi energy levels towards the valence band. This phenomenon contributes to a reduction in the energy band gap of the entire system. Furthermore, the strong hybridization between Ti-3d and O-2p orbitals, as well as between Co-4d and O-2p orbitals, suggests that this is the factor responsible for the enhancement of ferroelectricity in the BZCT materials[42]. ## 3.3 Electrical properties Fig. 5 shows the energy difference and polarization function of pure BTO and BSZT, respectively. The energy polarization curve isitted by the phenomenological Landau–Ginzburg–Devonshire theory, and the equation is as follows. where ΔGis the energy difference between the ferroelectric phase and the paraelectric phase, and α, β, γ are coefficient constants. The potential curves for both BTO and BCZT are well-fitted by the image-only Landau equation, respectively. The two minima in the double-trap potential curves correspond to the two stable polarisation states, the P+ state and the P- state,and, for each of these minima,the depth of the trap with respect to P=0 corresponds to the effective barrier for the reversal of the polarization. The P=0 state represents the paraelectric phase, which has no Ti displacement. For both cases of pure BTO and BZCT, the P+ and P- states are identical. It is generally accepted that the magnitude of the double-trap depth is proportional to the magnitude of the ferroelectric polarisation[43]. Therefore, analyzing the variation of the ferroelectric double-well potential depth will help to predict the ferroelectric phase transition and the response to the external electric field. As can be seen from Table 2, the double-well potential depth of BTO is 0.397 meV. The double-potential depth of BZCT is 0.766 meV, which is larger than that of BTO, and we predictthat it is more difficult for BZCT to achieve polarisation switching. This is due to the larger ionic shift and stronger hybridization of Co-3d with O-2p orbitals and Ti-3d with O-2p orbitals. The spontaneous polarisation intensity of BZCT is 32.41 μC/cm<sup>2</sup> , which is higher than that of pure BTO, which is 26.86 μC/cm<sup>2</sup> . The changes in the spontaneous polarisation and the depth of the traps suggest that the spontaneous ferroelectricity of BCZT is higher than that of pure BTO ferroelectric is enhanced. ![Fig. 5 Double-well potentials of (a) pure BTO and (b) BZCT. The red dots are the total energy calculated by DFT and theblue line is from the phenomenological Landau model.](path) In the realm of energy storage devices, superior performance is primarily determined by two critical metrics: enhanced energy storage density and optimal energy storage efficiency. The underpinnings of these attributes are typically reflected in the material's high spontaneous polarization and significant breakdown strength, according to the existing studies. Fig. 6 illustrates the transport properties of four distinct BTO systems, for which we have ascertained the conductivities. These calculations were grounded in the semi-empirical framework of the Boltzmann transport theory. where ƒμ(T,ε) is the Fermi-Dirac distribution function, μ is the chemical potential, and T is the temperature. When a constant relaxation time isknown, it can be shown that the conductivity at RTA and RBA can be obtained for a certain chemical potential and temperature by calculating the energy band structure. Only their conductivities have been discussed qualitatively, so their relaxation time constants have not been estimated here. ![Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT](path) between the pure BTO and doped BTO systems. A comparative analysis of the conductivity and relaxation time between the individual systems is illustrated in figure, where the horizontal coordinates are the maximum and minimum values of the chemical potential with respect to the Fermi energy level, We can see that the (Zn,Co) co-doped BZCT shows lower conductivity than the other three systems in a more stable state, and fewer carriers are transferred in the BZCT; and when the electrons in the system are in a higher energy state, i.e. the BZCT is in an excited state or there is an external energy input, such as sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, promoting carrier migration, which has good potential applications in the field of photovoltaics, When the electrons in the system are in a higher energy state, i.e., the BZCT is in an excited state or there is an external energy input, e.g., sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, which promotes carrier migration, which has good potential applications in the field of photovoltaics. This promotes carrier migration, which has good potential applications in the field of photovoltaics and is also consistentwith the previous results of electronic energy band analysis. Conductivity describes the conductivity of the material, from the overall point of view, the conductivity of BZCT is weaker than the pure BTO system, indicating that the breakdown strength of BZCT is higher than that of pure BTO system, BZCT has a higher spontaneous polarization and higher breakdown strength than that of pure BTO[45], and BZCT is expected to be used in the field of energy storage. ## 3.4 Optical properties In recent years, the effective utilization of solar energy has attracted the attention of many researchers [46, 47]. Ferroelectric photovoltaic materials have been demonstrated to exhibit an excellent photovoltaic effect, with photogenerated voltage not limited by the forbidden bandwidth (band gap) ofthe material itself. Furthermore, the photogenerated current can be regulated by the built-in electric field[48]. Accordingly, we have investigated the impact of modified ferroelectric characteristics on the optical attributes of the materials and explored the prospective applications of perovskite materials in photorefractive, optoelectronic, and solar cell(photovoltaic) domains. ![Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.](path) In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. 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Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. Wang, H. Lu, H. Huyan, S. Lindemann, S. Ryu, H. Lee, T.H. Kim, In-plane quasi-single-domain BaTiO<sup>3</sup> via interfacial symmetry engineering, Nature Communications (2021). [8] Hoffmann, Michael, J., Glaum, Julia, Genenko, Yuri, A., Albe, Karsten, Mechanisms of aging and fatigue in ferroelectrics, Materials Science & Engineering B Solid State Materials for Advanced Technology (2015). [9] H. Shen, K. Xia, P. Wang, R. Tan, The electronic, structural, ferroelectric and optical properties of strontium and zirconium co-doped BaTiO3: First-principles calculations, Solid State Communications (2022). [10] D.S. Fu, S. Hao, J.L. Li, L.S. 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Fig. 2 (100)plane charge density plots of(a)BTO,(b)BZCT, and (c) (110)plane charge density plots of BZCT.	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. ![Fig. 1 Supercell lattice model of (a)BTO and (b)BZCT (light blue for barium atoms, orange spheres for titanium atoms, purple spheres for cobalt atoms, magenta spheres for zinc atoms, yellow spheres for oxygen atoms)](path) It is evident that doping results in a reduction in the lattice parameter of BZCT, an increase in the c lattice parameter, a slight rise in the c/a ratio, and an enhancement of the tetragonal nature of the lattice. Concurrently, the β angle, which is associated with the Zn atoms, undergoes a slight decline, and the lattice symmetry is diminished. The lattice aberration induced by (Zn, Co) doping is particularly pronounced. This is attributed to the difference in ion size between the dopant elements, Zn and Co, and the substituted ions. Additionally, the cell volume of the doped system is smaller than that of the pure BTO, indicating that the dopantelements induce lattice shrinkage. Following atomic relaxation, the optimized structure exhibits a smaller lattice volume, enhanced tetragonality, and relatively weakened symmetry, which is a contributing factor to the enhanced intrinsic polarization observed in BZCT ferroelectrics. ## 3.2. Electronic properties density of states(DOS), can reflect a multitude of physical properties and bonding characteristics. The study of the electronic energy band structure provides useful information for realizing paraelectricity and ferroelectricity, including semiconductor behavior. In order to gain a deeper understanding of the underlying mechanism of lattice distortion, we selected the 100-plane and the 110-plane, which contain dopant elements in the form of Zn and Co atoms, for a charge density analysis. This is illustrated in Fig. 2, where the upward direction corresponds to the c-axis. As illustrated in Fig. 2, the Ti and Co ions in BZCT exhibit relaxation along the c-axis direction, deviating from the Ti-O facets in comparison to the pure BTO structure. Additionally, as illustrated in Fig. 2(c), there is an overlapping region of electron densities between the Zn ions and the adjacent O ions, indicating a robust interaction between Zn and O. This observation implies the presence of covalent bonding character in the Zn-O linkage. These results in the adjacent O2 atoms of the Zn ions undergoing reverse relaxation along the C-axis, which causes a significant zigzagging of the Ti-O plane in this layer of the O2 ions and an increase in the distortion of the oxygen octahedron. Consequently, this causes an increase in the displacement of Ti atoms from the center of the oxygen octahedron, which accounts for the observed enhancement in the intrinsic polarization of the doped system. Furthermore, Fig. 2(b) illustrates that the electronic charge densities of Ti ions and Co ions overlap with the oxygen ions, forming covalent bonds. This phenomenon is analogous to the bonding observed between Zn ions and corresponding O2 ions, as depicted in Fig. 2(c). Furthermore, the nature of the bonding between the Ti ions and the O2 ions is markedly different, with no overlap of electronic layers between Ti-O2. It can thusbe postulated that the promotion of oxygen octahedral distortion by the Ti-O planar fluctuation of BZCT,caused by Co-O chemical bonding and the formation of covalent bonds between Zn-O, represents the primary factors responsible for the enhancement of ferroelectricity and the improvement of other properties of BZCT. ![Fig. 2 (100)plane charge density plots of(a)BTO,(b)BZCT, and (c) (110)plane charge density plots of BZCT.](path) To study the electronic structure of BTO perovskites and their doping modification changes, the electronic energy band structures of pure BTO and BZCT in the Brillouin zone along the high symmetry direction were calculated. The Fermi energy level was set to zero, which is shown as a gray dashed line in the figure. From Fig. 3(a), it can be seen that the CBM is located at the G-point due to the dominance of the Ti-3d state, while the VBM is located at the G-point at FL(0 eV), which is guided by the O-2p-state. The valence band tops (VBM) and conduction band bottoms (CBM) of the BTOs are located at highly symmetric G-points, indicating that the BTOs are direct bandgap semiconductors with an energy band value of 1.7918 eV, which is in close agreement with the reported values of 1.723 eV[36] and 1.778 eV[37]. Fig. 3(b) shows the electrified energy band structure of BZCT, where it is observed that the valence band top (VBM), predominantly constituted by the Ti-3d state, is situated at the X point, whereas the conduction band bottom (CBM), primarily comprising the O-2p state, is located at the Y point. This suggests that ![Fig. 3 Band structure of (a)BTO,(b)BZCT without Hubbard U and band structure of (c)BTO,(d)BZCT withHubbard U](path) The previous experimental study indicated that the band gap of BTO is approximately 3.2 eV[38]. This discrepancy can be attributed to the fact that the generalized gradient approximation (GGA) methodology employed in the calculations of p-d repulsion for cations and anions, as well as the estimation of band gaps, often results in an underestimation of the latter[39]. The observed trend of decreasing band gaps for BTO is consistent with the typicalunderestimation of density functional theory observed for another perovskite material, SrTiO3[40]. To rectify this discrepancy, we employed the Hubbard+U algorithm to compute the revised electronic energy band structure of pure BTO and BZCT. The outcomes of this calculation are illustrated in Fig. 3(c) and Fig. 3(d). Following the correction by the Hubbard+U algorithm, the electronic energy band gap of pure BTO is 3.21 eV, which is in agreement with other experimental findings [38] ,[41]. The U-added algorithm of BZCT demonstrates that its energy bandwidth bandgap is 2.34 eV, a value that is smaller than that ofpure BTO. The electrons and holes can be excited by lower electron energy, and due to its status as a direct bandgap semiconductor, the material is conducive to carrier migration. Consequently, the BZCT canbe applied to optoelectronic materials. As this paper is concerned with the comparative alterations in properties prior to and following doping, along with the underlying mechanisms, the data will be used without the inclusion of U in the subsequent investigation. ![Fig. 4 The total and partial density of states of(a) BTO-TDOS, (b) BZCT-TDOS, (c) BZCT-PDOS.](path) The total density of states (TDOS) and partial wave density of states (PDOS) for BTO and BCTO were calculated and are presented in Fig. 4. In the case of pure BTO, the energy range for the density of states (DOS) was selected to be between -6 eV and 6 eV. In the valence band (VB), the O-2p state is the primary contributor. In the conduction band (CB) region, the primary contributions are made by the Ti-3d and O-2p states. The hybridization of Ti-3d and O-2p states in the valence band (VB) and conduction band (CB) regions is a key factor contributing to the ferroelectricity observed in pure BTO[42]. Furthermore, the introduction of Zn and Co elements into the BTO cell results in the emergence of a new peak situated in close proximity to the Fermi energy level, specifically at 0 eV. The DOS plots of Fig. 4(b)(c) demonstrate that the primary contributions to this new state are Co-3d and O-2p states. The emergence of this peak results in the Fermi energy level of BCZT being situated in close proximity to the valence band. Concurrently, Co introduces a new electronic state at the base of the conduction band of BTO, thereby reducing the band gap of the energy band of BZCT. The main contributors to the valence band and the conduction band of BCTO remain the Ti-3d and O-2p states, respectively. The impact of Zn doping is primarily manifested in the peak at -6eV in the Zn-3d valence band. In conclusion, the introduction of (Zn,Co) co-doping results in the emergence of new impurity energy levels within the material, leading to a downward shift in the conduction band and a shift in the Fermi energy levels towards the valence band. This phenomenon contributes to a reduction in the energy band gap of the entire system. Furthermore, the strong hybridization between Ti-3d and O-2p orbitals, as well as between Co-4d and O-2p orbitals, suggests that this is the factor responsible for the enhancement of ferroelectricity in the BZCT materials[42]. ## 3.3 Electrical properties Fig. 5 shows the energy difference and polarization function of pure BTO and BSZT, respectively. The energy polarization curve isitted by the phenomenological Landau–Ginzburg–Devonshire theory, and the equation is as follows. where ΔGis the energy difference between the ferroelectric phase and the paraelectric phase, and α, β, γ are coefficient constants. The potential curves for both BTO and BCZT are well-fitted by the image-only Landau equation, respectively. The two minima in the double-trap potential curves correspond to the two stable polarisation states, the P+ state and the P- state,and, for each of these minima,the depth of the trap with respect to P=0 corresponds to the effective barrier for the reversal of the polarization. The P=0 state represents the paraelectric phase, which has no Ti displacement. For both cases of pure BTO and BZCT, the P+ and P- states are identical. It is generally accepted that the magnitude of the double-trap depth is proportional to the magnitude of the ferroelectric polarisation[43]. Therefore, analyzing the variation of the ferroelectric double-well potential depth will help to predict the ferroelectric phase transition and the response to the external electric field. As can be seen from Table 2, the double-well potential depth of BTO is 0.397 meV. The double-potential depth of BZCT is 0.766 meV, which is larger than that of BTO, and we predictthat it is more difficult for BZCT to achieve polarisation switching. This is due to the larger ionic shift and stronger hybridization of Co-3d with O-2p orbitals and Ti-3d with O-2p orbitals. The spontaneous polarisation intensity of BZCT is 32.41 μC/cm<sup>2</sup> , which is higher than that of pure BTO, which is 26.86 μC/cm<sup>2</sup> . The changes in the spontaneous polarisation and the depth of the traps suggest that the spontaneous ferroelectricity of BCZT is higher than that of pure BTO ferroelectric is enhanced. ![Fig. 5 Double-well potentials of (a) pure BTO and (b) BZCT. The red dots are the total energy calculated by DFT and theblue line is from the phenomenological Landau model.](path) In the realm of energy storage devices, superior performance is primarily determined by two critical metrics: enhanced energy storage density and optimal energy storage efficiency. The underpinnings of these attributes are typically reflected in the material's high spontaneous polarization and significant breakdown strength, according to the existing studies. Fig. 6 illustrates the transport properties of four distinct BTO systems, for which we have ascertained the conductivities. These calculations were grounded in the semi-empirical framework of the Boltzmann transport theory. where ƒμ(T,ε) is the Fermi-Dirac distribution function, μ is the chemical potential, and T is the temperature. When a constant relaxation time isknown, it can be shown that the conductivity at RTA and RBA can be obtained for a certain chemical potential and temperature by calculating the energy band structure. Only their conductivities have been discussed qualitatively, so their relaxation time constants have not been estimated here. ![Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT](path) between the pure BTO and doped BTO systems. A comparative analysis of the conductivity and relaxation time between the individual systems is illustrated in figure, where the horizontal coordinates are the maximum and minimum values of the chemical potential with respect to the Fermi energy level, We can see that the (Zn,Co) co-doped BZCT shows lower conductivity than the other three systems in a more stable state, and fewer carriers are transferred in the BZCT; and when the electrons in the system are in a higher energy state, i.e. the BZCT is in an excited state or there is an external energy input, such as sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, promoting carrier migration, which has good potential applications in the field of photovoltaics, When the electrons in the system are in a higher energy state, i.e., the BZCT is in an excited state or there is an external energy input, e.g., sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, which promotes carrier migration, which has good potential applications in the field of photovoltaics. This promotes carrier migration, which has good potential applications in the field of photovoltaics and is also consistentwith the previous results of electronic energy band analysis. Conductivity describes the conductivity of the material, from the overall point of view, the conductivity of BZCT is weaker than the pure BTO system, indicating that the breakdown strength of BZCT is higher than that of pure BTO system, BZCT has a higher spontaneous polarization and higher breakdown strength than that of pure BTO[45], and BZCT is expected to be used in the field of energy storage. ## 3.4 Optical properties In recent years, the effective utilization of solar energy has attracted the attention of many researchers [46, 47]. Ferroelectric photovoltaic materials have been demonstrated to exhibit an excellent photovoltaic effect, with photogenerated voltage not limited by the forbidden bandwidth (band gap) ofthe material itself. Furthermore, the photogenerated current can be regulated by the built-in electric field[48]. Accordingly, we have investigated the impact of modified ferroelectric characteristics on the optical attributes of the materials and explored the prospective applications of perovskite materials in photorefractive, optoelectronic, and solar cell(photovoltaic) domains. ![Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.](path) In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. 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Fig. 1 The schem matic view o of cg-N cry ystal structu ure.	# One Pot Synthesis of Cubic Gauche Polymeric Nitrogen ### Abstract The long sought cubic gauche polymeric nitrogen (cg-N) consisting of N-N single bonds has been synthesized by a simple route using sodium azide as a precursor at ambient conditions. The recrystallization process was designed to expose crystal faces with low activation energy that facilitates initiating the polymeric reaction at ambient conditions. The azide was considered as a precursor due to the low energy barrier in transforming double bonded N=N to single bonded cg-N. Raman spectrum measurements detected the emerging vibron peaks at 635 cm-1 for the polymerized sodium azide samples, demonstrating the formation of cg-N with N-N single bonds. Different from traditional high pressure technique and recently developed plasma enhanced chemical vapor deposition method, the route achieves the quantitative synthesis of cg-N at ambient conditions. The simple method to synthesize cg-N offers potential for further scale up production as well as practical applications of polymeric nitrogen based materials as high energy density materials. ### Introduction The N≡N triple bond is one of the strongest chemical bonds, with a very high bond energy of 954 kJ/mol [1] , while the N-N single bond is much weaker, with a bond energy of 160 kJ/mol [2, 3] . Due to the huge difference in energy between triply bonded dinitrogen and singly bonded nitrogen, the polymerized nitrogen materials with N-N single bonds are proposed as high energy density materials and expected to release a high amount of chemical energy while transforming into a triple bonded dinitrogen molecules. It was anticipated that the greatest utility of fully single-bonded nitrogen would produce a tenfold improvement in detonation pressure over HMX [4] and therefore polymerized nitrogen is highly sought as a high energy density material. It was predicted in 1985 that molecular nitrogen would polymerize into atomic solid at high pressure [5] . Later, the cubic gauche (cg-N) lattice with a single bonded diamond like structure was proposed[6] . Subsequently, large amounts of theoretical work reported on the high pressure approach to polymerized nitrogen with N-N single bonded forms [7- 23] . It was not until 2004 that Eremets et. al. prepared cg-N directly from molecular nitrogen at pressures above 110 GPa (1GPa≈10,000 atmospheric pressure) and temperatures above 2000 K using laser heated diamond cell technique [24] . The cubic gauche structure was confirmed by growing the single crystal in 2007[25] . However synthesis pressures above 110 GPa are highly required. Later on polymerized nitrogen with layered structure (LP-N) [26] , hexagonal layered structure (HLP-N) [27] , black phosphorus nitrogen (BP-N) [28] , and Panda nitrogen [29] were successfully prepared at 120 GPa, 244 GPa, 146 GPa & 161 GPa, respectively. Nevertheless, the high pressure is commonly regarded as a necessary route to synthesize polymerized nitrogen. All these polymerized nitrogen samples decomposed in the process of releasing pressure. Some strategies, such as chemical doping, were attempted at high pressure by introducing metals in order to stabilize the polymerized nitrogen lattice at lower pressure [30-34] but it remains challenging to recover the polymerized nitrogen samples at ambient pressure. In addition, introducing non-energetic element, especially the heavy metal seriously decreases energetic density of polymerized nitrogen materials, detrimental to the natural applications. In the past few years, plasma enhanced chemical vapor deposition (PECVD) technique has been developed to take advantage of the non-equilibrium plasma environment to synthesize polymerized nitrogen materials. The relative thermodynamic instability of polymerized nitrogen can be overcome by kinetic stability provided through energy band hybridization by a charge transfer mechanism, which makes the high energy lattice gain stability and avoid the highly energetic conversion to molecular N2 [35] . The cg-N and N8 cluster were successfully prepared with or without the assistance of carbon nanotube using sodium azide as precursor [35-41] . However the plasma generally can only penetrate several nanometers into the solid surface [42] , leading to limited quantity of cg-N. Besides, high energy plasma particles also could dissociate the cg-N produced in the experiment, which is backed up by the fact that the optimized deposition time is demanded for a very small amount of cg-N sample prepared by the PECVD method [38] . Theory proposes that the transition path to polymerized nitrogen might pass through different molecular structures with limited thermodynamic stability fields [43] . Additionally earlier experiments aimed at synthesizing polymerized nitrogen only reached the amorphous state at ambient temperature compression, indicating that increasing reaction temperature to activate the precursor might play a crucial role in the formation of the atomic structures [44, 45] . For example LiN5 and K2N6 were synthesized at lower pressures [30, 32, 33] with the assistance of laser heating. We here for the first time report a simple route to synthesize cg-N by using sodium azide as a precursor. The cg-N was successfully obtained through recrystallization reaction, followed by transforming azide ions with N=N double bonds into single bonded cg-N that can stabilize at ambient pressure conditions. To date it is the simplest synthesis route to realize cg-N that can be further developed for scale up production and potential practical applications. #### Materials & Methods High pure sodium azide (NaN3) (≥99.9%) was chosen as nitrogen source to synthesize cg-N. The recrystallization process was performed first. A solution of NaN3 (2 mol/L) was pre-prepared and dropped into a crucible. Then the crucible was transferred into a tubular furnace. The vacuum operation was performed to realize the recrystallization of the NaN3 powder. The reaction temperatures and times within the 200- 300°C and up to 10 hours were tried in the synthesis processes. The optimized synthesis condition with reaction temperature 240 ~ 260 <sup>o</sup> C and heating time 5 hours was employed to synthesize the cg-N. The sample was named as polymerized sodium azide followed by synthesis temperature, time, such as PSA-240o C-5h. According to aforementioned conditions, heating NaN3 without recrystallization process to prepare cg-N was tried and it was named as unpolymerized sodium azide (UPSA). The Raman spectra were recorded in range of 100~1500 cm-1 with a spectral resolu micro laser ( ution of 1 cm scope and m (λ0 =532 nm m-<sup>1</sup> using a multichanne m). an integrated el air cooled d laser Ram d CCD dete man system ector. The e (Renishaw excitation so w) with a co ource is an onfocal Ar ion #### Resul lts ![#### Fig. 1 The schem matic view o of cg-N cry ystal structu ure.](path) T I213 s be 1.4 the cry with N lone p diamo with o of Fig spectr metast The cg-N w space group 40 Å and th ystal structu N-N single pair electro ond (left in F others formi g. 1. The t rum calcula table at amb was proposed p with a latt he bond ang ure of cg-N, bonds at eq ons. Thus c Fig.1). The ing a three theoretical w ation [12] , in bient pressu d to adopt a ice constan le is 114.0° , in which e qual distanc cg-N exhib single bond dimensiona work on cg ndicating th ure. a body cent nt of 3.773 Å ° at ambient each nitroge ces and the its a near ded nitrogen al network s g-N didn't hat the enti tered cubic Å. The N-N t pressure c en atom bon e remaining tetrahedral n atoms form structure [47] show imag ire covalent Bravais lat N bond leng conditions [6 nds to three p orbital is structure s m fused ring ], as shown ginary freq t network o ttice, belong gth is predic 6, 46] . Fig. 1 nearest nei s occupied similar to t gs, which c in the right quency in p of cg-struc ging to cted to shows ghbors by the that in onnect t panel phonon ture is T precur single The r activa date th to obt charac The azide i rsor for syn e bonded cg recrystalliza ation energy he amount o tain ideal X cterize the c ion is comp nthesizing th g-N. In this ation proce y. Then the of cg-N obt X-ray diffrac chemical bo posed of N he cg-N du work the so ess is expe polymeric tained exper ction pattern ond transfor N=N doubl ue to the low odium azide ected to ex reaction ca rimentally a ns. Therefo rmation in t e bonds th w energy b e is used in xposing so an be initiat at ambient p ore the Ram the present hat can be barriers for n present sim me crystal ed at ambie pressure is man spectrum work that i used as an transformin mple experi l faces wit ent conditio still not suf m was adop is commonl n ideal ng into iments. th low ons. To fficient pted to ly used ![Fig. 2 The Raman spectra of polymerized sodium azide (PSA) synthesized at 200 ~300 <sup>o</sup> C with a reaction time of 3 hours, where the fingerprint peak at 635 cm-1 for cg-N is detected. The vibron intensity of cg-N shows optimized synthesis temperature between 240~260 <sup>o</sup> C.](path) Fig. 2 shows the Raman spectra of polymerized NaN3 (PSA) synthesized at 200~300 o C with a reaction time of 3 hours. The sharp peaks of NaN3 near 120, 1267 & 1358 cm-1 are assigned to the vibrational lattice mode, the first overtone of the IR active bending ν2 mode and the symmetric stretching ν1 mode of the azide ion, respectively, which matches well with the Raman spectra of free standing NaN3 [41] . For comparison the NaN3 raw material and a sample heated only at 250 <sup>o</sup> C for 3 hours without the recrystallization process were characterized and showed same Raman vibron profiles without any signals at 635 cm-1 (see Fig. S1). The lines at 635 cm-1 for polymerized sodium nitrogen sample (PSA) are comparable to the theoretical results extrapolated to ambient pressure because the Raman modes tend to soften with decreasing pressure [24, 46, 48] . It also agrees with our recent work on polymerized cg-N based on potassium azide. Therefore the emergent intense vibron at 635 cm-<sup>1</sup> is assigned to the pore breathing A symmetry and is regarded as the finger print peak of cg-N, which unambiguously indicates the successful synthesis of cg-N [46] . The synthesis conditions of cg-N were first systematically investigated by controlling the reaction temperature with a fixed reaction time of 3 hours. As shown in Fig. 2, the vibron intensity of cg-N increases as the reaction temperature raises from 200 <sup>o</sup> C to 240 <sup>o</sup> C compared to that at 1358 cm-<sup>1</sup> for the unreacted or remnant NaN3 in the PSA samples. As temperature increases up to 270 <sup>o</sup> C, the Raman intensity of cg-N remains comprehensive but decreases clearly at 300o C. Additionally, it is worth noting that the intensity of vibrational lattice mode at 120 cm-1 for the unreacted NaN3 in PSA-240 sample dramatically decreases compared to those prepared at other temperatures. The PSA sample exhibits uniformly dark blue color and should be related to the by product formed during the synthesis of cg-N and will be discussed below. Therefore, the optimized temperature range of 240~260 <sup>o</sup> C was adopted for cg-N preparation. A quantitative comparison was made by calculating the intensity ratios of the characteristic Raman peaks to measure the abundance of cg-N phase in the PSA samples. Table I summarizes the partial line intensities of NaN3 and cg-N. The amount of cg-N synthesized is reflected by the peak intensity at 635 cm-1 , while those at 120 and 1358 cm-<sup>1</sup> represent the unreacted NaN3. Herein, a simplified conversion degree from NaN3 to cg-N is defined by the peak intensity ratio. The intensities of NaN3 at 120 cm-1 and 1358 cm-1 , calculated by (I200-Ibg)/(I1358-Ibg) for the NaN3 raw material (SA-R), NaN3 heated only (UPSA) and polymerized NaN3 (PSA) samples at different temperatures are between 2 and 3 and remain little changed. For the SA-R and UPSA samples no trace of cg-N is detected. After recrystallization process the vibron intensity of cg-N enhances dramatically for PSA samples and the ratio of cg-N to NaN3 is 1.26-1.39 calculated by (I635-Ibg)/(I1358-Ibg), and 0.42~ 0.60 calculated by (I635-Ibg)/(I120-Ibg). Therefore it indicates that the crystallization process strongly promotes the formation of cg-N and the optimized synthesis temperature range is 240~260 <sup>o</sup> C. ![Fig. 3 The reaction time dependence of the conversion ratios of cg-N to NaN3 for the PSA samples prepared at 240, 250 & 260 <sup>o</sup> C. The optimized synthesis conditions for the PSA samples are 240~260 <sup>o</sup> C with a reaction time of 5 hours.](path) Further the preparation conditions were explored by synthesizing PSA samples at 240~260 <sup>o</sup> C with variable reaction time. The Raman vibron intensity of cg-N in the PSA-240 samples increases gradually as the reaction time extends up to 5 hours and then decreases, which is shown in Fig. S2. Samples prepared at 250 and 260 <sup>o</sup> C show the same results (Fig. S3). For comparison, the intensity ratios of the cg-N to unreacted NaN3 in theses PSA samples were calculated in Table SI. The synthesis time dependence of the ratios of cg-N to NaN3 at 240-260 <sup>o</sup> C was plotted in Fig. 3. All samples show consistent trends: the PSA samples with a reaction time of 5 hours show the highest conversion degree. Therefore, the optimized preparation conditions for polymerized sodium azide samples are a temperature range of 240~260 <sup>o</sup> C and a reaction time of 5 hours. #### Discussion been proposed, which suffer from the challenge that samples could not recover to ambient pressure and enhancing yield is difficult. Here a simple route was designed for synthesizing cg-N under atmospheric conditions considering the thermodynamic barriers related to transforming to N-N single bonds and activation energy of precursor. In our experiments it was found that recrystallization process played a crucial role to form cg-N. For azides the crystal face with low activation energy may facilitate initiating the transformation of the double bonds N=N in azide ions to single bonded cg-N. The optimized temperatures for cg-N preparation by NaN3 are primarily restricted by the melting points NaN3 (~275°C). These results align well with the fact that the cg-N forms on the crystal face with lower activation energy of azides precursors oriented by the recrystallization process. Regarding the formation mechanism of cg-N it was observed that the polymerized sodium azide turned dark blue in our experiments that should be related to the formation of by-product Na3N during the synthesis of cg-N. This is consistent with the reported results that pH value of the solution increases after synthesis reaction as Na3N displays alkalinity when dissolved in water [38] . A more detailed conversion mechanism is currently at investigation. ### Conclusion The polymeric nitrogen with cubic gauche structure was successfully synthesized by a simple route using sodium azide as a precursor. A high yield may be obtained by separating the by products formed on the surface of the produced cg-N. The route is promising for further scale up or applied to synthesize other metastable materials. #### Conflict of Interest There are no financial conflicts of interest to disclose. #### Acknowledgments The work was supported by the National Natural Science Foundation of China, Ministry of Science & Technology, and CAS Project for Young Scientists in Basic Research of China. ## Author Contributions Changqing Jin & Jun Zhang designed and supervised the whole research. Runteng Chen, Jun Zhang, Zelong Wang, Ke Lu, Yi Peng, Jianfa Zhao, Shaomin Feng and Changqing Jin performed the experiments. Jun Zhang & Changqing Jin wrote the manuscript with contributions from all authors. #### References
Fig. 2 The Raman spectra of polymerized sodium azide (PSA) synthesized at 200 ~300 <sup>o</sup> C with a reaction time of 3 hours, where the fingerprint peak at 635 cm-1 for cg-N is detected. The vibron intensity of cg-N shows optimized synthesis temperature between 240~260 <sup>o</sup> C.	# One Pot Synthesis of Cubic Gauche Polymeric Nitrogen ### Abstract The long sought cubic gauche polymeric nitrogen (cg-N) consisting of N-N single bonds has been synthesized by a simple route using sodium azide as a precursor at ambient conditions. The recrystallization process was designed to expose crystal faces with low activation energy that facilitates initiating the polymeric reaction at ambient conditions. The azide was considered as a precursor due to the low energy barrier in transforming double bonded N=N to single bonded cg-N. Raman spectrum measurements detected the emerging vibron peaks at 635 cm-1 for the polymerized sodium azide samples, demonstrating the formation of cg-N with N-N single bonds. Different from traditional high pressure technique and recently developed plasma enhanced chemical vapor deposition method, the route achieves the quantitative synthesis of cg-N at ambient conditions. The simple method to synthesize cg-N offers potential for further scale up production as well as practical applications of polymeric nitrogen based materials as high energy density materials. ### Introduction The N≡N triple bond is one of the strongest chemical bonds, with a very high bond energy of 954 kJ/mol [1] , while the N-N single bond is much weaker, with a bond energy of 160 kJ/mol [2, 3] . Due to the huge difference in energy between triply bonded dinitrogen and singly bonded nitrogen, the polymerized nitrogen materials with N-N single bonds are proposed as high energy density materials and expected to release a high amount of chemical energy while transforming into a triple bonded dinitrogen molecules. It was anticipated that the greatest utility of fully single-bonded nitrogen would produce a tenfold improvement in detonation pressure over HMX [4] and therefore polymerized nitrogen is highly sought as a high energy density material. It was predicted in 1985 that molecular nitrogen would polymerize into atomic solid at high pressure [5] . Later, the cubic gauche (cg-N) lattice with a single bonded diamond like structure was proposed[6] . Subsequently, large amounts of theoretical work reported on the high pressure approach to polymerized nitrogen with N-N single bonded forms [7- 23] . It was not until 2004 that Eremets et. al. prepared cg-N directly from molecular nitrogen at pressures above 110 GPa (1GPa≈10,000 atmospheric pressure) and temperatures above 2000 K using laser heated diamond cell technique [24] . The cubic gauche structure was confirmed by growing the single crystal in 2007[25] . However synthesis pressures above 110 GPa are highly required. Later on polymerized nitrogen with layered structure (LP-N) [26] , hexagonal layered structure (HLP-N) [27] , black phosphorus nitrogen (BP-N) [28] , and Panda nitrogen [29] were successfully prepared at 120 GPa, 244 GPa, 146 GPa & 161 GPa, respectively. Nevertheless, the high pressure is commonly regarded as a necessary route to synthesize polymerized nitrogen. All these polymerized nitrogen samples decomposed in the process of releasing pressure. Some strategies, such as chemical doping, were attempted at high pressure by introducing metals in order to stabilize the polymerized nitrogen lattice at lower pressure [30-34] but it remains challenging to recover the polymerized nitrogen samples at ambient pressure. In addition, introducing non-energetic element, especially the heavy metal seriously decreases energetic density of polymerized nitrogen materials, detrimental to the natural applications. In the past few years, plasma enhanced chemical vapor deposition (PECVD) technique has been developed to take advantage of the non-equilibrium plasma environment to synthesize polymerized nitrogen materials. The relative thermodynamic instability of polymerized nitrogen can be overcome by kinetic stability provided through energy band hybridization by a charge transfer mechanism, which makes the high energy lattice gain stability and avoid the highly energetic conversion to molecular N2 [35] . The cg-N and N8 cluster were successfully prepared with or without the assistance of carbon nanotube using sodium azide as precursor [35-41] . However the plasma generally can only penetrate several nanometers into the solid surface [42] , leading to limited quantity of cg-N. Besides, high energy plasma particles also could dissociate the cg-N produced in the experiment, which is backed up by the fact that the optimized deposition time is demanded for a very small amount of cg-N sample prepared by the PECVD method [38] . Theory proposes that the transition path to polymerized nitrogen might pass through different molecular structures with limited thermodynamic stability fields [43] . Additionally earlier experiments aimed at synthesizing polymerized nitrogen only reached the amorphous state at ambient temperature compression, indicating that increasing reaction temperature to activate the precursor might play a crucial role in the formation of the atomic structures [44, 45] . For example LiN5 and K2N6 were synthesized at lower pressures [30, 32, 33] with the assistance of laser heating. We here for the first time report a simple route to synthesize cg-N by using sodium azide as a precursor. The cg-N was successfully obtained through recrystallization reaction, followed by transforming azide ions with N=N double bonds into single bonded cg-N that can stabilize at ambient pressure conditions. To date it is the simplest synthesis route to realize cg-N that can be further developed for scale up production and potential practical applications. #### Materials & Methods High pure sodium azide (NaN3) (≥99.9%) was chosen as nitrogen source to synthesize cg-N. The recrystallization process was performed first. A solution of NaN3 (2 mol/L) was pre-prepared and dropped into a crucible. Then the crucible was transferred into a tubular furnace. The vacuum operation was performed to realize the recrystallization of the NaN3 powder. The reaction temperatures and times within the 200- 300°C and up to 10 hours were tried in the synthesis processes. The optimized synthesis condition with reaction temperature 240 ~ 260 <sup>o</sup> C and heating time 5 hours was employed to synthesize the cg-N. The sample was named as polymerized sodium azide followed by synthesis temperature, time, such as PSA-240o C-5h. According to aforementioned conditions, heating NaN3 without recrystallization process to prepare cg-N was tried and it was named as unpolymerized sodium azide (UPSA). The Raman spectra were recorded in range of 100~1500 cm-1 with a spectral resolu micro laser ( ution of 1 cm scope and m (λ0 =532 nm m-<sup>1</sup> using a multichanne m). an integrated el air cooled d laser Ram d CCD dete man system ector. The e (Renishaw excitation so w) with a co ource is an onfocal Ar ion #### Resul lts Fig. 2 shows the Raman spectra of polymerized NaN3 (PSA) synthesized at 200~300 o C with a reaction time of 3 hours. The sharp peaks of NaN3 near 120, 1267 & 1358 cm-1 are assigned to the vibrational lattice mode, the first overtone of the IR active bending ν2 mode and the symmetric stretching ν1 mode of the azide ion, respectively, which matches well with the Raman spectra of free standing NaN3 [41] . For comparison the NaN3 raw material and a sample heated only at 250 <sup>o</sup> C for 3 hours without the recrystallization process were characterized and showed same Raman vibron profiles without any signals at 635 cm-1 (see Fig. S1). The lines at 635 cm-1 for polymerized sodium nitrogen sample (PSA) are comparable to the theoretical results extrapolated to ambient pressure because the Raman modes tend to soften with decreasing pressure [24, 46, 48] . It also agrees with our recent work on polymerized cg-N based on potassium azide. Therefore the emergent intense vibron at 635 cm-<sup>1</sup> is assigned to the pore breathing A symmetry and is regarded as the finger print peak of cg-N, which unambiguously indicates the successful synthesis of cg-N [46] . The synthesis conditions of cg-N were first systematically investigated by controlling the reaction temperature with a fixed reaction time of 3 hours. As shown in Fig. 2, the vibron intensity of cg-N increases as the reaction temperature raises from 200 <sup>o</sup> C to 240 <sup>o</sup> C compared to that at 1358 cm-<sup>1</sup> for the unreacted or remnant NaN3 in the PSA samples. As temperature increases up to 270 <sup>o</sup> C, the Raman intensity of cg-N remains comprehensive but decreases clearly at 300o C. Additionally, it is worth noting that the intensity of vibrational lattice mode at 120 cm-1 for the unreacted NaN3 in PSA-240 sample dramatically decreases compared to those prepared at other temperatures. The PSA sample exhibits uniformly dark blue color and should be related to the by product formed during the synthesis of cg-N and will be discussed below. Therefore, the optimized temperature range of 240~260 <sup>o</sup> C was adopted for cg-N preparation. A quantitative comparison was made by calculating the intensity ratios of the characteristic Raman peaks to measure the abundance of cg-N phase in the PSA samples. Table I summarizes the partial line intensities of NaN3 and cg-N. The amount of cg-N synthesized is reflected by the peak intensity at 635 cm-1 , while those at 120 and 1358 cm-<sup>1</sup> represent the unreacted NaN3. Herein, a simplified conversion degree from NaN3 to cg-N is defined by the peak intensity ratio. The intensities of NaN3 at 120 cm-1 and 1358 cm-1 , calculated by (I200-Ibg)/(I1358-Ibg) for the NaN3 raw material (SA-R), NaN3 heated only (UPSA) and polymerized NaN3 (PSA) samples at different temperatures are between 2 and 3 and remain little changed. For the SA-R and UPSA samples no trace of cg-N is detected. After recrystallization process the vibron intensity of cg-N enhances dramatically for PSA samples and the ratio of cg-N to NaN3 is 1.26-1.39 calculated by (I635-Ibg)/(I1358-Ibg), and 0.42~ 0.60 calculated by (I635-Ibg)/(I120-Ibg). Therefore it indicates that the crystallization process strongly promotes the formation of cg-N and the optimized synthesis temperature range is 240~260 <sup>o</sup> C. ![Fig. 3 The reaction time dependence of the conversion ratios of cg-N to NaN3 for the PSA samples prepared at 240, 250 & 260 <sup>o</sup> C. The optimized synthesis conditions for the PSA samples are 240~260 <sup>o</sup> C with a reaction time of 5 hours.](path) Further the preparation conditions were explored by synthesizing PSA samples at 240~260 <sup>o</sup> C with variable reaction time. The Raman vibron intensity of cg-N in the PSA-240 samples increases gradually as the reaction time extends up to 5 hours and then decreases, which is shown in Fig. S2. Samples prepared at 250 and 260 <sup>o</sup> C show the same results (Fig. S3). For comparison, the intensity ratios of the cg-N to unreacted NaN3 in theses PSA samples were calculated in Table SI. The synthesis time dependence of the ratios of cg-N to NaN3 at 240-260 <sup>o</sup> C was plotted in Fig. 3. All samples show consistent trends: the PSA samples with a reaction time of 5 hours show the highest conversion degree. Therefore, the optimized preparation conditions for polymerized sodium azide samples are a temperature range of 240~260 <sup>o</sup> C and a reaction time of 5 hours. #### Discussion been proposed, which suffer from the challenge that samples could not recover to ambient pressure and enhancing yield is difficult. Here a simple route was designed for synthesizing cg-N under atmospheric conditions considering the thermodynamic barriers related to transforming to N-N single bonds and activation energy of precursor. In our experiments it was found that recrystallization process played a crucial role to form cg-N. For azides the crystal face with low activation energy may facilitate initiating the transformation of the double bonds N=N in azide ions to single bonded cg-N. The optimized temperatures for cg-N preparation by NaN3 are primarily restricted by the melting points NaN3 (~275°C). These results align well with the fact that the cg-N forms on the crystal face with lower activation energy of azides precursors oriented by the recrystallization process. Regarding the formation mechanism of cg-N it was observed that the polymerized sodium azide turned dark blue in our experiments that should be related to the formation of by-product Na3N during the synthesis of cg-N. This is consistent with the reported results that pH value of the solution increases after synthesis reaction as Na3N displays alkalinity when dissolved in water [38] . A more detailed conversion mechanism is currently at investigation. ### Conclusion The polymeric nitrogen with cubic gauche structure was successfully synthesized by a simple route using sodium azide as a precursor. A high yield may be obtained by separating the by products formed on the surface of the produced cg-N. The route is promising for further scale up or applied to synthesize other metastable materials. #### Conflict of Interest There are no financial conflicts of interest to disclose. #### Acknowledgments The work was supported by the National Natural Science Foundation of China, Ministry of Science & Technology, and CAS Project for Young Scientists in Basic Research of China. ## Author Contributions Changqing Jin & Jun Zhang designed and supervised the whole research. Runteng Chen, Jun Zhang, Zelong Wang, Ke Lu, Yi Peng, Jianfa Zhao, Shaomin Feng and Changqing Jin performed the experiments. Jun Zhang & Changqing Jin wrote the manuscript with contributions from all authors. #### References
Figure 3: Neutron scattering data taken in the [110] x [001] scattering plane of the 100 nm thick BiFeO3 thin film taken at a magnetic field of 10 T and a temperature of 150 K. Figure (a) and (b) show the RSMs around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc Bragg reflections, respectively. The black lines represent the corresponding linescans in [112�]pc direction as shown in Figs. (c) and (d), respectively. Gaussian lineshapes were fitted to the experimental data, as indicated by the dashed and solid lines.	# Continuous Collapse of the Spin Cycloid in BiFeO3 Thin Films under an Applied Magnetic Field probed by Neutron Scattering <sup>1</sup> School of Physics, The University of New South Wales, NSW 2052, Australia 2 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA. Ningbo Institute of Materials Technology & Engineering, Chinese Academy of Sciences, Ningbo 315201, China 4 Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA 5 School of Engineering, Brown University, Providence, RI, 02912, USA. 6 Department of Physics, University of California, Berkeley, CA, 94720, USA. Md. Firoz Pervez, ORCID: [0000-0002-1453-8704](https://orcid.org/0000-0002-1453-8704) Clemens Ulrich, ORCID: 0000-0002-6829-9374 Hongrui Zhang, ORCID: [0000-0001-7896-019X](https://orcid.org/0000-0001-7896-019X) Yen-Lin Huang, ORCID: 0000-0002-6129-8547 Lucas Caretta, ORCID: 0000-0001-7229-7980 Ramamoorthy Ramesh, ORCID: 0000-0003-0524-1332 ## ABSTRACT Bismuth ferrite (BiFeO3) is one of the rare materials that exhibits multiferroic properties already at room-temperature. Therefore, it offers tremendous potential for future technological applications, such as memory and logic. However, a weak magnetoelectric coupling together with the presence of a noncollinear cycloidal spin order restricts various practical applications of BiFeO3. Therefore, there is a large interest in the search for suitable methods for the modulation of the spin cycloid in BiFeO3. By performing neutron diffraction experiments using a triple-axis instrument we have determined that the spin cycloid can be systematically suppressed by applying a high magnetic field of 10 T in a BiFeO3 thin film of about 100 nm grown on a (110)-oriented SrTiO3 substrate. As predicted by previous theoretical calculations, we observed that the required critical magnetic field to suppress the spin cycloid in a BiFeO3 thin film was lower as compared to the previously reported critical magnetic field for bulk BiFeO3 single crystals. Our experiment reveals that the spin cycloid continuously expands with increasing magnetic field before the complete transformation into a G-type antiferromagnetic spin order. Such tuning of the length of the spin cycloid up to a complete suppression offers new functionalities for future technological applications as in spintronics or magnonics. ## 1. INTROCUCTION Multiferroics are a fascinating class of materials that manifest multiple, simultaneous ferroic orders such as ferroelectric and magnetic polarizations. Magnetoelectric coupling between the ferroelectric and magnetic orders, including ferromagnetic, ferrimagnetic, antiferromagnetic or more complex noncollinear spin structures has been observed in single or multiphase multiferroic materials. Often this magnetoelectric coupling emerges from the simultaneous breaking of spatial inversion symmetry and time-reversal symmetry [1-2]. A magnetoelectric coupling in multiferroics allows for an efficient method of the electric control of the magnetic order as well as the magnetic control of electric polarization. Therefore, the magnetoelectric coupling offers an enormous potential for the application in next-generation spintronics and magnonic devices (e.g., memory devices, magnetic switches, magnetic sensors, high-frequency magnetic devices, spin valve devices, etc.) at much lower power consumption and faster operation as compared to the present conventional electronic devices [1-13]. Bismuth ferrite (BiFeO3) is one of the rare room temperature type-I, single-phase multiferroic oxides where primarily independent magnetic and ferroelectric order emerge [1,9]. However, a weak direct coupling between magnetic and ferroelectric order is present in BiFeO<sup>3</sup> [7,14,15]. Bulk BiFeO<sup>3</sup> is a rhombohedrally distorted cubic perovskite with the polar space group 3 − 3 6 [16] and a lattice parameter of 3.968 Å in the pseudocubic (pc) notation. This notation will be used throughout this article. In the case of BiFeO<sup>3</sup> thin films, epitaxial tensile or compressive strain results in a further distortion of the in-plane and out-of-plane lattice parameters. As consequence, different crystal structures including tetragonal (4), monoclinic ( ), monoclinic ( ), rhombohedral (3) and orthorhombic ( 2) are possible in BiFeO<sup>3</sup> thin films [17,18]. In the BiFeO<sup>3</sup> single crystals, the displacement of the A-site Bi 3+ ions and the lone pair selectrons cause the rhombohedral distortion which leads to the spontaneous ferroelectric polarization of about 100 μC cm−2 along the direction of the body diagonal [111]pc. This spontaneous ferroelectric polarization persists up to high temperatures (TC~1123 K) [19,20]. Below 643 K BiFeO<sup>3</sup> possesses a complex noncollinear magnetic structure arising from the Fe3+ ions along with nearly zero average magnetization [21]. The Fe3+ magnetic moments of the nearest neighbors of the adjacent (111)pc planes are ordered in a predominantly G-type antiferromagnetic spin structure. The Dzyaloshinskii-Moriya interaction, which is caused by both, the spin-orbit coupling and a broken inversion symmetry, tends to favor a particular spin rotation in BiFeO<sup>3</sup> which leads to a cycloidal spin order with a propagation length of about λ = 63 nm. In bulk BiFeO<sup>3</sup> the spin cycloid propagates along one of three crystallographic directions: [1�10]pc, [101�]pc or [011�]pc where the rotation axis of the spins is perpendicular to the plane defined by the propagation vector and the direction of the electric polarization along [111]pc [20-25]. Through the existence of s spin cycloid in BiFeO3 thin films was long time under debate, in 2010 Ke et al. found a type-I spin cycloid (~64 nm ) propagating along the [11�0]pc direction in a partially relaxed BiFeO<sup>3</sup> thin film with a thickness of about 800 nm deposited on a (100)-oriented SrTiO<sup>3</sup> substrate [26]. In 2011, Ratcliff et al. discovered the presence of a spin cycloid (~62 nm) in a 1 μm thin BiFeO<sup>3</sup> film deposited on a (110)-SrTiO<sup>3</sup> substrate [27], propagating along a unique [112�]pc direction, which is different from the direction [11�0]pc in the bulk single crystals [24,28]. In a neutron scattering study, Bertinshaw et al. discovered a spin cycloid in a BiFeO<sup>3</sup> thin film of just 100 nm when grown on a (110)-oriented SrTiO<sup>3</sup> substrate with a thin SrRuO<sup>3</sup> intermediate layer [29]. This report also showed that the length of the spin cycloid extends and diverges to infinity at the magnetic phase transition temperature of 650 ± 10 K. Finally, the report of Burns et al. revealed a dependence of the length of the spin cycloid on the film thickness. They observed that the length of spin cycloid increases for decreasing film thicknesses down to 20 nm [30]. Various factors can influence the spin cycloid in BiFeO3 thin films such as temperature, external electric and magnetic fields, film thickness, epitaxial strain, an intermediate layer, doping, etc. [8,10,26-34]. One of the key experimental strategies for understanding the fundamental physical interactions and domain states in magnetic materials is applying an external magnetic field. Several experimental techniques, including magnetization measurements and electric polarization measurements [7,14,15,35-38], electron spin resonance experiments (ESR) [39] and Raman light scattering experiments [40] have been performed to investigate the influence of an applied magnetic field (pulsed and static) on BiFeO3. They showed that applying a high magnetic field can suppress spin cycloid in BiFeO<sup>3</sup> single crystals or polycrystalline ceramics at a magnetic field of 16 T to 20 T [7,14,15,35-38,41,42]. A theoretical study by Gareeva et al. predicted the lowering of the required critical magnetic field for strained BiFeO<sup>3</sup> thin films as compared to BiFeO<sup>3</sup> single crystals [43,44]. Moreover, the Raman experiments by Agbelele et al. on BiFeO<sup>3</sup> thin films [40] reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T−6 T for BiFeO3 thin films grown on different substrates. Neutron scattering experiments would unambiguously reveal the nature of magnetic phase transition and would determine the precise magnetic spin structure in BiFeO3 thin films. Therefore, we have performed neutron diffraction experiments on a 100 nm thick BiFeO<sup>3</sup> film deposited by the PLD technique on a (110)-oriented SrTiO3 substrate. The experiment was performed using a triple-axis spectrometer at high magnetic fields of up to 10 T, in order to investigate the overall effect of a magnetic field on the spin structure in BiFeO3 thin films. ## 2. EXPERIMENTAL DETAILS The BiFeO<sup>3</sup> thin film sample with a film thickness of ~100 nm was grown by the Pulsed Laser Deposition on a (110)-oriented single-sided polished, 10 mm × 10 mm × 0.5 mm SrTiO<sup>3</sup> substrate (SHINKOSHA CO., LTD.). As an SrRuO3 intermediate layers can become magnetic at low temperature, the sample was grown without an intermediate layer. The lattice mismatch between SrTiO3 (apc = 3.905 Å [45]) and BiFeO3 (apc= 3.968 Å [16]) leads to an out-of-plane elongated lattice parameter, resulting in a monoclinically distorted structure of the BiFeO<sup>3</sup> films [18,27,29,30,46,47]. The triple-axis spectrometer instrument TAIPAN [48], located at the Australian Centre for Neutron Scattering (ACNS) at the Australian Nuclear Science and Technology Organisation (ANSTO) in Sydney, Australia, has demonstrated to be the ideal choice for the measurement of magnetic Bragg peaks of transition metal oxide thin films due to its enhanced resolution and the improved signal-to-background ratio as compared to conventional neutron diffraction instruments [29,30,49]. In order to apply a high magnetic field, a 12 T superconducting magnet was used. The magnetic field was applied in the direction perpendicular to the scattering plane, i.e. along the [11�0] in-plane direction of the film. In order to suppress contaminations from second order reflections from the SrTiO3 substrate at the � 1 2 1 2 1 2 � Bragg peak position, two pyrolytic-graphite, PG (002), filters with a total thickness of 60 mm were placed behind the sample. An incident energy of 14.86 eV (λ = 2.3462 Å) was used in the standard elastic scattering mode. This was provided by a PG-monochromator and a PG-analyzer with vertical focusing but horizontally flat configuration. In order to enhance the resolution and to suppress the background, 40′′ collimators were placed in the neutron beam before and after the sample (see also Ref. [29,30,49]). A linescan was performed around the � 1 2 1 2 1 2 � Bragg peak of the second order contamination of the SrTiO3 substrate without the two PG (002) filters to determine the instrumental resolution of a Full Width at Half Maximum of 0.0032 reciprocal lattice units (r.l.u.). The substrate SrTiO3 possesses a crystallographic phase transition at 105 K. In order to avoid corresponding effects on the BiFeO<sup>3</sup> thin film, the neutron scattering measurements were performed at the temperature of 150 K [49], which also provides a stronger magnetic signal from the BiFeO3thin film compared to room temperature [29]. ## 3. RESULTS AND DISCUSSION Figure 1 shows the neutron diffraction data of the 100 nm thick BiFeO3 film taken at a temperature of 150 K and zero magnetic field. In order to access the magnetic Bragg peaks arising from the spin cycloid, the film was grown on a (110)-oriented SrTiO3 substrate and mounted in the [110] x [001] scattering plane. Figure 1(a) shows the neutron scattering reciprocal space map (RSM) around the � 1 2 1 2 1 2 �pc Bragg reflection. The peak at precisely � 1 2 1 2 1 2 �pc arises from the second order contamination of the (111) structural Bragg reflection of the SrTiO3 substrate and is still visible despite the use of two PG (002) filters. The two Bragg peaks at around QHK = 0.485 can be attributed to magnetic scattering from the spin cycloid of the BiFeO3 thin film. The reduced QHK value is a consequence of the epitaxial strain, which leads to an increase of the out-of-plane lattice parameter [29,30]. The distance between the peak maxima indicates the length of the spin cycloid (λ = 2π/δ) and the direction of their separation, as shown by the black line, corresponds to the propagation direction of the spin cycloid along [112�]pc. This propagation direction was also observed by previous neutron diffraction experiments on BiFeO3 thin films [26,27,29,30]. Theoretical studies have predicted that a spin cycloid can possibly propagate either along the [112�]pc, [12�1]pc or [2�11]pc directions in a BiFeO3 thin film deposited on a (110)-oriented SrTiO<sup>3</sup> substrate as compared to the [1�10]pc, [101�]pc or [011�]pc directions in bulk BiFeO3 [27,50]. The change in the propagation direction is a consequence of the in-plane strain of the BiFeO3 film. Figure 1(b) shows the RSM around the � 1 2 1 2 1 2 ̅ �pc. Only one magnetic Bragg peak appears. Due to the orientation of the instrumental resolution ellipsoid the two magnetic Bragg peaks of the spin cycloid are overlapping in this measurement geometry (see Suppl. Mater of Ref. [30]). Figures 1(c) and 1(d) show the corresponding line scans along the black lines in Figs. 1(a) and (b), respectively. The red lines are Gaussian lineshapes fitted to the experimental data. For the fit the linewidth was kept constant for both magnetic Bragg peaks. The separation of the two magnetic Bragg peaks in Fig. 1(c) is = Δ = Δ = Δ = 0.00415, which corresponds to a length of the spin cycloid of 65.7 ± 1.0 nm. This is in good agreement to the value of about 63 nm obtained for bulk BiFeO3 [20-25] and for BiFeO3 thin films [27,29,30]. Figure 1(d) shows the corresponding linescan over the � 1 2 1 2 1 2 ̅ �pc RSM as indicated by the black line in Fig. 1(b). Note, to avoid the presence of the λ/2 contamination from the SrTiO3 substrate, the linescan was not taken precisely through the peak maximum but slightly shifted as indicated by the black line in Fig. 1(b). Only one sharp peak is observed. Its integrated intensity corresponds roughly to the integrated intensity of the two magnetic Bragg peaks in Fig. 1(b) [27,29,30]. The magnetic field dependent data are shown in Fig. 2. The data were taken around the � 1 2 1 2 1 2 �pc Bragg reflection following the black line of the RSM as shown in Fig. 1(a), i.e. along the [112�]pc direction. The magnetic field was increased in 1 T steps to a strength of 10 T (Fig. 2(a)) and then decreased to zero field again (Fig. 2(b)). The obtained data were analyzed by fitting two Gaussian lineshapes to the experimental data, where the linewidth and peak intensity of each of the two Gaussian peaks were kept constant using the values obtained for the 0 T measurement. With increasing magnetic field the distance between the two magnetic Bragg peaks decreases, which converge into one single peak at a magnetic field of 10 T (see Fig. 2(a)). The intensity of the peak maximum increases with increasing magnetic field to a value of about twice the maximum intensity at 0 T. However, the integrated intensity over both peaks remains almost constant. The merging into one single Bragg peak is an indication that the length of the spin cycloid increases continuously and finally diverges to infinity at a magnetic field of 10 T. The resulting spin structure is G-type antiferromagnetic, i.e. the spin cycloid collapses into a G-type antiferromagnetic spin arrangement. Upon decreasing magnetic field the peak splits again into two well separated peaks and the overall height of the peak maximum decreases with decreasing magnetic field see Fig. 2(b)). The initial lineshape, both in peak intensity and peak splitting has recovered when reaching 0 T again, i.e. the length of the spin cycloid reaches the same value of 65.7 ± 1.0 nm after the magnetic field cycle. In order to confirm that only one magnetic Bragg peak remains at 10 T, RSMs were taken around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc positions at 10 T and (see Figs. 3(a) and 3(b), respectively). The corresponding linescans along the [112�]pc direction through the peak maxima are shown in Figs. 3(c) and 3(d). At zero magnetic field two magnetic Bragg peaks were present (see Figs. 1(a) and 1(c)) which indicates the presence of a spin cycloid in the BiFeO3 thin film. At a magnetic field of 10 T the two magnetic Bragg peaks have converged into one single peak. The intensity of its peak maximum is almost twice the intensity of an initial single Bragg peak. Note, due to the orientation of the elliptical resolution function for the � 1 2 1 2 1 2 ̅ �pc Bragg reflections, in Figs (b) and (d) only one single peak is expected even in the presence of a peak splitting. Figure 4 shows the overall result of the magnetic field dependence of the neutron diffraction signal of the 100 nm thick BiFeO3 thin film. The magnetic field dependence of the peak maximum is shown in Fig. 4(a) for an increasing (blue symbols) and a decreasing (red symbols) magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. In Fig. 4(b) the magnetic field dependence of the length of the spin cycloid as extracted from the splitting of the two magnetic Bragg peaks, is presented. The distance between both Bragg peaks decreases continuously and converges towards zero at 10 T. This indicates that the length of the spin cycloid increases systematically and diverges to infinity at a magnetic field of about 10 T. The spin cycloid expands and finally collapses into a commensurate G-type antiferromagnetic spin structure. Since only one set of magnetic Bragg peaks was measured in this experiment, no conclusion can be drawn if the final antiferromagnetic structure is canted. It should be noted that both, the peak intensity and the length of the spin cycloid recover with decreasing magnetic field and exhibited a slight hysteresis behavior between 4 T and 10 T. For bulk BiFeO3 an abrupt phase transition was observed between about 16 T and 20 T. Based on measurements of the electric polarization, magnetization, and ESR in combination with theoretical modelling, the resulting phase was attributed to a canted antiferromagnetic structure [7,14,37-39]. In addition, an intermediate phase was observed by ESR and neutron diffraction experiments between about 10 T and 20 T [39,42]. The magnetic structure of this phase was described as conical. It is important to note that the neutron diffraction experiments on BiFeO3 single crystals under high magnetic field did not indicate any change in the incommensurable splitting in both phases, i.e. the length of the spin cycloid did not change with magnetic field [41,42]. This is in contrast to our neutron diffraction experiments on a 100 nm BiFeO3 thin film where the length of the spin cycloid increases systematically and diverges at 10 T. The present experiment on a BiFeO3 thin film does not provide any information about the conical intermediate phase. With a maximum of 10 T this phase was perhaps not reached. In addition, a conical spin structure would cause two magnetic Bragg peaks in the direction perpendicular to the scattering plane. Due to the finite out-of-plane instrumental resolution, they would appear as a single magnetic Bragg peak in the centre. Since the incommensurate splitting of the two magnetic Bragg peaks did converge to zero, we would consider this scenario as unlikely. An external magnetic field aligns the spins along the direction of the field. As the spin cycloid is partly anisotropic on a large scale, the applied magnetic field direction and magnetization direction are not same [15]. With increasing the magnetic field the Zeeman interaction strength can become comparable to the DM interaction. At a critical magnetic field this can lead to a suppression of the spin cycloidal. This can be expressed through the Landau-Ginzburg formalism of its free energy [25,38,39]. The Lifshitz invariant (a term in the Landau-Ginzburg theory that describes a coupling between the magnetization and its spatial variation due to the cycloidal spin structure) is increased by applying a magnetic field. For BiFeO<sup>3</sup> this can lead to the transformation of spin cycloid order into a simpler collinear antiferromagnetic state as observed in bulk BiFeO3 [7,14,37-39,41.42]. Theoretical studies have suggested that a lower critical magnetic field is required to suppress the cycloidal spin order in epitaxially grown BiFeO<sup>3</sup> thin films [43,44]. This was confirmed by the Raman experiments of Agbelele et al. in BiFeO<sup>3</sup> thin films [40], where they reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T − 6 T (depending on the chosen substrate) for BiFeO<sup>3</sup> thin films with a thickness of 70 nm grown on different (001)-oriented substrates, i.e. DyScO3, GdScO3, and SmScO3. In comparison to our BiFeO3 thin film grown on (110)-oriented SrTiO3 substrate they had a significantly smaller epitaxial strain (see Refs. [30] and [40], respectively). Therefore, our critical value of 10 T for the collapse of the spin cycloid in a BeFeO3 thin film is in accordance with the results of Agbelele et al. on differently strained BiFeO3 thin films [40]. ## 4. CONCLUSION In summary, our neutron diffraction experiment using a triple-axis spectrometer revealed a magnetic field-dependent change of the spin cycloid in a 100 nm BiFeO<sup>3</sup> film deposited by PLD on a (110)-oriented SrTiO<sup>3</sup> substrate. The experimental result show a continuous change of the spin cycloidal upon applying a magnetic field. The length of the spin cycloid expands systematically and diverges to infinity at an applied magnetic field of 10 T where the spin cycloidal transforms into a commensurate G-type antiferromagnetic spin structure. A transition to a canted antiferromagnetic spin structure was observed before for polycrystalline BiFeO<sup>3</sup> and single crystals but at much higher magnetic fields in the range of 16 T to 20 T. In contrast, for bulk BiFeO3 the incommensurable splitting of the magnetic Bragg peaks did not change with the magnetic field. Raman light scattering experiments showed that BiFeO3 thin films with lower epitaxial strain when grown on different (001)-oriented substrates had a transition from the cycloidal phase to a collinear spin structure at a lower magnetic field of 4-6 T, depending on the substrate. Our neutron diffraction experiments directly proof that the length of the cycloid systematically extends and transitions continuously into a G-type antiferromagnetic structure. This provides important information about the electronic correlations and microscopic physical effects behind the origin of the spin cycloid in BiFeO3. Furthermore, our results demonstrate that the application of a magnetic field offers a further route for the manipulation of the spin cycloid in multiferroics. Therefore, our experimental result will help to develop strategies to improve the linear magnetoelectric coupling in BiFeO<sup>3</sup> thin films with larger electric polarization, which is crucial for future technological applications. ## ACKNOWLEDGEMENT We would like to thank ANSTO for providing the neutron scattering beamtime at the instrument TAIPAN through the proposals P7934 and P9961, and Guochu Deng for his support as beamline scientist. ## REFERENCES - [1] M. Fiebig, T. [Lottermoser,](https://www.nature.com/articles/natrevmats201646#auth-Thomas-Lottermoser-Aff1) [D. Meier,](https://www.nature.com/articles/natrevmats201646#auth-Dennis-Meier-Aff1) and [M. Trassin,](javascript:;) Nature Rev. Mater. 1, 16046 (2016), doi: 10.1038/natrevmats.2016.46 - [8] W. Ratcliff, J. W. Lynn, V. [Kiryukhin,](https://www.nature.com/articles/npjquantmats20163#auth-Valery-Kiryukhin-Aff2) P. Jain, and [M. R. 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Series. 340, 012003 (2012), doi: 10.1088/1742- 6596/340/1/012003 ![Figure 4: (a) Intensity of the pek maximum of the neuron scattering data as a function of magnetic field as shown in Fig. 2. The blue symbols indicate the data obtained for an increasing magnetic field and the red symbols present the data of a decreasing magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. (b) length of the spin cycloid extracted from the splitting of the two magnetic Bragg peaks. The length of the spin cycloid is 65.7 ± 1.0 nm at zero magnetic field, expands systematically with increasing magnetic field and diverges towards infinity at a magnetic field of 10 T. The inset shows the zoomed-in data of the cycloidal length and the dashed horizontal line indicates the spin cycloid length of 63 nm obtained previously on bulk and thin film BiFeO3 samples. The solid lines serve as a guides to the eye.](path)
Figure 2: Neutron scattering data of the magnetic field dependence of the two incommensurate magnetic Bragg peaks in a 100 nm thick BiFeO3 thin films. The linescans follow the black line shown in Fig. 1(a) and were measured at a temperature of 150 K in the magnetic field range from 0 T to 10 T, i.e. in (a) for an increasing magnetic field and in (b) for a decreasing magnetic field back to 0 T. The dashed and solid lines correspond to the result of a fit of two Gaussian lineshapes to the experimental data. For the fit the linewidth was kept constant at the value determined at zero magnetic field and the intensity of both Bragg peaks was kept constant at each magnetic field.	# Continuous Collapse of the Spin Cycloid in BiFeO3 Thin Films under an Applied Magnetic Field probed by Neutron Scattering <sup>1</sup> School of Physics, The University of New South Wales, NSW 2052, Australia 2 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA. Ningbo Institute of Materials Technology & Engineering, Chinese Academy of Sciences, Ningbo 315201, China 4 Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA 5 School of Engineering, Brown University, Providence, RI, 02912, USA. 6 Department of Physics, University of California, Berkeley, CA, 94720, USA. Md. Firoz Pervez, ORCID: [0000-0002-1453-8704](https://orcid.org/0000-0002-1453-8704) Clemens Ulrich, ORCID: 0000-0002-6829-9374 Hongrui Zhang, ORCID: [0000-0001-7896-019X](https://orcid.org/0000-0001-7896-019X) Yen-Lin Huang, ORCID: 0000-0002-6129-8547 Lucas Caretta, ORCID: 0000-0001-7229-7980 Ramamoorthy Ramesh, ORCID: 0000-0003-0524-1332 ## ABSTRACT Bismuth ferrite (BiFeO3) is one of the rare materials that exhibits multiferroic properties already at room-temperature. Therefore, it offers tremendous potential for future technological applications, such as memory and logic. However, a weak magnetoelectric coupling together with the presence of a noncollinear cycloidal spin order restricts various practical applications of BiFeO3. Therefore, there is a large interest in the search for suitable methods for the modulation of the spin cycloid in BiFeO3. By performing neutron diffraction experiments using a triple-axis instrument we have determined that the spin cycloid can be systematically suppressed by applying a high magnetic field of 10 T in a BiFeO3 thin film of about 100 nm grown on a (110)-oriented SrTiO3 substrate. As predicted by previous theoretical calculations, we observed that the required critical magnetic field to suppress the spin cycloid in a BiFeO3 thin film was lower as compared to the previously reported critical magnetic field for bulk BiFeO3 single crystals. Our experiment reveals that the spin cycloid continuously expands with increasing magnetic field before the complete transformation into a G-type antiferromagnetic spin order. Such tuning of the length of the spin cycloid up to a complete suppression offers new functionalities for future technological applications as in spintronics or magnonics. ## 1. INTROCUCTION Multiferroics are a fascinating class of materials that manifest multiple, simultaneous ferroic orders such as ferroelectric and magnetic polarizations. Magnetoelectric coupling between the ferroelectric and magnetic orders, including ferromagnetic, ferrimagnetic, antiferromagnetic or more complex noncollinear spin structures has been observed in single or multiphase multiferroic materials. Often this magnetoelectric coupling emerges from the simultaneous breaking of spatial inversion symmetry and time-reversal symmetry [1-2]. A magnetoelectric coupling in multiferroics allows for an efficient method of the electric control of the magnetic order as well as the magnetic control of electric polarization. Therefore, the magnetoelectric coupling offers an enormous potential for the application in next-generation spintronics and magnonic devices (e.g., memory devices, magnetic switches, magnetic sensors, high-frequency magnetic devices, spin valve devices, etc.) at much lower power consumption and faster operation as compared to the present conventional electronic devices [1-13]. Bismuth ferrite (BiFeO3) is one of the rare room temperature type-I, single-phase multiferroic oxides where primarily independent magnetic and ferroelectric order emerge [1,9]. However, a weak direct coupling between magnetic and ferroelectric order is present in BiFeO<sup>3</sup> [7,14,15]. Bulk BiFeO<sup>3</sup> is a rhombohedrally distorted cubic perovskite with the polar space group 3 − 3 6 [16] and a lattice parameter of 3.968 Å in the pseudocubic (pc) notation. This notation will be used throughout this article. In the case of BiFeO<sup>3</sup> thin films, epitaxial tensile or compressive strain results in a further distortion of the in-plane and out-of-plane lattice parameters. As consequence, different crystal structures including tetragonal (4), monoclinic ( ), monoclinic ( ), rhombohedral (3) and orthorhombic ( 2) are possible in BiFeO<sup>3</sup> thin films [17,18]. In the BiFeO<sup>3</sup> single crystals, the displacement of the A-site Bi 3+ ions and the lone pair selectrons cause the rhombohedral distortion which leads to the spontaneous ferroelectric polarization of about 100 μC cm−2 along the direction of the body diagonal [111]pc. This spontaneous ferroelectric polarization persists up to high temperatures (TC~1123 K) [19,20]. Below 643 K BiFeO<sup>3</sup> possesses a complex noncollinear magnetic structure arising from the Fe3+ ions along with nearly zero average magnetization [21]. The Fe3+ magnetic moments of the nearest neighbors of the adjacent (111)pc planes are ordered in a predominantly G-type antiferromagnetic spin structure. The Dzyaloshinskii-Moriya interaction, which is caused by both, the spin-orbit coupling and a broken inversion symmetry, tends to favor a particular spin rotation in BiFeO<sup>3</sup> which leads to a cycloidal spin order with a propagation length of about λ = 63 nm. In bulk BiFeO<sup>3</sup> the spin cycloid propagates along one of three crystallographic directions: [1�10]pc, [101�]pc or [011�]pc where the rotation axis of the spins is perpendicular to the plane defined by the propagation vector and the direction of the electric polarization along [111]pc [20-25]. Through the existence of s spin cycloid in BiFeO3 thin films was long time under debate, in 2010 Ke et al. found a type-I spin cycloid (~64 nm ) propagating along the [11�0]pc direction in a partially relaxed BiFeO<sup>3</sup> thin film with a thickness of about 800 nm deposited on a (100)-oriented SrTiO<sup>3</sup> substrate [26]. In 2011, Ratcliff et al. discovered the presence of a spin cycloid (~62 nm) in a 1 μm thin BiFeO<sup>3</sup> film deposited on a (110)-SrTiO<sup>3</sup> substrate [27], propagating along a unique [112�]pc direction, which is different from the direction [11�0]pc in the bulk single crystals [24,28]. In a neutron scattering study, Bertinshaw et al. discovered a spin cycloid in a BiFeO<sup>3</sup> thin film of just 100 nm when grown on a (110)-oriented SrTiO<sup>3</sup> substrate with a thin SrRuO<sup>3</sup> intermediate layer [29]. This report also showed that the length of the spin cycloid extends and diverges to infinity at the magnetic phase transition temperature of 650 ± 10 K. Finally, the report of Burns et al. revealed a dependence of the length of the spin cycloid on the film thickness. They observed that the length of spin cycloid increases for decreasing film thicknesses down to 20 nm [30]. Various factors can influence the spin cycloid in BiFeO3 thin films such as temperature, external electric and magnetic fields, film thickness, epitaxial strain, an intermediate layer, doping, etc. [8,10,26-34]. One of the key experimental strategies for understanding the fundamental physical interactions and domain states in magnetic materials is applying an external magnetic field. Several experimental techniques, including magnetization measurements and electric polarization measurements [7,14,15,35-38], electron spin resonance experiments (ESR) [39] and Raman light scattering experiments [40] have been performed to investigate the influence of an applied magnetic field (pulsed and static) on BiFeO3. They showed that applying a high magnetic field can suppress spin cycloid in BiFeO<sup>3</sup> single crystals or polycrystalline ceramics at a magnetic field of 16 T to 20 T [7,14,15,35-38,41,42]. A theoretical study by Gareeva et al. predicted the lowering of the required critical magnetic field for strained BiFeO<sup>3</sup> thin films as compared to BiFeO<sup>3</sup> single crystals [43,44]. Moreover, the Raman experiments by Agbelele et al. on BiFeO<sup>3</sup> thin films [40] reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T−6 T for BiFeO3 thin films grown on different substrates. Neutron scattering experiments would unambiguously reveal the nature of magnetic phase transition and would determine the precise magnetic spin structure in BiFeO3 thin films. Therefore, we have performed neutron diffraction experiments on a 100 nm thick BiFeO<sup>3</sup> film deposited by the PLD technique on a (110)-oriented SrTiO3 substrate. The experiment was performed using a triple-axis spectrometer at high magnetic fields of up to 10 T, in order to investigate the overall effect of a magnetic field on the spin structure in BiFeO3 thin films. ## 2. EXPERIMENTAL DETAILS The BiFeO<sup>3</sup> thin film sample with a film thickness of ~100 nm was grown by the Pulsed Laser Deposition on a (110)-oriented single-sided polished, 10 mm × 10 mm × 0.5 mm SrTiO<sup>3</sup> substrate (SHINKOSHA CO., LTD.). As an SrRuO3 intermediate layers can become magnetic at low temperature, the sample was grown without an intermediate layer. The lattice mismatch between SrTiO3 (apc = 3.905 Å [45]) and BiFeO3 (apc= 3.968 Å [16]) leads to an out-of-plane elongated lattice parameter, resulting in a monoclinically distorted structure of the BiFeO<sup>3</sup> films [18,27,29,30,46,47]. The triple-axis spectrometer instrument TAIPAN [48], located at the Australian Centre for Neutron Scattering (ACNS) at the Australian Nuclear Science and Technology Organisation (ANSTO) in Sydney, Australia, has demonstrated to be the ideal choice for the measurement of magnetic Bragg peaks of transition metal oxide thin films due to its enhanced resolution and the improved signal-to-background ratio as compared to conventional neutron diffraction instruments [29,30,49]. In order to apply a high magnetic field, a 12 T superconducting magnet was used. The magnetic field was applied in the direction perpendicular to the scattering plane, i.e. along the [11�0] in-plane direction of the film. In order to suppress contaminations from second order reflections from the SrTiO3 substrate at the � 1 2 1 2 1 2 � Bragg peak position, two pyrolytic-graphite, PG (002), filters with a total thickness of 60 mm were placed behind the sample. An incident energy of 14.86 eV (λ = 2.3462 Å) was used in the standard elastic scattering mode. This was provided by a PG-monochromator and a PG-analyzer with vertical focusing but horizontally flat configuration. In order to enhance the resolution and to suppress the background, 40′′ collimators were placed in the neutron beam before and after the sample (see also Ref. [29,30,49]). A linescan was performed around the � 1 2 1 2 1 2 � Bragg peak of the second order contamination of the SrTiO3 substrate without the two PG (002) filters to determine the instrumental resolution of a Full Width at Half Maximum of 0.0032 reciprocal lattice units (r.l.u.). The substrate SrTiO3 possesses a crystallographic phase transition at 105 K. In order to avoid corresponding effects on the BiFeO<sup>3</sup> thin film, the neutron scattering measurements were performed at the temperature of 150 K [49], which also provides a stronger magnetic signal from the BiFeO3thin film compared to room temperature [29]. ## 3. RESULTS AND DISCUSSION Figure 1 shows the neutron diffraction data of the 100 nm thick BiFeO3 film taken at a temperature of 150 K and zero magnetic field. In order to access the magnetic Bragg peaks arising from the spin cycloid, the film was grown on a (110)-oriented SrTiO3 substrate and mounted in the [110] x [001] scattering plane. Figure 1(a) shows the neutron scattering reciprocal space map (RSM) around the � 1 2 1 2 1 2 �pc Bragg reflection. The peak at precisely � 1 2 1 2 1 2 �pc arises from the second order contamination of the (111) structural Bragg reflection of the SrTiO3 substrate and is still visible despite the use of two PG (002) filters. The two Bragg peaks at around QHK = 0.485 can be attributed to magnetic scattering from the spin cycloid of the BiFeO3 thin film. The reduced QHK value is a consequence of the epitaxial strain, which leads to an increase of the out-of-plane lattice parameter [29,30]. The distance between the peak maxima indicates the length of the spin cycloid (λ = 2π/δ) and the direction of their separation, as shown by the black line, corresponds to the propagation direction of the spin cycloid along [112�]pc. This propagation direction was also observed by previous neutron diffraction experiments on BiFeO3 thin films [26,27,29,30]. Theoretical studies have predicted that a spin cycloid can possibly propagate either along the [112�]pc, [12�1]pc or [2�11]pc directions in a BiFeO3 thin film deposited on a (110)-oriented SrTiO<sup>3</sup> substrate as compared to the [1�10]pc, [101�]pc or [011�]pc directions in bulk BiFeO3 [27,50]. The change in the propagation direction is a consequence of the in-plane strain of the BiFeO3 film. Figure 1(b) shows the RSM around the � 1 2 1 2 1 2 ̅ �pc. Only one magnetic Bragg peak appears. Due to the orientation of the instrumental resolution ellipsoid the two magnetic Bragg peaks of the spin cycloid are overlapping in this measurement geometry (see Suppl. Mater of Ref. [30]). Figures 1(c) and 1(d) show the corresponding line scans along the black lines in Figs. 1(a) and (b), respectively. The red lines are Gaussian lineshapes fitted to the experimental data. For the fit the linewidth was kept constant for both magnetic Bragg peaks. The separation of the two magnetic Bragg peaks in Fig. 1(c) is = Δ = Δ = Δ = 0.00415, which corresponds to a length of the spin cycloid of 65.7 ± 1.0 nm. This is in good agreement to the value of about 63 nm obtained for bulk BiFeO3 [20-25] and for BiFeO3 thin films [27,29,30]. Figure 1(d) shows the corresponding linescan over the � 1 2 1 2 1 2 ̅ �pc RSM as indicated by the black line in Fig. 1(b). Note, to avoid the presence of the λ/2 contamination from the SrTiO3 substrate, the linescan was not taken precisely through the peak maximum but slightly shifted as indicated by the black line in Fig. 1(b). Only one sharp peak is observed. Its integrated intensity corresponds roughly to the integrated intensity of the two magnetic Bragg peaks in Fig. 1(b) [27,29,30]. The magnetic field dependent data are shown in Fig. 2. The data were taken around the � 1 2 1 2 1 2 �pc Bragg reflection following the black line of the RSM as shown in Fig. 1(a), i.e. along the [112�]pc direction. The magnetic field was increased in 1 T steps to a strength of 10 T (Fig. 2(a)) and then decreased to zero field again (Fig. 2(b)). The obtained data were analyzed by fitting two Gaussian lineshapes to the experimental data, where the linewidth and peak intensity of each of the two Gaussian peaks were kept constant using the values obtained for the 0 T measurement. With increasing magnetic field the distance between the two magnetic Bragg peaks decreases, which converge into one single peak at a magnetic field of 10 T (see Fig. 2(a)). The intensity of the peak maximum increases with increasing magnetic field to a value of about twice the maximum intensity at 0 T. However, the integrated intensity over both peaks remains almost constant. The merging into one single Bragg peak is an indication that the length of the spin cycloid increases continuously and finally diverges to infinity at a magnetic field of 10 T. The resulting spin structure is G-type antiferromagnetic, i.e. the spin cycloid collapses into a G-type antiferromagnetic spin arrangement. Upon decreasing magnetic field the peak splits again into two well separated peaks and the overall height of the peak maximum decreases with decreasing magnetic field see Fig. 2(b)). The initial lineshape, both in peak intensity and peak splitting has recovered when reaching 0 T again, i.e. the length of the spin cycloid reaches the same value of 65.7 ± 1.0 nm after the magnetic field cycle. In order to confirm that only one magnetic Bragg peak remains at 10 T, RSMs were taken around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc positions at 10 T and (see Figs. 3(a) and 3(b), respectively). The corresponding linescans along the [112�]pc direction through the peak maxima are shown in Figs. 3(c) and 3(d). At zero magnetic field two magnetic Bragg peaks were present (see Figs. 1(a) and 1(c)) which indicates the presence of a spin cycloid in the BiFeO3 thin film. At a magnetic field of 10 T the two magnetic Bragg peaks have converged into one single peak. The intensity of its peak maximum is almost twice the intensity of an initial single Bragg peak. Note, due to the orientation of the elliptical resolution function for the � 1 2 1 2 1 2 ̅ �pc Bragg reflections, in Figs (b) and (d) only one single peak is expected even in the presence of a peak splitting. Figure 4 shows the overall result of the magnetic field dependence of the neutron diffraction signal of the 100 nm thick BiFeO3 thin film. The magnetic field dependence of the peak maximum is shown in Fig. 4(a) for an increasing (blue symbols) and a decreasing (red symbols) magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. In Fig. 4(b) the magnetic field dependence of the length of the spin cycloid as extracted from the splitting of the two magnetic Bragg peaks, is presented. The distance between both Bragg peaks decreases continuously and converges towards zero at 10 T. This indicates that the length of the spin cycloid increases systematically and diverges to infinity at a magnetic field of about 10 T. The spin cycloid expands and finally collapses into a commensurate G-type antiferromagnetic spin structure. Since only one set of magnetic Bragg peaks was measured in this experiment, no conclusion can be drawn if the final antiferromagnetic structure is canted. It should be noted that both, the peak intensity and the length of the spin cycloid recover with decreasing magnetic field and exhibited a slight hysteresis behavior between 4 T and 10 T. For bulk BiFeO3 an abrupt phase transition was observed between about 16 T and 20 T. Based on measurements of the electric polarization, magnetization, and ESR in combination with theoretical modelling, the resulting phase was attributed to a canted antiferromagnetic structure [7,14,37-39]. In addition, an intermediate phase was observed by ESR and neutron diffraction experiments between about 10 T and 20 T [39,42]. The magnetic structure of this phase was described as conical. It is important to note that the neutron diffraction experiments on BiFeO3 single crystals under high magnetic field did not indicate any change in the incommensurable splitting in both phases, i.e. the length of the spin cycloid did not change with magnetic field [41,42]. This is in contrast to our neutron diffraction experiments on a 100 nm BiFeO3 thin film where the length of the spin cycloid increases systematically and diverges at 10 T. The present experiment on a BiFeO3 thin film does not provide any information about the conical intermediate phase. With a maximum of 10 T this phase was perhaps not reached. In addition, a conical spin structure would cause two magnetic Bragg peaks in the direction perpendicular to the scattering plane. Due to the finite out-of-plane instrumental resolution, they would appear as a single magnetic Bragg peak in the centre. Since the incommensurate splitting of the two magnetic Bragg peaks did converge to zero, we would consider this scenario as unlikely. An external magnetic field aligns the spins along the direction of the field. As the spin cycloid is partly anisotropic on a large scale, the applied magnetic field direction and magnetization direction are not same [15]. With increasing the magnetic field the Zeeman interaction strength can become comparable to the DM interaction. At a critical magnetic field this can lead to a suppression of the spin cycloidal. This can be expressed through the Landau-Ginzburg formalism of its free energy [25,38,39]. The Lifshitz invariant (a term in the Landau-Ginzburg theory that describes a coupling between the magnetization and its spatial variation due to the cycloidal spin structure) is increased by applying a magnetic field. For BiFeO<sup>3</sup> this can lead to the transformation of spin cycloid order into a simpler collinear antiferromagnetic state as observed in bulk BiFeO3 [7,14,37-39,41.42]. Theoretical studies have suggested that a lower critical magnetic field is required to suppress the cycloidal spin order in epitaxially grown BiFeO<sup>3</sup> thin films [43,44]. This was confirmed by the Raman experiments of Agbelele et al. in BiFeO<sup>3</sup> thin films [40], where they reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T − 6 T (depending on the chosen substrate) for BiFeO<sup>3</sup> thin films with a thickness of 70 nm grown on different (001)-oriented substrates, i.e. DyScO3, GdScO3, and SmScO3. In comparison to our BiFeO3 thin film grown on (110)-oriented SrTiO3 substrate they had a significantly smaller epitaxial strain (see Refs. [30] and [40], respectively). Therefore, our critical value of 10 T for the collapse of the spin cycloid in a BeFeO3 thin film is in accordance with the results of Agbelele et al. on differently strained BiFeO3 thin films [40]. ## 4. CONCLUSION In summary, our neutron diffraction experiment using a triple-axis spectrometer revealed a magnetic field-dependent change of the spin cycloid in a 100 nm BiFeO<sup>3</sup> film deposited by PLD on a (110)-oriented SrTiO<sup>3</sup> substrate. The experimental result show a continuous change of the spin cycloidal upon applying a magnetic field. The length of the spin cycloid expands systematically and diverges to infinity at an applied magnetic field of 10 T where the spin cycloidal transforms into a commensurate G-type antiferromagnetic spin structure. A transition to a canted antiferromagnetic spin structure was observed before for polycrystalline BiFeO<sup>3</sup> and single crystals but at much higher magnetic fields in the range of 16 T to 20 T. In contrast, for bulk BiFeO3 the incommensurable splitting of the magnetic Bragg peaks did not change with the magnetic field. Raman light scattering experiments showed that BiFeO3 thin films with lower epitaxial strain when grown on different (001)-oriented substrates had a transition from the cycloidal phase to a collinear spin structure at a lower magnetic field of 4-6 T, depending on the substrate. Our neutron diffraction experiments directly proof that the length of the cycloid systematically extends and transitions continuously into a G-type antiferromagnetic structure. This provides important information about the electronic correlations and microscopic physical effects behind the origin of the spin cycloid in BiFeO3. Furthermore, our results demonstrate that the application of a magnetic field offers a further route for the manipulation of the spin cycloid in multiferroics. Therefore, our experimental result will help to develop strategies to improve the linear magnetoelectric coupling in BiFeO<sup>3</sup> thin films with larger electric polarization, which is crucial for future technological applications. ## ACKNOWLEDGEMENT We would like to thank ANSTO for providing the neutron scattering beamtime at the instrument TAIPAN through the proposals P7934 and P9961, and Guochu Deng for his support as beamline scientist. ## REFERENCES - [1] M. Fiebig, T. [Lottermoser,](https://www.nature.com/articles/natrevmats201646#auth-Thomas-Lottermoser-Aff1) [D. Meier,](https://www.nature.com/articles/natrevmats201646#auth-Dennis-Meier-Aff1) and [M. Trassin,](javascript:;) Nature Rev. Mater. 1, 16046 (2016), doi: 10.1038/natrevmats.2016.46 - [8] W. Ratcliff, J. W. Lynn, V. [Kiryukhin,](https://www.nature.com/articles/npjquantmats20163#auth-Valery-Kiryukhin-Aff2) P. Jain, and [M. R. 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Series. 340, 012003 (2012), doi: 10.1088/1742- 6596/340/1/012003 ![Figure 3: Neutron scattering data taken in the [110] x [001] scattering plane of the 100 nm thick BiFeO3 thin film taken at a magnetic field of 10 T and a temperature of 150 K. Figure (a) and (b) show the RSMs around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc Bragg reflections, respectively. The black lines represent the corresponding linescans in [112�]pc direction as shown in Figs. (c) and (d), respectively. Gaussian lineshapes were fitted to the experimental data, as indicated by the dashed and solid lines.](path) ![Figure 4: (a) Intensity of the pek maximum of the neuron scattering data as a function of magnetic field as shown in Fig. 2. The blue symbols indicate the data obtained for an increasing magnetic field and the red symbols present the data of a decreasing magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. (b) length of the spin cycloid extracted from the splitting of the two magnetic Bragg peaks. The length of the spin cycloid is 65.7 ± 1.0 nm at zero magnetic field, expands systematically with increasing magnetic field and diverges towards infinity at a magnetic field of 10 T. The inset shows the zoomed-in data of the cycloidal length and the dashed horizontal line indicates the spin cycloid length of 63 nm obtained previously on bulk and thin film BiFeO3 samples. The solid lines serve as a guides to the eye.](path)
Figure 4: (a) Intensity of the pek maximum of the neuron scattering data as a function of magnetic field as shown in Fig. 2. The blue symbols indicate the data obtained for an increasing magnetic field and the red symbols present the data of a decreasing magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. (b) length of the spin cycloid extracted from the splitting of the two magnetic Bragg peaks. The length of the spin cycloid is 65.7 ± 1.0 nm at zero magnetic field, expands systematically with increasing magnetic field and diverges towards infinity at a magnetic field of 10 T. The inset shows the zoomed-in data of the cycloidal length and the dashed horizontal line indicates the spin cycloid length of 63 nm obtained previously on bulk and thin film BiFeO3 samples. The solid lines serve as a guides to the eye.	# Continuous Collapse of the Spin Cycloid in BiFeO3 Thin Films under an Applied Magnetic Field probed by Neutron Scattering <sup>1</sup> School of Physics, The University of New South Wales, NSW 2052, Australia 2 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA. Ningbo Institute of Materials Technology & Engineering, Chinese Academy of Sciences, Ningbo 315201, China 4 Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA 5 School of Engineering, Brown University, Providence, RI, 02912, USA. 6 Department of Physics, University of California, Berkeley, CA, 94720, USA. Md. Firoz Pervez, ORCID: [0000-0002-1453-8704](https://orcid.org/0000-0002-1453-8704) Clemens Ulrich, ORCID: 0000-0002-6829-9374 Hongrui Zhang, ORCID: [0000-0001-7896-019X](https://orcid.org/0000-0001-7896-019X) Yen-Lin Huang, ORCID: 0000-0002-6129-8547 Lucas Caretta, ORCID: 0000-0001-7229-7980 Ramamoorthy Ramesh, ORCID: 0000-0003-0524-1332 ## ABSTRACT Bismuth ferrite (BiFeO3) is one of the rare materials that exhibits multiferroic properties already at room-temperature. Therefore, it offers tremendous potential for future technological applications, such as memory and logic. However, a weak magnetoelectric coupling together with the presence of a noncollinear cycloidal spin order restricts various practical applications of BiFeO3. Therefore, there is a large interest in the search for suitable methods for the modulation of the spin cycloid in BiFeO3. By performing neutron diffraction experiments using a triple-axis instrument we have determined that the spin cycloid can be systematically suppressed by applying a high magnetic field of 10 T in a BiFeO3 thin film of about 100 nm grown on a (110)-oriented SrTiO3 substrate. As predicted by previous theoretical calculations, we observed that the required critical magnetic field to suppress the spin cycloid in a BiFeO3 thin film was lower as compared to the previously reported critical magnetic field for bulk BiFeO3 single crystals. Our experiment reveals that the spin cycloid continuously expands with increasing magnetic field before the complete transformation into a G-type antiferromagnetic spin order. Such tuning of the length of the spin cycloid up to a complete suppression offers new functionalities for future technological applications as in spintronics or magnonics. ## 1. INTROCUCTION Multiferroics are a fascinating class of materials that manifest multiple, simultaneous ferroic orders such as ferroelectric and magnetic polarizations. Magnetoelectric coupling between the ferroelectric and magnetic orders, including ferromagnetic, ferrimagnetic, antiferromagnetic or more complex noncollinear spin structures has been observed in single or multiphase multiferroic materials. Often this magnetoelectric coupling emerges from the simultaneous breaking of spatial inversion symmetry and time-reversal symmetry [1-2]. A magnetoelectric coupling in multiferroics allows for an efficient method of the electric control of the magnetic order as well as the magnetic control of electric polarization. Therefore, the magnetoelectric coupling offers an enormous potential for the application in next-generation spintronics and magnonic devices (e.g., memory devices, magnetic switches, magnetic sensors, high-frequency magnetic devices, spin valve devices, etc.) at much lower power consumption and faster operation as compared to the present conventional electronic devices [1-13]. Bismuth ferrite (BiFeO3) is one of the rare room temperature type-I, single-phase multiferroic oxides where primarily independent magnetic and ferroelectric order emerge [1,9]. However, a weak direct coupling between magnetic and ferroelectric order is present in BiFeO<sup>3</sup> [7,14,15]. Bulk BiFeO<sup>3</sup> is a rhombohedrally distorted cubic perovskite with the polar space group 3 − 3 6 [16] and a lattice parameter of 3.968 Å in the pseudocubic (pc) notation. This notation will be used throughout this article. In the case of BiFeO<sup>3</sup> thin films, epitaxial tensile or compressive strain results in a further distortion of the in-plane and out-of-plane lattice parameters. As consequence, different crystal structures including tetragonal (4), monoclinic ( ), monoclinic ( ), rhombohedral (3) and orthorhombic ( 2) are possible in BiFeO<sup>3</sup> thin films [17,18]. In the BiFeO<sup>3</sup> single crystals, the displacement of the A-site Bi 3+ ions and the lone pair selectrons cause the rhombohedral distortion which leads to the spontaneous ferroelectric polarization of about 100 μC cm−2 along the direction of the body diagonal [111]pc. This spontaneous ferroelectric polarization persists up to high temperatures (TC~1123 K) [19,20]. Below 643 K BiFeO<sup>3</sup> possesses a complex noncollinear magnetic structure arising from the Fe3+ ions along with nearly zero average magnetization [21]. The Fe3+ magnetic moments of the nearest neighbors of the adjacent (111)pc planes are ordered in a predominantly G-type antiferromagnetic spin structure. The Dzyaloshinskii-Moriya interaction, which is caused by both, the spin-orbit coupling and a broken inversion symmetry, tends to favor a particular spin rotation in BiFeO<sup>3</sup> which leads to a cycloidal spin order with a propagation length of about λ = 63 nm. In bulk BiFeO<sup>3</sup> the spin cycloid propagates along one of three crystallographic directions: [1�10]pc, [101�]pc or [011�]pc where the rotation axis of the spins is perpendicular to the plane defined by the propagation vector and the direction of the electric polarization along [111]pc [20-25]. Through the existence of s spin cycloid in BiFeO3 thin films was long time under debate, in 2010 Ke et al. found a type-I spin cycloid (~64 nm ) propagating along the [11�0]pc direction in a partially relaxed BiFeO<sup>3</sup> thin film with a thickness of about 800 nm deposited on a (100)-oriented SrTiO<sup>3</sup> substrate [26]. In 2011, Ratcliff et al. discovered the presence of a spin cycloid (~62 nm) in a 1 μm thin BiFeO<sup>3</sup> film deposited on a (110)-SrTiO<sup>3</sup> substrate [27], propagating along a unique [112�]pc direction, which is different from the direction [11�0]pc in the bulk single crystals [24,28]. In a neutron scattering study, Bertinshaw et al. discovered a spin cycloid in a BiFeO<sup>3</sup> thin film of just 100 nm when grown on a (110)-oriented SrTiO<sup>3</sup> substrate with a thin SrRuO<sup>3</sup> intermediate layer [29]. This report also showed that the length of the spin cycloid extends and diverges to infinity at the magnetic phase transition temperature of 650 ± 10 K. Finally, the report of Burns et al. revealed a dependence of the length of the spin cycloid on the film thickness. They observed that the length of spin cycloid increases for decreasing film thicknesses down to 20 nm [30]. Various factors can influence the spin cycloid in BiFeO3 thin films such as temperature, external electric and magnetic fields, film thickness, epitaxial strain, an intermediate layer, doping, etc. [8,10,26-34]. One of the key experimental strategies for understanding the fundamental physical interactions and domain states in magnetic materials is applying an external magnetic field. Several experimental techniques, including magnetization measurements and electric polarization measurements [7,14,15,35-38], electron spin resonance experiments (ESR) [39] and Raman light scattering experiments [40] have been performed to investigate the influence of an applied magnetic field (pulsed and static) on BiFeO3. They showed that applying a high magnetic field can suppress spin cycloid in BiFeO<sup>3</sup> single crystals or polycrystalline ceramics at a magnetic field of 16 T to 20 T [7,14,15,35-38,41,42]. A theoretical study by Gareeva et al. predicted the lowering of the required critical magnetic field for strained BiFeO<sup>3</sup> thin films as compared to BiFeO<sup>3</sup> single crystals [43,44]. Moreover, the Raman experiments by Agbelele et al. on BiFeO<sup>3</sup> thin films [40] reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T−6 T for BiFeO3 thin films grown on different substrates. Neutron scattering experiments would unambiguously reveal the nature of magnetic phase transition and would determine the precise magnetic spin structure in BiFeO3 thin films. Therefore, we have performed neutron diffraction experiments on a 100 nm thick BiFeO<sup>3</sup> film deposited by the PLD technique on a (110)-oriented SrTiO3 substrate. The experiment was performed using a triple-axis spectrometer at high magnetic fields of up to 10 T, in order to investigate the overall effect of a magnetic field on the spin structure in BiFeO3 thin films. ## 2. EXPERIMENTAL DETAILS The BiFeO<sup>3</sup> thin film sample with a film thickness of ~100 nm was grown by the Pulsed Laser Deposition on a (110)-oriented single-sided polished, 10 mm × 10 mm × 0.5 mm SrTiO<sup>3</sup> substrate (SHINKOSHA CO., LTD.). As an SrRuO3 intermediate layers can become magnetic at low temperature, the sample was grown without an intermediate layer. The lattice mismatch between SrTiO3 (apc = 3.905 Å [45]) and BiFeO3 (apc= 3.968 Å [16]) leads to an out-of-plane elongated lattice parameter, resulting in a monoclinically distorted structure of the BiFeO<sup>3</sup> films [18,27,29,30,46,47]. The triple-axis spectrometer instrument TAIPAN [48], located at the Australian Centre for Neutron Scattering (ACNS) at the Australian Nuclear Science and Technology Organisation (ANSTO) in Sydney, Australia, has demonstrated to be the ideal choice for the measurement of magnetic Bragg peaks of transition metal oxide thin films due to its enhanced resolution and the improved signal-to-background ratio as compared to conventional neutron diffraction instruments [29,30,49]. In order to apply a high magnetic field, a 12 T superconducting magnet was used. The magnetic field was applied in the direction perpendicular to the scattering plane, i.e. along the [11�0] in-plane direction of the film. In order to suppress contaminations from second order reflections from the SrTiO3 substrate at the � 1 2 1 2 1 2 � Bragg peak position, two pyrolytic-graphite, PG (002), filters with a total thickness of 60 mm were placed behind the sample. An incident energy of 14.86 eV (λ = 2.3462 Å) was used in the standard elastic scattering mode. This was provided by a PG-monochromator and a PG-analyzer with vertical focusing but horizontally flat configuration. In order to enhance the resolution and to suppress the background, 40′′ collimators were placed in the neutron beam before and after the sample (see also Ref. [29,30,49]). A linescan was performed around the � 1 2 1 2 1 2 � Bragg peak of the second order contamination of the SrTiO3 substrate without the two PG (002) filters to determine the instrumental resolution of a Full Width at Half Maximum of 0.0032 reciprocal lattice units (r.l.u.). The substrate SrTiO3 possesses a crystallographic phase transition at 105 K. In order to avoid corresponding effects on the BiFeO<sup>3</sup> thin film, the neutron scattering measurements were performed at the temperature of 150 K [49], which also provides a stronger magnetic signal from the BiFeO3thin film compared to room temperature [29]. ## 3. RESULTS AND DISCUSSION Figure 1 shows the neutron diffraction data of the 100 nm thick BiFeO3 film taken at a temperature of 150 K and zero magnetic field. In order to access the magnetic Bragg peaks arising from the spin cycloid, the film was grown on a (110)-oriented SrTiO3 substrate and mounted in the [110] x [001] scattering plane. Figure 1(a) shows the neutron scattering reciprocal space map (RSM) around the � 1 2 1 2 1 2 �pc Bragg reflection. The peak at precisely � 1 2 1 2 1 2 �pc arises from the second order contamination of the (111) structural Bragg reflection of the SrTiO3 substrate and is still visible despite the use of two PG (002) filters. The two Bragg peaks at around QHK = 0.485 can be attributed to magnetic scattering from the spin cycloid of the BiFeO3 thin film. The reduced QHK value is a consequence of the epitaxial strain, which leads to an increase of the out-of-plane lattice parameter [29,30]. The distance between the peak maxima indicates the length of the spin cycloid (λ = 2π/δ) and the direction of their separation, as shown by the black line, corresponds to the propagation direction of the spin cycloid along [112�]pc. This propagation direction was also observed by previous neutron diffraction experiments on BiFeO3 thin films [26,27,29,30]. Theoretical studies have predicted that a spin cycloid can possibly propagate either along the [112�]pc, [12�1]pc or [2�11]pc directions in a BiFeO3 thin film deposited on a (110)-oriented SrTiO<sup>3</sup> substrate as compared to the [1�10]pc, [101�]pc or [011�]pc directions in bulk BiFeO3 [27,50]. The change in the propagation direction is a consequence of the in-plane strain of the BiFeO3 film. Figure 1(b) shows the RSM around the � 1 2 1 2 1 2 ̅ �pc. Only one magnetic Bragg peak appears. Due to the orientation of the instrumental resolution ellipsoid the two magnetic Bragg peaks of the spin cycloid are overlapping in this measurement geometry (see Suppl. Mater of Ref. [30]). Figures 1(c) and 1(d) show the corresponding line scans along the black lines in Figs. 1(a) and (b), respectively. The red lines are Gaussian lineshapes fitted to the experimental data. For the fit the linewidth was kept constant for both magnetic Bragg peaks. The separation of the two magnetic Bragg peaks in Fig. 1(c) is = Δ = Δ = Δ = 0.00415, which corresponds to a length of the spin cycloid of 65.7 ± 1.0 nm. This is in good agreement to the value of about 63 nm obtained for bulk BiFeO3 [20-25] and for BiFeO3 thin films [27,29,30]. Figure 1(d) shows the corresponding linescan over the � 1 2 1 2 1 2 ̅ �pc RSM as indicated by the black line in Fig. 1(b). Note, to avoid the presence of the λ/2 contamination from the SrTiO3 substrate, the linescan was not taken precisely through the peak maximum but slightly shifted as indicated by the black line in Fig. 1(b). Only one sharp peak is observed. Its integrated intensity corresponds roughly to the integrated intensity of the two magnetic Bragg peaks in Fig. 1(b) [27,29,30]. The magnetic field dependent data are shown in Fig. 2. The data were taken around the � 1 2 1 2 1 2 �pc Bragg reflection following the black line of the RSM as shown in Fig. 1(a), i.e. along the [112�]pc direction. The magnetic field was increased in 1 T steps to a strength of 10 T (Fig. 2(a)) and then decreased to zero field again (Fig. 2(b)). The obtained data were analyzed by fitting two Gaussian lineshapes to the experimental data, where the linewidth and peak intensity of each of the two Gaussian peaks were kept constant using the values obtained for the 0 T measurement. With increasing magnetic field the distance between the two magnetic Bragg peaks decreases, which converge into one single peak at a magnetic field of 10 T (see Fig. 2(a)). The intensity of the peak maximum increases with increasing magnetic field to a value of about twice the maximum intensity at 0 T. However, the integrated intensity over both peaks remains almost constant. The merging into one single Bragg peak is an indication that the length of the spin cycloid increases continuously and finally diverges to infinity at a magnetic field of 10 T. The resulting spin structure is G-type antiferromagnetic, i.e. the spin cycloid collapses into a G-type antiferromagnetic spin arrangement. Upon decreasing magnetic field the peak splits again into two well separated peaks and the overall height of the peak maximum decreases with decreasing magnetic field see Fig. 2(b)). The initial lineshape, both in peak intensity and peak splitting has recovered when reaching 0 T again, i.e. the length of the spin cycloid reaches the same value of 65.7 ± 1.0 nm after the magnetic field cycle. In order to confirm that only one magnetic Bragg peak remains at 10 T, RSMs were taken around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc positions at 10 T and (see Figs. 3(a) and 3(b), respectively). The corresponding linescans along the [112�]pc direction through the peak maxima are shown in Figs. 3(c) and 3(d). At zero magnetic field two magnetic Bragg peaks were present (see Figs. 1(a) and 1(c)) which indicates the presence of a spin cycloid in the BiFeO3 thin film. At a magnetic field of 10 T the two magnetic Bragg peaks have converged into one single peak. The intensity of its peak maximum is almost twice the intensity of an initial single Bragg peak. Note, due to the orientation of the elliptical resolution function for the � 1 2 1 2 1 2 ̅ �pc Bragg reflections, in Figs (b) and (d) only one single peak is expected even in the presence of a peak splitting. Figure 4 shows the overall result of the magnetic field dependence of the neutron diffraction signal of the 100 nm thick BiFeO3 thin film. The magnetic field dependence of the peak maximum is shown in Fig. 4(a) for an increasing (blue symbols) and a decreasing (red symbols) magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. In Fig. 4(b) the magnetic field dependence of the length of the spin cycloid as extracted from the splitting of the two magnetic Bragg peaks, is presented. The distance between both Bragg peaks decreases continuously and converges towards zero at 10 T. This indicates that the length of the spin cycloid increases systematically and diverges to infinity at a magnetic field of about 10 T. The spin cycloid expands and finally collapses into a commensurate G-type antiferromagnetic spin structure. Since only one set of magnetic Bragg peaks was measured in this experiment, no conclusion can be drawn if the final antiferromagnetic structure is canted. It should be noted that both, the peak intensity and the length of the spin cycloid recover with decreasing magnetic field and exhibited a slight hysteresis behavior between 4 T and 10 T. For bulk BiFeO3 an abrupt phase transition was observed between about 16 T and 20 T. Based on measurements of the electric polarization, magnetization, and ESR in combination with theoretical modelling, the resulting phase was attributed to a canted antiferromagnetic structure [7,14,37-39]. In addition, an intermediate phase was observed by ESR and neutron diffraction experiments between about 10 T and 20 T [39,42]. The magnetic structure of this phase was described as conical. It is important to note that the neutron diffraction experiments on BiFeO3 single crystals under high magnetic field did not indicate any change in the incommensurable splitting in both phases, i.e. the length of the spin cycloid did not change with magnetic field [41,42]. This is in contrast to our neutron diffraction experiments on a 100 nm BiFeO3 thin film where the length of the spin cycloid increases systematically and diverges at 10 T. The present experiment on a BiFeO3 thin film does not provide any information about the conical intermediate phase. With a maximum of 10 T this phase was perhaps not reached. In addition, a conical spin structure would cause two magnetic Bragg peaks in the direction perpendicular to the scattering plane. Due to the finite out-of-plane instrumental resolution, they would appear as a single magnetic Bragg peak in the centre. Since the incommensurate splitting of the two magnetic Bragg peaks did converge to zero, we would consider this scenario as unlikely. An external magnetic field aligns the spins along the direction of the field. As the spin cycloid is partly anisotropic on a large scale, the applied magnetic field direction and magnetization direction are not same [15]. With increasing the magnetic field the Zeeman interaction strength can become comparable to the DM interaction. At a critical magnetic field this can lead to a suppression of the spin cycloidal. This can be expressed through the Landau-Ginzburg formalism of its free energy [25,38,39]. The Lifshitz invariant (a term in the Landau-Ginzburg theory that describes a coupling between the magnetization and its spatial variation due to the cycloidal spin structure) is increased by applying a magnetic field. For BiFeO<sup>3</sup> this can lead to the transformation of spin cycloid order into a simpler collinear antiferromagnetic state as observed in bulk BiFeO3 [7,14,37-39,41.42]. Theoretical studies have suggested that a lower critical magnetic field is required to suppress the cycloidal spin order in epitaxially grown BiFeO<sup>3</sup> thin films [43,44]. This was confirmed by the Raman experiments of Agbelele et al. in BiFeO<sup>3</sup> thin films [40], where they reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T − 6 T (depending on the chosen substrate) for BiFeO<sup>3</sup> thin films with a thickness of 70 nm grown on different (001)-oriented substrates, i.e. DyScO3, GdScO3, and SmScO3. In comparison to our BiFeO3 thin film grown on (110)-oriented SrTiO3 substrate they had a significantly smaller epitaxial strain (see Refs. [30] and [40], respectively). Therefore, our critical value of 10 T for the collapse of the spin cycloid in a BeFeO3 thin film is in accordance with the results of Agbelele et al. on differently strained BiFeO3 thin films [40]. ## 4. CONCLUSION In summary, our neutron diffraction experiment using a triple-axis spectrometer revealed a magnetic field-dependent change of the spin cycloid in a 100 nm BiFeO<sup>3</sup> film deposited by PLD on a (110)-oriented SrTiO<sup>3</sup> substrate. The experimental result show a continuous change of the spin cycloidal upon applying a magnetic field. The length of the spin cycloid expands systematically and diverges to infinity at an applied magnetic field of 10 T where the spin cycloidal transforms into a commensurate G-type antiferromagnetic spin structure. A transition to a canted antiferromagnetic spin structure was observed before for polycrystalline BiFeO<sup>3</sup> and single crystals but at much higher magnetic fields in the range of 16 T to 20 T. In contrast, for bulk BiFeO3 the incommensurable splitting of the magnetic Bragg peaks did not change with the magnetic field. Raman light scattering experiments showed that BiFeO3 thin films with lower epitaxial strain when grown on different (001)-oriented substrates had a transition from the cycloidal phase to a collinear spin structure at a lower magnetic field of 4-6 T, depending on the substrate. Our neutron diffraction experiments directly proof that the length of the cycloid systematically extends and transitions continuously into a G-type antiferromagnetic structure. This provides important information about the electronic correlations and microscopic physical effects behind the origin of the spin cycloid in BiFeO3. Furthermore, our results demonstrate that the application of a magnetic field offers a further route for the manipulation of the spin cycloid in multiferroics. Therefore, our experimental result will help to develop strategies to improve the linear magnetoelectric coupling in BiFeO<sup>3</sup> thin films with larger electric polarization, which is crucial for future technological applications. ## ACKNOWLEDGEMENT We would like to thank ANSTO for providing the neutron scattering beamtime at the instrument TAIPAN through the proposals P7934 and P9961, and Guochu Deng for his support as beamline scientist. ## REFERENCES - [1] M. Fiebig, T. [Lottermoser,](https://www.nature.com/articles/natrevmats201646#auth-Thomas-Lottermoser-Aff1) [D. Meier,](https://www.nature.com/articles/natrevmats201646#auth-Dennis-Meier-Aff1) and [M. Trassin,](javascript:;) Nature Rev. Mater. 1, 16046 (2016), doi: 10.1038/natrevmats.2016.46 - [8] W. Ratcliff, J. W. Lynn, V. [Kiryukhin,](https://www.nature.com/articles/npjquantmats20163#auth-Valery-Kiryukhin-Aff2) P. Jain, and [M. R. 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Figure 1: (a) and (c) Neutron scattering reciprocal space maps (RSMs) of the [110]-[001] plane of a 100 nm thick BiFeO<sup>3</sup> thin film grown on a (110)-oriented SrTiO3 substrate. The data were taken at 150 K and 0 T. The Bragg reflection at � 1 2 1 2 1 2 �pc corresponds to second order contamination from the SrTiO3 substrate. (a) RSM around the � 1 2 1 2 1 2 �pc Bragg reflection. Two additional Bragg peaks are observed. They originate from the spin cycloid of BiFeO3. Their spacing, as highlighted by the black line, indicates the propagation direction, i.e. [112�]pc and the length of the spin cycloid. (b) RSM around the � 1 2 1 2 1 2 ̅ �pc. Due to the elliptical instrumental resolution only one additional Bragg peak is visible. (c) and (d): corresponding line scans along the black lines in Figs. 1(a) and (b). The red lines are Gaussian lineshapes fitted to the experimental data with fixed linewidth and intensities for both magnetic Bragg reflections.	# Continuous Collapse of the Spin Cycloid in BiFeO3 Thin Films under an Applied Magnetic Field probed by Neutron Scattering <sup>1</sup> School of Physics, The University of New South Wales, NSW 2052, Australia 2 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA. Ningbo Institute of Materials Technology & Engineering, Chinese Academy of Sciences, Ningbo 315201, China 4 Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA 5 School of Engineering, Brown University, Providence, RI, 02912, USA. 6 Department of Physics, University of California, Berkeley, CA, 94720, USA. Md. Firoz Pervez, ORCID: [0000-0002-1453-8704](https://orcid.org/0000-0002-1453-8704) Clemens Ulrich, ORCID: 0000-0002-6829-9374 Hongrui Zhang, ORCID: [0000-0001-7896-019X](https://orcid.org/0000-0001-7896-019X) Yen-Lin Huang, ORCID: 0000-0002-6129-8547 Lucas Caretta, ORCID: 0000-0001-7229-7980 Ramamoorthy Ramesh, ORCID: 0000-0003-0524-1332 ## ABSTRACT Bismuth ferrite (BiFeO3) is one of the rare materials that exhibits multiferroic properties already at room-temperature. Therefore, it offers tremendous potential for future technological applications, such as memory and logic. However, a weak magnetoelectric coupling together with the presence of a noncollinear cycloidal spin order restricts various practical applications of BiFeO3. Therefore, there is a large interest in the search for suitable methods for the modulation of the spin cycloid in BiFeO3. By performing neutron diffraction experiments using a triple-axis instrument we have determined that the spin cycloid can be systematically suppressed by applying a high magnetic field of 10 T in a BiFeO3 thin film of about 100 nm grown on a (110)-oriented SrTiO3 substrate. As predicted by previous theoretical calculations, we observed that the required critical magnetic field to suppress the spin cycloid in a BiFeO3 thin film was lower as compared to the previously reported critical magnetic field for bulk BiFeO3 single crystals. Our experiment reveals that the spin cycloid continuously expands with increasing magnetic field before the complete transformation into a G-type antiferromagnetic spin order. Such tuning of the length of the spin cycloid up to a complete suppression offers new functionalities for future technological applications as in spintronics or magnonics. ## 1. INTROCUCTION Multiferroics are a fascinating class of materials that manifest multiple, simultaneous ferroic orders such as ferroelectric and magnetic polarizations. Magnetoelectric coupling between the ferroelectric and magnetic orders, including ferromagnetic, ferrimagnetic, antiferromagnetic or more complex noncollinear spin structures has been observed in single or multiphase multiferroic materials. Often this magnetoelectric coupling emerges from the simultaneous breaking of spatial inversion symmetry and time-reversal symmetry [1-2]. A magnetoelectric coupling in multiferroics allows for an efficient method of the electric control of the magnetic order as well as the magnetic control of electric polarization. Therefore, the magnetoelectric coupling offers an enormous potential for the application in next-generation spintronics and magnonic devices (e.g., memory devices, magnetic switches, magnetic sensors, high-frequency magnetic devices, spin valve devices, etc.) at much lower power consumption and faster operation as compared to the present conventional electronic devices [1-13]. Bismuth ferrite (BiFeO3) is one of the rare room temperature type-I, single-phase multiferroic oxides where primarily independent magnetic and ferroelectric order emerge [1,9]. However, a weak direct coupling between magnetic and ferroelectric order is present in BiFeO<sup>3</sup> [7,14,15]. Bulk BiFeO<sup>3</sup> is a rhombohedrally distorted cubic perovskite with the polar space group 3 − 3 6 [16] and a lattice parameter of 3.968 Å in the pseudocubic (pc) notation. This notation will be used throughout this article. In the case of BiFeO<sup>3</sup> thin films, epitaxial tensile or compressive strain results in a further distortion of the in-plane and out-of-plane lattice parameters. As consequence, different crystal structures including tetragonal (4), monoclinic ( ), monoclinic ( ), rhombohedral (3) and orthorhombic ( 2) are possible in BiFeO<sup>3</sup> thin films [17,18]. In the BiFeO<sup>3</sup> single crystals, the displacement of the A-site Bi 3+ ions and the lone pair selectrons cause the rhombohedral distortion which leads to the spontaneous ferroelectric polarization of about 100 μC cm−2 along the direction of the body diagonal [111]pc. This spontaneous ferroelectric polarization persists up to high temperatures (TC~1123 K) [19,20]. Below 643 K BiFeO<sup>3</sup> possesses a complex noncollinear magnetic structure arising from the Fe3+ ions along with nearly zero average magnetization [21]. The Fe3+ magnetic moments of the nearest neighbors of the adjacent (111)pc planes are ordered in a predominantly G-type antiferromagnetic spin structure. The Dzyaloshinskii-Moriya interaction, which is caused by both, the spin-orbit coupling and a broken inversion symmetry, tends to favor a particular spin rotation in BiFeO<sup>3</sup> which leads to a cycloidal spin order with a propagation length of about λ = 63 nm. In bulk BiFeO<sup>3</sup> the spin cycloid propagates along one of three crystallographic directions: [1�10]pc, [101�]pc or [011�]pc where the rotation axis of the spins is perpendicular to the plane defined by the propagation vector and the direction of the electric polarization along [111]pc [20-25]. Through the existence of s spin cycloid in BiFeO3 thin films was long time under debate, in 2010 Ke et al. found a type-I spin cycloid (~64 nm ) propagating along the [11�0]pc direction in a partially relaxed BiFeO<sup>3</sup> thin film with a thickness of about 800 nm deposited on a (100)-oriented SrTiO<sup>3</sup> substrate [26]. In 2011, Ratcliff et al. discovered the presence of a spin cycloid (~62 nm) in a 1 μm thin BiFeO<sup>3</sup> film deposited on a (110)-SrTiO<sup>3</sup> substrate [27], propagating along a unique [112�]pc direction, which is different from the direction [11�0]pc in the bulk single crystals [24,28]. In a neutron scattering study, Bertinshaw et al. discovered a spin cycloid in a BiFeO<sup>3</sup> thin film of just 100 nm when grown on a (110)-oriented SrTiO<sup>3</sup> substrate with a thin SrRuO<sup>3</sup> intermediate layer [29]. This report also showed that the length of the spin cycloid extends and diverges to infinity at the magnetic phase transition temperature of 650 ± 10 K. Finally, the report of Burns et al. revealed a dependence of the length of the spin cycloid on the film thickness. They observed that the length of spin cycloid increases for decreasing film thicknesses down to 20 nm [30]. Various factors can influence the spin cycloid in BiFeO3 thin films such as temperature, external electric and magnetic fields, film thickness, epitaxial strain, an intermediate layer, doping, etc. [8,10,26-34]. One of the key experimental strategies for understanding the fundamental physical interactions and domain states in magnetic materials is applying an external magnetic field. Several experimental techniques, including magnetization measurements and electric polarization measurements [7,14,15,35-38], electron spin resonance experiments (ESR) [39] and Raman light scattering experiments [40] have been performed to investigate the influence of an applied magnetic field (pulsed and static) on BiFeO3. They showed that applying a high magnetic field can suppress spin cycloid in BiFeO<sup>3</sup> single crystals or polycrystalline ceramics at a magnetic field of 16 T to 20 T [7,14,15,35-38,41,42]. A theoretical study by Gareeva et al. predicted the lowering of the required critical magnetic field for strained BiFeO<sup>3</sup> thin films as compared to BiFeO<sup>3</sup> single crystals [43,44]. Moreover, the Raman experiments by Agbelele et al. on BiFeO<sup>3</sup> thin films [40] reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T−6 T for BiFeO3 thin films grown on different substrates. Neutron scattering experiments would unambiguously reveal the nature of magnetic phase transition and would determine the precise magnetic spin structure in BiFeO3 thin films. Therefore, we have performed neutron diffraction experiments on a 100 nm thick BiFeO<sup>3</sup> film deposited by the PLD technique on a (110)-oriented SrTiO3 substrate. The experiment was performed using a triple-axis spectrometer at high magnetic fields of up to 10 T, in order to investigate the overall effect of a magnetic field on the spin structure in BiFeO3 thin films. ## 2. EXPERIMENTAL DETAILS The BiFeO<sup>3</sup> thin film sample with a film thickness of ~100 nm was grown by the Pulsed Laser Deposition on a (110)-oriented single-sided polished, 10 mm × 10 mm × 0.5 mm SrTiO<sup>3</sup> substrate (SHINKOSHA CO., LTD.). As an SrRuO3 intermediate layers can become magnetic at low temperature, the sample was grown without an intermediate layer. The lattice mismatch between SrTiO3 (apc = 3.905 Å [45]) and BiFeO3 (apc= 3.968 Å [16]) leads to an out-of-plane elongated lattice parameter, resulting in a monoclinically distorted structure of the BiFeO<sup>3</sup> films [18,27,29,30,46,47]. The triple-axis spectrometer instrument TAIPAN [48], located at the Australian Centre for Neutron Scattering (ACNS) at the Australian Nuclear Science and Technology Organisation (ANSTO) in Sydney, Australia, has demonstrated to be the ideal choice for the measurement of magnetic Bragg peaks of transition metal oxide thin films due to its enhanced resolution and the improved signal-to-background ratio as compared to conventional neutron diffraction instruments [29,30,49]. In order to apply a high magnetic field, a 12 T superconducting magnet was used. The magnetic field was applied in the direction perpendicular to the scattering plane, i.e. along the [11�0] in-plane direction of the film. In order to suppress contaminations from second order reflections from the SrTiO3 substrate at the � 1 2 1 2 1 2 � Bragg peak position, two pyrolytic-graphite, PG (002), filters with a total thickness of 60 mm were placed behind the sample. An incident energy of 14.86 eV (λ = 2.3462 Å) was used in the standard elastic scattering mode. This was provided by a PG-monochromator and a PG-analyzer with vertical focusing but horizontally flat configuration. In order to enhance the resolution and to suppress the background, 40′′ collimators were placed in the neutron beam before and after the sample (see also Ref. [29,30,49]). A linescan was performed around the � 1 2 1 2 1 2 � Bragg peak of the second order contamination of the SrTiO3 substrate without the two PG (002) filters to determine the instrumental resolution of a Full Width at Half Maximum of 0.0032 reciprocal lattice units (r.l.u.). The substrate SrTiO3 possesses a crystallographic phase transition at 105 K. In order to avoid corresponding effects on the BiFeO<sup>3</sup> thin film, the neutron scattering measurements were performed at the temperature of 150 K [49], which also provides a stronger magnetic signal from the BiFeO3thin film compared to room temperature [29]. ## 3. RESULTS AND DISCUSSION Figure 1 shows the neutron diffraction data of the 100 nm thick BiFeO3 film taken at a temperature of 150 K and zero magnetic field. In order to access the magnetic Bragg peaks arising from the spin cycloid, the film was grown on a (110)-oriented SrTiO3 substrate and mounted in the [110] x [001] scattering plane. Figure 1(a) shows the neutron scattering reciprocal space map (RSM) around the � 1 2 1 2 1 2 �pc Bragg reflection. The peak at precisely � 1 2 1 2 1 2 �pc arises from the second order contamination of the (111) structural Bragg reflection of the SrTiO3 substrate and is still visible despite the use of two PG (002) filters. The two Bragg peaks at around QHK = 0.485 can be attributed to magnetic scattering from the spin cycloid of the BiFeO3 thin film. The reduced QHK value is a consequence of the epitaxial strain, which leads to an increase of the out-of-plane lattice parameter [29,30]. The distance between the peak maxima indicates the length of the spin cycloid (λ = 2π/δ) and the direction of their separation, as shown by the black line, corresponds to the propagation direction of the spin cycloid along [112�]pc. This propagation direction was also observed by previous neutron diffraction experiments on BiFeO3 thin films [26,27,29,30]. Theoretical studies have predicted that a spin cycloid can possibly propagate either along the [112�]pc, [12�1]pc or [2�11]pc directions in a BiFeO3 thin film deposited on a (110)-oriented SrTiO<sup>3</sup> substrate as compared to the [1�10]pc, [101�]pc or [011�]pc directions in bulk BiFeO3 [27,50]. The change in the propagation direction is a consequence of the in-plane strain of the BiFeO3 film. Figure 1(b) shows the RSM around the � 1 2 1 2 1 2 ̅ �pc. Only one magnetic Bragg peak appears. Due to the orientation of the instrumental resolution ellipsoid the two magnetic Bragg peaks of the spin cycloid are overlapping in this measurement geometry (see Suppl. Mater of Ref. [30]). Figures 1(c) and 1(d) show the corresponding line scans along the black lines in Figs. 1(a) and (b), respectively. The red lines are Gaussian lineshapes fitted to the experimental data. For the fit the linewidth was kept constant for both magnetic Bragg peaks. The separation of the two magnetic Bragg peaks in Fig. 1(c) is = Δ = Δ = Δ = 0.00415, which corresponds to a length of the spin cycloid of 65.7 ± 1.0 nm. This is in good agreement to the value of about 63 nm obtained for bulk BiFeO3 [20-25] and for BiFeO3 thin films [27,29,30]. Figure 1(d) shows the corresponding linescan over the � 1 2 1 2 1 2 ̅ �pc RSM as indicated by the black line in Fig. 1(b). Note, to avoid the presence of the λ/2 contamination from the SrTiO3 substrate, the linescan was not taken precisely through the peak maximum but slightly shifted as indicated by the black line in Fig. 1(b). Only one sharp peak is observed. Its integrated intensity corresponds roughly to the integrated intensity of the two magnetic Bragg peaks in Fig. 1(b) [27,29,30]. The magnetic field dependent data are shown in Fig. 2. The data were taken around the � 1 2 1 2 1 2 �pc Bragg reflection following the black line of the RSM as shown in Fig. 1(a), i.e. along the [112�]pc direction. The magnetic field was increased in 1 T steps to a strength of 10 T (Fig. 2(a)) and then decreased to zero field again (Fig. 2(b)). The obtained data were analyzed by fitting two Gaussian lineshapes to the experimental data, where the linewidth and peak intensity of each of the two Gaussian peaks were kept constant using the values obtained for the 0 T measurement. With increasing magnetic field the distance between the two magnetic Bragg peaks decreases, which converge into one single peak at a magnetic field of 10 T (see Fig. 2(a)). The intensity of the peak maximum increases with increasing magnetic field to a value of about twice the maximum intensity at 0 T. However, the integrated intensity over both peaks remains almost constant. The merging into one single Bragg peak is an indication that the length of the spin cycloid increases continuously and finally diverges to infinity at a magnetic field of 10 T. The resulting spin structure is G-type antiferromagnetic, i.e. the spin cycloid collapses into a G-type antiferromagnetic spin arrangement. Upon decreasing magnetic field the peak splits again into two well separated peaks and the overall height of the peak maximum decreases with decreasing magnetic field see Fig. 2(b)). The initial lineshape, both in peak intensity and peak splitting has recovered when reaching 0 T again, i.e. the length of the spin cycloid reaches the same value of 65.7 ± 1.0 nm after the magnetic field cycle. In order to confirm that only one magnetic Bragg peak remains at 10 T, RSMs were taken around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc positions at 10 T and (see Figs. 3(a) and 3(b), respectively). The corresponding linescans along the [112�]pc direction through the peak maxima are shown in Figs. 3(c) and 3(d). At zero magnetic field two magnetic Bragg peaks were present (see Figs. 1(a) and 1(c)) which indicates the presence of a spin cycloid in the BiFeO3 thin film. At a magnetic field of 10 T the two magnetic Bragg peaks have converged into one single peak. The intensity of its peak maximum is almost twice the intensity of an initial single Bragg peak. Note, due to the orientation of the elliptical resolution function for the � 1 2 1 2 1 2 ̅ �pc Bragg reflections, in Figs (b) and (d) only one single peak is expected even in the presence of a peak splitting. Figure 4 shows the overall result of the magnetic field dependence of the neutron diffraction signal of the 100 nm thick BiFeO3 thin film. The magnetic field dependence of the peak maximum is shown in Fig. 4(a) for an increasing (blue symbols) and a decreasing (red symbols) magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. In Fig. 4(b) the magnetic field dependence of the length of the spin cycloid as extracted from the splitting of the two magnetic Bragg peaks, is presented. The distance between both Bragg peaks decreases continuously and converges towards zero at 10 T. This indicates that the length of the spin cycloid increases systematically and diverges to infinity at a magnetic field of about 10 T. The spin cycloid expands and finally collapses into a commensurate G-type antiferromagnetic spin structure. Since only one set of magnetic Bragg peaks was measured in this experiment, no conclusion can be drawn if the final antiferromagnetic structure is canted. It should be noted that both, the peak intensity and the length of the spin cycloid recover with decreasing magnetic field and exhibited a slight hysteresis behavior between 4 T and 10 T. For bulk BiFeO3 an abrupt phase transition was observed between about 16 T and 20 T. Based on measurements of the electric polarization, magnetization, and ESR in combination with theoretical modelling, the resulting phase was attributed to a canted antiferromagnetic structure [7,14,37-39]. In addition, an intermediate phase was observed by ESR and neutron diffraction experiments between about 10 T and 20 T [39,42]. The magnetic structure of this phase was described as conical. It is important to note that the neutron diffraction experiments on BiFeO3 single crystals under high magnetic field did not indicate any change in the incommensurable splitting in both phases, i.e. the length of the spin cycloid did not change with magnetic field [41,42]. This is in contrast to our neutron diffraction experiments on a 100 nm BiFeO3 thin film where the length of the spin cycloid increases systematically and diverges at 10 T. The present experiment on a BiFeO3 thin film does not provide any information about the conical intermediate phase. With a maximum of 10 T this phase was perhaps not reached. In addition, a conical spin structure would cause two magnetic Bragg peaks in the direction perpendicular to the scattering plane. Due to the finite out-of-plane instrumental resolution, they would appear as a single magnetic Bragg peak in the centre. Since the incommensurate splitting of the two magnetic Bragg peaks did converge to zero, we would consider this scenario as unlikely. An external magnetic field aligns the spins along the direction of the field. As the spin cycloid is partly anisotropic on a large scale, the applied magnetic field direction and magnetization direction are not same [15]. With increasing the magnetic field the Zeeman interaction strength can become comparable to the DM interaction. At a critical magnetic field this can lead to a suppression of the spin cycloidal. This can be expressed through the Landau-Ginzburg formalism of its free energy [25,38,39]. The Lifshitz invariant (a term in the Landau-Ginzburg theory that describes a coupling between the magnetization and its spatial variation due to the cycloidal spin structure) is increased by applying a magnetic field. For BiFeO<sup>3</sup> this can lead to the transformation of spin cycloid order into a simpler collinear antiferromagnetic state as observed in bulk BiFeO3 [7,14,37-39,41.42]. Theoretical studies have suggested that a lower critical magnetic field is required to suppress the cycloidal spin order in epitaxially grown BiFeO<sup>3</sup> thin films [43,44]. This was confirmed by the Raman experiments of Agbelele et al. in BiFeO<sup>3</sup> thin films [40], where they reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T − 6 T (depending on the chosen substrate) for BiFeO<sup>3</sup> thin films with a thickness of 70 nm grown on different (001)-oriented substrates, i.e. DyScO3, GdScO3, and SmScO3. In comparison to our BiFeO3 thin film grown on (110)-oriented SrTiO3 substrate they had a significantly smaller epitaxial strain (see Refs. [30] and [40], respectively). Therefore, our critical value of 10 T for the collapse of the spin cycloid in a BeFeO3 thin film is in accordance with the results of Agbelele et al. on differently strained BiFeO3 thin films [40]. ## 4. CONCLUSION In summary, our neutron diffraction experiment using a triple-axis spectrometer revealed a magnetic field-dependent change of the spin cycloid in a 100 nm BiFeO<sup>3</sup> film deposited by PLD on a (110)-oriented SrTiO<sup>3</sup> substrate. The experimental result show a continuous change of the spin cycloidal upon applying a magnetic field. The length of the spin cycloid expands systematically and diverges to infinity at an applied magnetic field of 10 T where the spin cycloidal transforms into a commensurate G-type antiferromagnetic spin structure. A transition to a canted antiferromagnetic spin structure was observed before for polycrystalline BiFeO<sup>3</sup> and single crystals but at much higher magnetic fields in the range of 16 T to 20 T. In contrast, for bulk BiFeO3 the incommensurable splitting of the magnetic Bragg peaks did not change with the magnetic field. Raman light scattering experiments showed that BiFeO3 thin films with lower epitaxial strain when grown on different (001)-oriented substrates had a transition from the cycloidal phase to a collinear spin structure at a lower magnetic field of 4-6 T, depending on the substrate. Our neutron diffraction experiments directly proof that the length of the cycloid systematically extends and transitions continuously into a G-type antiferromagnetic structure. This provides important information about the electronic correlations and microscopic physical effects behind the origin of the spin cycloid in BiFeO3. Furthermore, our results demonstrate that the application of a magnetic field offers a further route for the manipulation of the spin cycloid in multiferroics. Therefore, our experimental result will help to develop strategies to improve the linear magnetoelectric coupling in BiFeO<sup>3</sup> thin films with larger electric polarization, which is crucial for future technological applications. ## ACKNOWLEDGEMENT We would like to thank ANSTO for providing the neutron scattering beamtime at the instrument TAIPAN through the proposals P7934 and P9961, and Guochu Deng for his support as beamline scientist. ## REFERENCES - [1] M. Fiebig, T. [Lottermoser,](https://www.nature.com/articles/natrevmats201646#auth-Thomas-Lottermoser-Aff1) [D. Meier,](https://www.nature.com/articles/natrevmats201646#auth-Dennis-Meier-Aff1) and [M. Trassin,](javascript:;) Nature Rev. Mater. 1, 16046 (2016), doi: 10.1038/natrevmats.2016.46 - [8] W. Ratcliff, J. W. Lynn, V. [Kiryukhin,](https://www.nature.com/articles/npjquantmats20163#auth-Valery-Kiryukhin-Aff2) P. Jain, and [M. R. 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Series. 340, 012003 (2012), doi: 10.1088/1742- 6596/340/1/012003 ![Figure 2: Neutron scattering data of the magnetic field dependence of the two incommensurate magnetic Bragg peaks in a 100 nm thick BiFeO3 thin films. The linescans follow the black line shown in Fig. 1(a) and were measured at a temperature of 150 K in the magnetic field range from 0 T to 10 T, i.e. in (a) for an increasing magnetic field and in (b) for a decreasing magnetic field back to 0 T. The dashed and solid lines correspond to the result of a fit of two Gaussian lineshapes to the experimental data. For the fit the linewidth was kept constant at the value determined at zero magnetic field and the intensity of both Bragg peaks was kept constant at each magnetic field.](path) ![Figure 3: Neutron scattering data taken in the [110] x [001] scattering plane of the 100 nm thick BiFeO3 thin film taken at a magnetic field of 10 T and a temperature of 150 K. Figure (a) and (b) show the RSMs around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc Bragg reflections, respectively. The black lines represent the corresponding linescans in [112�]pc direction as shown in Figs. (c) and (d), respectively. Gaussian lineshapes were fitted to the experimental data, as indicated by the dashed and solid lines.](path) ![Figure 4: (a) Intensity of the pek maximum of the neuron scattering data as a function of magnetic field as shown in Fig. 2. The blue symbols indicate the data obtained for an increasing magnetic field and the red symbols present the data of a decreasing magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. (b) length of the spin cycloid extracted from the splitting of the two magnetic Bragg peaks. The length of the spin cycloid is 65.7 ± 1.0 nm at zero magnetic field, expands systematically with increasing magnetic field and diverges towards infinity at a magnetic field of 10 T. The inset shows the zoomed-in data of the cycloidal length and the dashed horizontal line indicates the spin cycloid length of 63 nm obtained previously on bulk and thin film BiFeO3 samples. The solid lines serve as a guides to the eye.](path)
Fig. 1 - The schematic unit cell of (a) Mo2GaB<sup>2</sup> and (b) Mo2GaB compound.	# Exploration of new 212 MAB phases: M2AB2 (M=Mo, Ta; A=Ga, Ge) via DFT calculations #### Abstract The recently developed MAB phases, an extension of the MAX phase, have sparked interest in research among scientists because of their better thermo-mechanical properties. In this paper, we have explored four new MAB phases M2AB2 (M=Mo, Ta and A=Ga, Ge) and studied the elastic, electronic, thermal, and optical properties to predict the possible applications. The stability of the new phases has been confirmed by calculating formation energy (Ef), formation enthalpy (∆H), phonon dispersion curve (PDC), and elastic constant (Cij). The study reveals that M2AB<sup>2</sup> (M=Mo, Ta and A=Ga, Ge) exhibit significantly higher elastic constants, elastic moduli, and Vickers hardness values than their counterpart 211 borides. Higher Vickers hardness values of Ta2AB2 (A=Ga, Ge) than Mo2AB2 (A=Ga, Ge) have been explained based on the values of the bond overlap population. The analysis of the density of states and electronic band structure revealed the metallic nature of the borides under examination. The thermodynamic characteristics of M2AB2 (M=Mo, Ta and A=Ga, Ge) under high temperatures (0–1000 K) are investigated using the quasi-harmonic Debye model. Critical thermal properties such as melting temperature (Tm), Grüneisen parameter (γ), minimum thermal conductivity (Kmin), Debye temperature (ΘD), and others are also computed. Compared with 211 MAX phases, the 212 phases exhibit higher values of (ΘD) and Tm, along with a lower value of Kmin. These findings suggest that the studied compounds exhibit superior thermal properties that are suitable for practical applications. The optical characteristics have been examined, and the reflectance spectrum indicates that the materials have the potential to mitigate solar heating across various energy regions. #First two authors contributed equally. ### 1. Introduction The MAX phase has garnered significant attention in the present era due to its outstanding mechanical and thermal characteristics at high temperatures, showcasing attributes shared by both metals and ceramics. The increased interest in MAX phase materials can be traced back to Barsoum's noteworthy contributions [1], [2]. The term MAX phase represents a family of multilayer solids where M is an earlier transition metal, A is an element from the IIIA or IVA group of the periodic table, and X is an atom of C/N/B [3]. MAX phase materials showcase metallic behavior due to alternate metallic A-layers and ceramic behavior attributed to the MX layers [4]. Like most metals and alloys, MAX phase materials have excellent thermal shock resistance, superior machinability, and enhanced thermal and electrical conductivities. On the other hand, they have high melting or decomposition temperatures and strong elastic stiffness, similar to many ceramics[5]. The unique combination of metallic and ceramic properties makes them versatile, with applications ranging from high-temperature coatings to nuclear accidenttolerant fuel (ATF), concentrated solar power (CSP), catalysis, and as precursors for MXenes [6], [7], [8], [9]. The diversity of MAX phases was confined to C and N as X elements for years (from 1960 to 2014). However, recent advancements have overcome this limitation by successfully synthesizing MAX phases that contain B [10]. The physical and chemical properties of B and Bcontaining compounds highlight the potential of MAX phase borides, which replace C/N with boron [11]. Due to the presence of B in the composition, they are also called the MAB phase. Furthermore, scientific communities are actively working to broaden the diversity of MAX phases by introducing structural changes, as seen in examples like Cr3AlB<sup>4</sup> (space group Immm), Cr4AlB<sup>6</sup> (space group Cmmm), Cr4AlB<sup>4</sup> [12], [13], [14]. Khazaei et al. reported on the first investigation of the theoretical MAX phase borides M2AlB (where M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta). They evaluated these compounds' electronic structure, mechanical characteristics, and dynamical stability in their investigation [15]. The diversity of MAX phases is a subject of research interest due to their structure variations and the number of atoms in the compounds, resulting in changes to their characteristics. In examples like the 211 MAX phase [16], 312 MAX phase [8][17], 212 MAX phase [3], and i-MAX phase [18], the recent additions to the MAX family include the 314 MAX phase and 212 MAX phase, where the element B has been incorporated as an X element [3] [19]. This introduces a novel aspect to the MAX phase family, contributing to its diversity. The unique structural features of MAB phases include both orthorhombic and hexagonal symmetry observed in crystals. These distinctive symmetries set MAB phases apart from the typical MAX phases. Yinqiao Liu's synthesis of the orthorhombic phase of M2AlB2, particularly the 212-MAB phases with M = Sc, Ti, Zr, Hf, V, Nb, Cr, Mo, W, Mn, Tc, Fe, Co, and Ni, has revealed unique structural stability and notable electrical and mechanical properties [20]. The MAB phase structure slightly differs from the standard MAX phases, which typically crystallize in the hexagonal system with a space group -6<sup>2</sup> (No. 187). Ali et al [3]. investigated the diverse physical properties of Zr2AB<sup>2</sup> (A = In, Tl). Martin Ade et al [13] synthesized ternary borides, namely Cr2AlB2, Cr3AlB4, and Cr4AlB6, and subsequently compared their mechanical properties. Qureshi et al. [19] investigated the 314 Zr3CdB<sup>4</sup> MAX phase boride, calculating its mechanical, thermodynamic, and optical properties. The 314 MAX phase Hf3PB<sup>4</sup> has been thoroughly studied using Density Functional Theory (DFT) and revealed that it was the hardest MAX phase compound discovered until that date [21]. The structure of 212 phases exhibits a slight deviation from 211 MAX phases. In the case of 212 phases, a 2D layer of B is situated between M layers, featuring an additional B atom at the X position, unlike in 211 phases [22] [23], [24]. The B-B bonding in 212 MAX phase borides has improved mechanical and thermal properties. To date, the physical properties of Zr2AB<sup>2</sup> (A = In, Tl) [25] and Hf2AB<sup>2</sup> (A = In, Sn) [26], M2AB (M= Ti, Zr, Hf; A=Al, Ga, In) [27] MAX phases, as well as Nb2AC (A = Ga, Ge, Tl, Zn, P, In, and Cd) MAX phases, have been investigated using density functional theory (DFT). In each instance, the mechanical properties of B-containing compounds show significant improvement compared to their traditional C/N containing 211 MAX phases. The Debye temperature and melting temperature are higher for boron-containing 212 phases than for 211 carbides/nitrides, while the minimum thermal conductivity is lower. The thermal expansion coefficient of borides remains well-suited for use as coating materials. Consequently, the superior thermomechanical properties of B-containing 212 MAX phases demonstrate their suitability for high-temperature technological applications, surpassing the commonly used 211 MAX phase carbides. It should be noted that the 212 MAB phase with a hexagonal structure has already been synthesized [24]. In addition, Ga and Ge-based MAX phases have also been synthesized previously [28]. Thus, the reports on the synthesis of 212 phase and Ga and Gebased MAX phases motivated us to select the 212 MAB phases: M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) for our present study, and we have performed an in-depth investigation of their physical properties through DFT method. Therefore, in this paper, the first-time prediction of the stability and mechanical, electronic, thermal, and optical properties of M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) phases has been presented. The results revealed that M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) compounds are stable and suitable for thermal barrier coating (TBC) and reflection coating applications. Additionally, to provide a comparison, the properties determined for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) are compared with those of other 212 and 211 compounds MAX phase borides. ### 2. Methods of calculations First principles density-functional theory (DFT) computations are conducted utilizing the Cambridge Serial Total Energy Package (CASTEP) module integrated within Materials Studio 2017 [29], [30]. The exchange-correlation function is estimated using the Generalized Gradient Approximation (GGA) method, originally suggested by Perdew, Burke, and Ernzerhof [31]. Pseudo-atomic simulations accounted for electronic orbitals corresponding to B (2s 2 2p 1 ), Ga (3d <sup>10</sup> 4s 2 4p 1 ), Ge (4s 2 4p 2 ), Mo (4s 2 4p 6 4d 5 5s 1 ), and Ta (5d 3 6s 2 ). The energy cutoff and k-point grids were established at 650 eV and 11 × 11 × 4, respectively. The structural relaxation was performed utilizing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique [32], while the electronic structure was computed employing density mixing. The parameters for relaxed structures incorporate the following tolerance thresholds: the self-consistent convergence of the total energy is set at 5 × 10-6 eV/atom, the maximum force exerted on the atom is limited to 0.01 eV/Å, the maximum ionic displacement is constrained to 5 × 10-4 Å, and a maximum stress threshold of 0.02 GPa is imposed. The finite strain method [33], grounded in density functional theory (DFT), is utilized to compute the elastic properties within this framework. All necessary equations for determining various properties are provided in the supplementary document. #### 3 Results and discussion #### 3.1 Structural properties The M2AB<sup>2</sup> compounds (where M=Mo or Ta; A=Ga or Ge) belong to the P6m<sup>2</sup> (No. 187)[23] space group and crystallize in the hexagonal system. Unlike conventional MAX phases, which typically belong to the P63/mmc (194) space group, the 212 MAX phases exhibit distinct characteristics. In Fig. 1, the unit cell structure of Mo2GaB<sup>2</sup> is depicted as a representative of M2AB<sup>2</sup> alongside Mo2GaB, facilitating a comparison to discern their differences easily. The atomic positions are as follows: M (Mo or Ta) at (0.3333, 0.6667, 0.6935), A at (0.6667, 0.3333, 0.0), and two B atoms positioned at (0.6667, 0.3333, 0.5) and (0.0, 0.0, 0.5). The B components are arranged at the corners of the unit cell in typical 211 MAX phases, but in 212 boride MAX phases, they form a 2D layer between the M layers. This structural arrangement results in B-B covalent bonds in the 2D layer, enhancing stability compared to conventional 211 MAX phases. The calculated lattice parameters of M2AB<sup>2</sup> are presented in Table 1 alongside those of other 211 and 212 MAX phases for comparison, demonstrating consistency with prior results [16], [34], [35] and affirming the accuracy of the computational methodology used. The primary distinction between 212- and 211-unit cell structures arises from differences in the lattice parameter c, where the c value for 211 exceeds that of 212. Moreover, the volumes of Mo- and Ta-based 211 MAX phase borides surpass those of 212 MAX phase borides. #### 3.2 Stability Examining a compound's stability is significant for multiple purposes, as it yields valuable information regarding the compound's synthesis parameters and aids in assessing the material's resilience across diverse environments, including thermal, compressive, and mechanical pressures. In this section, we delve into a comprehensive theoretical analysis concerning the chemical, dynamic, and mechanical stability of M2AB<sup>2</sup> compounds. The compound's chemical stability is determined by computing its formation energy by the following equation [36]: 2<sup>2</sup> = 22−( + + ) ++ . In the context provided, 22 represents the total energy of the compound after optimizing the unit cell. , , and denote the energies of the individual elements M, A, and B, respectively. The variables x, y, and z correspond to the number of atoms in the unit cell for M, Ga, and B, respectively. For Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, the calculated formation energy (Ef) are -1.8079 eV/atom, -1.8859 eV/atom, -2.1252 eV/atom, and -2.1933 eV/atom, respectively. The negative values signify the chemical stability of all compounds. The order of chemical stability can be expressed as Ta2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Mo2GaB2, indicating that Ta2GeB<sup>2</sup> is the most stable. Additionally, it's observed that MAX phase borides containing Ta are more stable than those containing Mo. Negative formation energy alone may not fully explain the chemical stability of M2AB2 (M=Mo, Ta and A=Ga, Ge). We calculated its formation enthalpy by examining potential pathways to evaluate its thermodynamic stability. For this analysis, we used the experimentally identified stable phases of MoB[37] and TaB[38], Ga4Mo[39], and B2Mo[40]. The potential decomposition pathways for our compounds, as determined from the Open Quantum Materials Database (OQMD), are outlined below. ![We have calculated the reaction energy as follows [41]:](path) Where, M=Mo, Ta and A=Ga, Ge. The following formula can calculate the decomposition energy associated with the reaction energy. = . Where n is the number of participating atoms. The calculated decomposition energies for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> are -0.16, -0.17, -0.26, and -0.28 meV/atom, respectively. So, we can state that the M2AB2 (M=Mo, Ta and A=Ga, Ge) system exhibits thermodynamic stability. Phonon dispersion curves (PDCs) have been calculated at the ground state utilizing the density functional perturbation theory (DFPT) linear-response approach to evaluate the dynamic stability of the MAX phase borides under investigation [42]. The PDCs, depicting the phonon dispersion along the high symmetry directions of the crystal Brillouin zone (BZ), along with the total phonon density of states (PHDOS) of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, are illustrated in Fig. 2(a, b, c and d). Analysis of the PDCs reveals no negative phonon frequencies for any of the compounds, indicating their dynamic stability. The PHDOS of the M2AB<sup>2</sup> compounds are derived from the PDCs and are presented alongside the PDCs in Fig. 2(a, b, c, and d), facilitating band identification through comparison of corresponding peaks. From Fig. 2, it is observed that in Mo2GaB2, the flatness of the bands for the Transverse Optical (TO) modes results in a prominent peak in the PHDOS, whereas non-flat bands for the Longitudinal Optical (LO) modes lead to weaker peaks in the PHDOS. Similar trends are observed in Mo2GeB2, Ta2GaB2, and Ta2GeB2. Notably, a distinct discrepancy arises between the optical and acoustic branches, with the top of the LO and bottom of the TO modes situated at the G point, with separations of 7.49 THz, 6.39 THz, 9.39 THz, and 8.51 THz for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, respectively. ![Fig. 2 – Phonon DOS and Phonon dispersion curves of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) Materials are subjected to various forces and loads in practical applications, necessitating understanding their mechanical stability. The mechanical stability of a compound can be assessed using stiffness constants. For a hexagonal system, the conditions for mechanical stability are as follows [43]: C<sup>11</sup> > 0, C<sup>11</sup> > C12, C<sup>44</sup> > 0, and (C<sup>11</sup> + C12)C<sup>33</sup> - 2(C13)² > 0. As indicated in Table 2, the Cij values of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) satisfy these conditions, thus confirming the mechanical stability of herein predicted phases: M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge). #### 3.3 Electronic properties Analyzing the electronic band structure (EBS) is crucial for gaining insights into the electronic behavior of a compound. The EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) MAX phases are depicted in Fig. 3(a, b, c, and d), with the Fermi energy (EF) level set at 0 eV, represented by a horizontal line. Observing the EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge), it is evident that the conduction band overlaps with the valence band, indicating the absence of a band gap. This observation confirms that the M2AB<sup>2</sup> compounds exhibit metallic behavior, which aligns with conventional MAX phases. The red lines illustrate the overlapping band at the Fermi level. Fig. 3 (a, b) shows that for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> compounds, the maximum band overlap occurs along the A-H path. Conversely, in Fig. 3(c, d), the maximum band overlap is observed along the G-M paths. Utilizing the band structure, we can analyze the electrical anisotropy of M2AB<sup>2</sup> MAX phase compounds. The anisotropic nature can be understood by analyzing the energy dispersion in the basal plane and along the c-axis. The paths G-A, H-K, and M-L show energy dispersion along the c-direction, and A-H, K-G, G-M, and L-H show energy dispersion in the basal plane. In comparison to the paths A-H, K-G, G-M, and L-H (basal plane), there is less energy dispersion along the lines G-A, H–K, and M-L (c-direction), as shown by Fig. 2(a, b, c and d). Lower energy dispersion in the c-direction results from a higher effective mass [44], indicating the strong electronic anisotropy of the M2AB<sup>2</sup> MAX phase compound. Consequently, conductivity along the c-axis is expected to be lower than in the basal planes. These findings are consistent with prior studies [34], [45]. ![Fig. 3 – Band structure of (a) Mo2GaB2, (b) Mo2GeB2, (c) Ta2GaB2, and (d) Ta2GeB<sup>2</sup> compounds.](path) To investigate the bonding nature and electronic conductivity, the total density of states (TDOS) and partial density of states (PDOS) of M2AB<sup>2</sup> compounds were computed. Figure 4 (a, b, c, and d) illustrates the TDOS and PDOS of these compounds, with the Fermi energy (EF) set at zero energy level, indicated by a straight line. These profiles exhibit typical characteristics of MAX phase materials. The Mo or Ta-d electronic states predominantly contribute to the Fermi level, with a minor contribution from the B-p and Ga or Ge-p electronic states. To ascertain the hybridization characteristics of various electronic states within the valence band, the energy spectrum of the valence band has been partitioned into two distinct segments. The first segment encompasses the lower valence band region spanning from -7 eV to -3.5 eV, originating from the hybridization of Mo-p, Mo-d, and B-s orbitals in the case of the Mo2GaB<sup>2</sup> compound. Conversely, for the Mo2GeB2 compound, the lower valence band region arises from the hybridization of Mo-d, Ge-p, and B-s orbitals. Notably, for both the Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compounds, the dominance of the B-p state characterizes the lower valence band region. The second segment pertains to the upper valence band region from -3.5 eV to 0 eV. In the case of the Mo2GaB<sup>2</sup> compound, this region arises from the hybridization of Mo-p and Mo-d orbitals. However, for the Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, the upper valence band region results from the hybridization of Mo-d orbitals and (Ga/Ge)-p orbitals. Notably, the Fermi level of M2AB<sup>2</sup> resides near the pseudogap in the TDOS profile, indicating a high level of electronic stability. Similar trends are observed in other compounds like Zr2AlN, V2AlN, Sc2AlB, Sc2GaB, Ta2GaB, and Hf2GaB<sup>2</sup> [15], [35], [45]. ![Fig. 4 – TDOS and PDOS of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) The electron charge density mapping is helpful in understanding the distribution of electron densities linked to chemical bonds. It delineates areas of positive and negative charge densities, signifying the development and exhaustion of electrical charges, respectively. As depicted in the map, covalent bonds become apparent by accumulating charges between two atoms. Furthermore, the presence of ionic bonds can be inferred from a balance between negative and positive charges at specific atom positions[46]. The valence electronic CDM, denoted in units of eÅ-3 , for M2AB<sup>2</sup> (where M = Mo, Ta; A = Ga, Ge) is showcased in Fig. 5(a, b, c, and d) along the (110) crystallographic plane. The accompanying scale illustrates the intensity of electronic charge density, with red and blue colors indicating low and high electronic charge density, respectively. As depicted in Fig. 5(a, b, c, and d), it is evident that charges accumulate in the regions between the B sites. Consequently, it is anticipated that strong covalent B‒B bonding occurs through the formation of two center-two electron (2c‒2e) bonds in the M2AB<sup>2</sup> (where M=Mo, Ta; A=Ga, Ge) compound, similar to other 212 MAX phase borides like Ti2PB2, Zr2PbB2, Nb2SB2, Zr2GaB<sup>2</sup> and Hf2GaB<sup>2</sup> [34], [45]. Mulliken analysis has corroborated the charge transfer from Mo/Ta atoms to B atoms. The charge received from Mo/Ta atoms is distributed among the B atoms positioned at the transitions and those located at the edges, facilitating the formation of a two center-two electron (2c‒2e) bond between B atoms within the 2D layer of B, as illustrated in Fig. 1(a). The hardness of each bond value presented in Table 5 also aligns with the results obtained from charge density mapping (CDM) and our analysis using moduli and elastic stiffness constants. ![Fig. 5 – The charge density mapping of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) ## 3.4.1 Mechanical properties The mechanical properties of materials play a pivotal role in determining their potential applications, serving as crucial indicators of their behavior and suitability in materials engineering endeavors. These properties are equally applicable to MAX phase materials. Initially, to assess the mechanical properties of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phases, we employed the stress-strain method within the CASTEP code to compute the elastic constants (Cij) [22], [47], [48]. These calculated elastic constants (Cij) are presented in Table 2, alongside those of other 212 and 211 MAX phases. Due to the hexagonal crystal structure of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phase borides, five stiffness constants emerge: C11, C12, C13, C33, and C<sup>44</sup> [49]. The mechanical stability was evaluated using these stiffness constants in the preceding section. For instance, C<sup>11</sup> and C<sup>33</sup> determine stiffness when stress is exerted along the (100) and (001) directions, respectively, whereas C<sup>44</sup> evaluates resistance to shear deformation on the (100) and (001) planes. The stiffness constants C<sup>11</sup> and C<sup>12</sup> directly reflect the strength of atomic bonds along the a- and c-axes. When C<sup>11</sup> exceeds C<sup>33</sup> (or vice versa), it signifies stronger atomic bonding along the a-axis (or c-axis). In Table 2, for Mo2GaB2, Mo2GeB2, and Ta2GeB<sup>2</sup> compounds, C<sup>11</sup> surpasses C33, indicating superior atomic bonding along the a-axis compared to the c-axis. This robust bonding along the a-axis suggests heightened resistance to a-axial deformation. Conversely, in Ta2GeB2, where C<sup>11</sup> < C33, stronger atomic bonding along the c-axis translates to increased resistance against c-axial deformation. Analysis of Table-2 reveals that the values of C<sup>11</sup> and C<sup>22</sup> are notably higher for 212 MAX phase borides than 211 MAX phase borides. Consequently, it can be inferred that 212 MAX phases exhibit stronger resistance to axial deformation when juxtaposed with 211 MAX phases. C<sup>44</sup> is commonly utilized to gauge shear deformation tolerance among the elastic constants. Notably, Ta2GaB exhibits superior shear deformation resistance owing to its highest C<sup>44</sup> value among the studied compounds. Furthermore, the individual elastic constants C12, C13, and C<sup>44</sup> denote shear deformation response under external stress. The observation that C<sup>11</sup> and C<sup>33</sup> possess larger magnitudes than C<sup>44</sup> implies that shear deformation is more facile than axial strain. Another crucial parameter derived from the stiffness constants is the Cauchy pressure (CP), calculated as C<sup>12</sup> - C44, which provides vital insights relevant to the practical applications of solids [51]. Pettifor [51] emphasized the significance of Cauchy pressure (CP) in discerning the chemical bonding and ductile/brittle properties of solids. A negative CP value indicates covalently bonded brittle solids, whereas a positive value signifies isotropic ionic ductile solids. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> fall into covalently bonded brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit metallic ductile characteristics. Similarly, like Ga-containing 211 MAX phase borides, Ga-containing 212 MAX phases also demonstrate negative CP values and brittle behavior [52][53]. The elastic constants obtained are used to produce several bulk elastic parameters that are used to characterize polycrystalline materials, such as Young's modulus (Y), bulk modulus (B), and shear modulus (G). In Table 2, the bulk modulus (B) and shear modulus (G) calculated using Hill's approximation [54] are also presented. Hill's values represent the average of the upper limit (Voigt [55]) and lower limit (Reuss [56]) of B. The necessary equations for these calculations are provided in the supplementary document (S1). Young's modulus (Y) is a crucial indicator of material stiffness. A higher Y value indicates a stiffer material. It can be observed from the table that Mo2GaB<sup>2</sup> exhibits the highest Y value, signifying its greater stiffness compared to others. According to the sequence of Y values, stiffness can be ranked as follows: Mo2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Ta2GeB2. Therefore, Mo2GaB2, with its higher Young's modulus, is anticipated to demonstrate superior mechanical stability and deformation resistance compared to the other compounds under investigation. This quality is essential for aircraft parts or high-performance machinery applications where critical dimensional stability and structural integrity are required [57], [58]. Table 2 shows that 212 MAX phase materials exhibit larger Young's modulus (Y) values than 211 MAX phases. This suggests that 212 MAX phase borides are stiffer than their 211 MAX phase counterparts. Young's modulus (E) also correlates well with thermal shock resistance (R): R ∝ 1/E [59]. Lower Young's modulus values correspond to higher thermal shock resistance. Therefore, materials with higher thermal shock resistance (i.e., lower Young's modulus) are more suitable for use as Thermal Barrier Coating (TBC) materials. Given that Ta2GaB<sup>2</sup> possesses the lowest Y value among the materials studied, it should be considered a superior candidate for TBC material due to its higher thermal shock resistance. The material's ability to withstand shape distortion is elucidated by the shear modulus (G). On the other hand, the bulk modulus (B) indicates the strength of a material's chemical bonds and its ability to withstand uniform compression or volume change. We computed Young's modulus (Y), bulk modulus (B), and shear modulus for our analysis. Based on the bulk modulus values presented in Table 2, the sequence of a material that resists compression when pressure is applied can be outlined as follows: Mo2GeB<sup>2</sup> > Ta2GeB<sup>2</sup> > Mo2GaB<sup>2</sup> > Ta2GeB2. Mo2GeB2, boasting a higher bulk modulus, may exhibit reduced plastic deformation and superior resistance to stress-induced deformation compared to other compounds examined. The ductile and brittle characteristics of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds can also be assessed through Poisson's ratio (ν) and Pugh's ratio (G/B). A compound demonstrates ductile (brittle) behavior if the ν value surpasses (falls below) 0.26 [60]. Furthermore, if the G/B value exceeds (is less than) 0.571, then the compound exhibits brittle (ductile) behavior [19]. According to both criteria, Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are classified as brittle compounds, whereas Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> are categorized as ductile compounds. The ductile nature of Ge-based MAX phases has been reported previously [61]. Fracture toughness (KIC) is a vital property that gauges a material's ability to resist crack propagation. In the case of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, Equation (S2) was employed to determine KIC. The KIC values are documented in Table 2, and notably, these values surpass those of 211 MAX phases [16], [35]. Another parameter, the "f-index," characterizes the isotropic nature and strength of atom-atom bonds within a single hexagonal crystalline lattice along the a- and c-directions. If the f-index is less than 1, chemical bonds exhibit greater rigidity along the c-axis; conversely, if f exceeds 1, bonds are more rigid in the ab-plane. When the value of f is set to 1, atomic bonds exhibit similar strength and uniformity in all directions [15]. The fvalue is computed using Equation (S3) and displayed in Table 2. Table 2 shows that the f-values of Mo2GeB2, Ta2GaB2, and Ta2GeB2, which are close to one, indicate a slight anisotropic bonding strength. Nevertheless, substances with robust bonds in the horizontal plane (ab plane) (f > 1), such as Mo2GaB2, are deemed optimal candidates for the exfoliation process. In engineering applications, the hardness of a solid material serves as a valuable criterion for designing various devices. The elastic properties of polycrystalline materials can be utilized to calculate hardness values, as the ability to resist indentation is closely linked to a material's hardness. Both micro-hardness (Hmicro) and macro-hardness (Hmacro) were computed using Equation (S4) and are presented in Table 2. Based on the values of Hmicro and Hmacro, Mo2GaB<sup>2</sup> emerges as the toughest among the studied phases. The order of hardness is as follows: Mo2GaB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GeB2. Interestingly, Ga-containing MAX phase borides exhibit superior hardness compared to Ge-containing MAX phase borides. In Table 2, we compare our findings with previously reported MAX phases and observe that our data align perfectly with the earlier results. #### 3.4.2 Elastic anisotropy Additionally, anisotropy is linked to other crucial events like anisotropic plastic deformation and the development and spread of micro-cracks within mechanical stress. By providing directiondependent elastic constants, the understanding of anisotropy also offers a framework for improving the mechanical stability of materials in extreme circumstances. The following formulae are used for hexagonal structures to calculate the various anisotropic variables from elastic constants Cij [62]. $A\_{1} = \frac{\frac{1}{6}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{44}};$ $A\_{2} = \frac{2\mathcal{C}\_{44}}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ and $A\_{3} = A\_{1}.$ $A\_{2} = \frac{\frac{1}{3}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ Table 3 lists every anisotropy parameter. Since the value of A<sup>i</sup> should be 1 to be isotropic, the computed value of A<sup>i</sup> (i = 1-3) indicates that all of the compounds under research exhibit anisotropic behavior [63]. An additional method for estimating elastic anisotropy is to use the percentage anisotropy to compressibility and shear (A<sup>B</sup> & AG). This gives polycrystalline materials a helpful way to measure elastic anisotropy. They have been described as [45]; Zero values for AB, AG, and the universal anisotropy factor (AU) show elastic isotropy; the maximum amount of anisotropy is represented by a value of 1. Table 3 shows that, compared to other compounds, the values of AB, AG, and A U for Mo2GeB<sup>2</sup> and Ta2GaB<sup>2</sup> are incredibly close to zero, suggesting that these compounds have nearly isotropic characteristics. #### 3.4.3 Mulliken Populations The Mulliken charge assigned to an atomic species quantifies the effective valence by calculating the absolute difference between the formal ionic charge and the Mulliken charge. Equations (S5) and (S6) are employed to ascertain the Mulliken charge for each atom (α). Table-4 provides the Mulliken atomic population and effective valence charge. Transition metals Mo and Ta in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have pure valence states of 4d<sup>5</sup> and 5d<sup>3</sup> , respectively. The d- orbital electrons of transition metals have been found to influence their effective valence charge significantly. A non-zero positive value indicates a combination of covalent and ionic attributes within chemical bonds. As this value decreases towards zero, it signifies a rise in ionicity. A zero value suggests an ideal ionic character in the bond. Conversely, a progression from zero with a positive value indicates an elevation in the covalency level of the bonds. Based on their effective valence, M atoms move from the left to the right in the periodic table, increasing the covalency of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). Table-4 reveals that the Mulliken atomic charge ascribed to the B atoms is solely negative. Conversely, positive Mulliken atomic charges are associated with transition metals (M) and A. This suggests a charge transfer from M and A to B for each compound within M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge), thereby fostering ionic chemical bonds among these atoms. Bond population serves as another indicator of bond covalency within a crystal, as a high value of bond population essentially signifies a heightened degree of covalency within the chemical bond. The bonding and anti-bonding states influence the populations with positive and negative bond overlap. As demonstrated in Table-5, the B-B bond exhibits greater covalency compared to any other bond in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The presence of an antibonding state between two relevant atoms, which subsequently decreases their chemical bonding, is indicated by a hostile bond overlap population. In Mo2GeB<sup>2</sup> and Ta2GeB2, a hostile bond overlap population is observed in the Ge-Mo and Ge-Ta bonds, indicating the presence of an antibonding state. Therefore, the existence of ionic bonding is guaranteed by electronic charge transfer. On the other hand, the high positive value of the bond overlap population (BOP) denotes the presence of covalent bonding, a feature shared by materials in the MAX phase. ## 3.4.4 Theoretical Vickers Hardness Vickers hardness, derived from the atomic bonds found in solids, indicates how resistant a material is to deformation in extreme circumstances. Several variables influence this feature, such as the crystal flaws, solid structure, atomic arrangement, and bond strength. The Vickers hardness of the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) MAX phases is determined using the Mulliken bond population method, as described by Gou et al. [66] using the formula (S7-S10). This method is particularly suitable for partial metallic systems such as MAX phases. Table-5 lists the computed Vickers hardness values for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The calculated values are 3.35 GPa, 4.76 GPa, 8.21 GPa, and 8.74 GPa for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, respectively. We observed that Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> have much higher hardness values than Mo2GaB<sup>2</sup> and Mo2GeB2. These values are also higher than that of other 212 phases, like, Ti2InB<sup>2</sup> (4.05 GPa) [50], Hf2InB<sup>2</sup> (3.94 GPa), and Hf2SnB<sup>2</sup> (4.41 GPa) [26], Zr2InB<sup>2</sup> (2.92 GPa) and Zr2TlB<sup>2</sup> (2.19 GPa) [25], Zr2GaB<sup>2</sup> (2.53GPa), Zr2GeB<sup>2</sup> (3.31GPa), Hf2GaB<sup>2</sup> (4.73GPa) and Hf2GeB2(4.83GPa) [67]The Hv calculated by the geometrical average of the individual bonding, where the bonding strength mainly determined by the BOP values. In the case of Ta2GaB<sup>2</sup> and Ta2GeB2, the BOP of M-B bonding is much higher compared to Mo2GaB<sup>2</sup> and Mo2GeB2. Even though the BOP of M-B for Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> is higher than that of the other 212 phases mentioned earlier. Therefore, higher values of Hv are expected for Ta2GaB<sup>2</sup> and Ta2GeB2. Additionally, we looked at the Vickers hardness value between 211 and 212 compounds and discovered that the 212 MAX phase compounds had a higher H<sup>V</sup> value. This is because a 2D layer of B atoms is positioned between the M atoms. The B atoms share two center-two electrons to form an extremely strong B-B bond [3]. Table 5 Calculated data for Mulliken bond number (n μ ), bond length (d μ ), bond overlap populations BOP, (P μ ), metallic populations (P μˊ), Vickers hardness (HV) M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. # 3.5 Thermal Properties MAX phases are ideal for high-temperature applications due to their exceptional mechanical qualities at elevated temperatures. As a result, researching the fundamental parameters necessary for predicting their application is of great interest, and these can be obtained from the vibrations of atoms or phonons. The Debye temperature (ΘD) of a solid is directly connected to its bonding strength, melting temperature, thermal expansion, and conductivity. Using the sound velocity and Anderson's technique [68], the Θ<sup>D</sup> of the phases under study has been computed using the formula (S11). Equation (S12) can be used to get the average sound velocity (Vm) from the longitudinal and transverse sound velocities. Equations (S13–14) were used to determine v<sup>l</sup> and vt. The calculated values of Debye's temperature are shown in Table 6, where Mo2GaB<sup>2</sup> has the highest Θ<sup>D</sup> and Ta2GeB<sup>2</sup> has the lowest. If we rank them, it is as follows: Mo2GaB<sup>2</sup> < Mo2GeB<sup>2</sup> < Ta2GaB2 < Ta2GeB2. Hadi et al. recently reported a MAX phase (V2SnC) as a TBC material with a Θ<sup>D</sup> value of 472 K [69]. Thus, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) exhibit encouraging potential as TBC materials, as shown in Table 6. Table 6-Data for density (ρ), longitudinal, transverse, and average sound velocities (vl, vt, and vm), Debye temperature (ΘD), minimum thermal conductivity (Kmin), Grüneisen parameter (γ), thermal expansion coefficient (TEC) at 300K and melting temperature (Tm) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The constant thermal conductivity value at high temperatures is the minimum thermal conductivity (Kmin). As the name implies, this conductivity is minimal because, at high temperatures, phonon coupling breaks. The formula (S15) for the minimum thermal conductivity of solids was derived using the Clarke model [70]. Table 6 lists the calculated value of kmin, with Ta2GaB<sup>2</sup> having the lowest value and Mo2GaB<sup>2</sup> having the highest. When selecting suitable materials for TBC applications, a minimum thermal conductivity of 1.25 W/mK is used as a screening criterion [71]. Our compounds exhibit lower minimum thermal conductivity values, holding promising potential as TBC materials. Gd2Zr2O<sup>7</sup> and Y2SiO5, two recently developed thermal barrier coating (TBC) materials, have minimal thermal conductivities (Kmin) of 1.22 W/m.K. and 1.3 W/m.K. [72], respectively, as confirmed by experiment. These numbers roughly match the values we computed for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The Grüneisen parameter (γ) is a crucial thermal parameter that helps explain the anharmonic effects of lattice dynamics; solids utilized at high temperatures are expected to have lower anharmonic effects. The Grüneisen parameter (γ) can be determined with the help of Poisson's ratio using equation (S16) [73]. The computed γ values, as shown in Table 6, suggest that the compounds under investigation exhibit a weak anharmonic effect. Additionally, for solids with a Poisson's ratio between 0.05 and 0.46, the values similarly fall within the range of 0.85 and 3.53 [74]. The melting temperature (Tm) of the compounds under investigation has been calculated using the following formula (S17). The strength of atomic bonding is the primary factor determining the melting temperature of solids; the higher the Tm, the stronger the atomic bonding. The order of T<sup>m</sup> for the titled phases is found to mirror the Y-based (Young's modulus) order, indicating a close link between T<sup>m</sup> and Y [75]. As observed in Table 6, our compounds roughly follow the Ybased ranking with Mo2GeB<sup>2</sup> < Mo2GaB<sup>2</sup> < Ta2GeB<sup>2</sup> < Ta2GaB2. The T<sup>m</sup> value for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) is also comparable with the TBC material Y4Al2O<sup>9</sup> (2000 K)[75]. ![Fig. 6: Temperature dependence of the calculated thermodynamic parameters. (a) Specific heat at constant pressure, C<sup>p</sup> (b) Specific heat at constant volume, C<sup>v</sup> (c) Free energy, F (d) Entropy, S (e) Internal energy, E and (f) Thermal expansion coefficient, TEC of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge).](path) The temperature dependence of specific heats, Cv, Cp, for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds was obtained using the formulas (S19-S20), as shown in Fig.6(a,b). Assuming that the quasi-harmonic model is accurate and that phase transitions are not anticipated for the compounds under study, these properties are approximated across a temperature range of 0 to 1000 K. Because phonon thermal softening occurs at higher temperatures, the heat capacity rises with temperature. Heat capacities increase quickly and follow the Debye-T 3 power law at lower temperatures. At higher temperature regimes, where C<sup>v</sup> and C<sup>p</sup> do not greatly depend on temperature, they approach the Dulong-Petit (3) limit.[76]. Using the quasi-harmonic approximation, various temperature-dependent thermodynamic potential functions for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have been estimated at zero pressure and displayed in Fig.6 [77]. These functions include the Helmholtz free energy (F), internal energy (E), and entropy (S) within 0-1000 K temperature using equation (S21-S23). The free energy progressively decreases as the temperature rises, as shown in Fig.6(c). Free energy typically declines, and this trend becomes increasingly negative as a natural process proceeds. As demonstrated in Fig. 6(d), the internal energy (E) shows a rising trend with temperature, in contrast to the free energy. Since thermal agitation creates disorder, a system's entropy rises as temperature rises. This is illustrated in Fig.6(d). A material's thermal expansion coefficient (TEC) is derived from the anharmonicity in the lattice dynamics and can be found using equation (S18). The measure of a material's capacity to expand or contract with heat or cold is called the Thermal Expansion Coefficient, or TEC. As observed in Fig. 5(f), the Thermal Expansion Coefficient (TEC) increases rapidly up to 365 K. Then it approaches a constant value, which indicates lower saturations in the materials with temperature changes. The materials under study have a very low TEC value, a crucial characteristic of materials intended for application in high-temperature technology. To be effective as thermal barrier coating (TBC) materials, compounds must exhibit a low thermal conductivity (Kmin) to impede heat transfer, a high melting temperature to withstand extreme heat, and a low thermal expansion coefficient (TEC) to maintain dimensional stability under thermal stress. The compounds M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge) possess these properties, making them suitable candidates for use as TBC materials #### 3.6 Optical Properties Different materials exhibit unique behaviors when exposed to electromagnetic radiation. The optical constants determine the overall response of the sample to the incident radiation. The complex dielectric function, defined as ε(ω)=ε1(ω)+iε2(ω), is one of the main optical characteristics of solids. The following formula determines the imaginary part of the dielectric function ε2(ω) from the momentum matrix element between the occupied and unoccupied electronic states. In this formula, e stands for an electronic charge, ω for light angular frequency, u for the polarization vector of the incident electric field, and and for the conduction and valence band wave functions, respectively, at k. The Kramers-Kronig equation can estimate the real part of the dielectric function, ε1(ω). In contrast, ε2(ω) and ε1(ω) are utilized to evaluate all other optical parameters, such as the absorption coefficient, photoconductivity, reflectivity, and loss function [78]. In this part, several energy-dependent optical properties of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) 212 MAX phases are calculated and analyzed in detail for the photon energy range of 0 to 30 eV, for [100] plane enabling the first assessment of the compounds' practical applicability. Given the metallic conductivity of the MAX compounds' electronic structure, additional parameters were chosen to analyze the optical properties. These include a plasma frequency of 3 eV, damping of 0.05 eV, and Gaussian smearing of 0.5 eV [79]. Figure 7(a) shows the real component of the dielectric function, ε1, which exhibits metallic behavior. In metallic systems, ε<sup>1</sup> has a considerably high negative value in the low-energy range, with the real component reaching negative, which aligns with the band structure finding. Fig. 7(b) depicts the imaginary part of the dielectric function, ε2(ω), representing dielectric losses about frequency. Mo2GeB<sup>2</sup> demonstrates the highest peak in the low-energy region, with all compounds approaching zero from above at around 17 eV. This observation confirms Drude's behavior. ![Fig. 7- (a) Real part of dielectric constant, ε<sup>1</sup> (b) Imaginary part of dielectric constant, ε<sup>2</sup> (c) Reflectivity, R and (d) Loss function, (e) The coefficient of absorption, (α) (f) Refractive index, (n) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) for [100] electric field directions.](path) The reflectivity of a material measures the percentage of incident light energy reflected off it. Equation (S24) was employed to compute reflectivity using the dielectric function, as shown in Fig.7(c). The spectra's visible and infrared (IR) portions consistently exhibit values exceeding 55%. Consequently, the materials under examination are expected to appear metallic gray. In the infrared (IR) region, the maximum reflectivity for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> is 65% and 74%, respectively, occurring at 0.33 eV. Conversely, for Ta2GaB2, the maximum reflectivity is 59% at 14.02 eV, while for Ta2GeB2, it is 65% at 14.78 eV in the UV region. After 18 eV, the reflectance decreases significantly. According to reports, substances with an average reflectivity value above 44% in the visible light region can effectively reduce solar heating by reflecting a significant amount of the solar spectrum. Among all the compounds we examined, Ta2GaB<sup>2</sup> has a reflectivity in the visible range of more than 44% [78]. Therefore, Ta2GaB<sup>2</sup> can be utilized as a coating material and should be able to mitigate solar heating. plasma frequencies (ωp). The plasma frequency for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is observed at 23 eV, 25 eV, 21 eV, and 23 eV, respectively. At this specific frequency, the absorption coefficient rapidly decreases, ε<sup>1</sup> crosses zero from the negative side, and the reflectance R(ω) displays a falling tail. Above this distinctive frequency, the materials are transparent to the incident electromagnetic radiation. Fig. 7(e) illustrates the absorbance coefficient (α) of compounds M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) determined using equation (S26). As α begins to rise from zero photon energy, the metallic nature of the substances under study is again indicated. The visible light region experiences a sharp increase in absorption, peaking in the UV region at around 14 eV and then progressively declining. The IR region exhibits negligible absorption. However, the materials mentioned above appear to have a significant absorption band primarily located in the visible and ultraviolet spectrum. This suggests that the materials can be used in UV surface-disinfection devices, medical sterilization equipment, and other optoelectronic device designs. Its refractive index is crucial to a material's potential application in optical devices like waveguides and photonic crystals. Equations (S27) and (S28) were used to derive the refractive index (n) and extinction coefficient (k) for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), as depicted in Fig.7(c) and Fig.S-1(a). The variations of n and k in MAX phase carbides with incident photon energy closely resemble ε1(ω) and ε2(ω). The static refractive index n(0) for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is 9.4,13.07,7 and 9.2, respectively, and decreases gradually with the increase in photon energy. The extinction coefficient for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2 gradually increases in the IR region, reaching their maximum values of 2.56, 4.27, 2.42, and 3.28, respectively. Following this, they slowly decrease in the visible and UV regions. Figure S-1(b) illustrates the photoconductivity (σ) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) across various photon energies. The photoconductivity (σ) parameter quantifies the impact of photon irradiation on a material's electrical conductivity. Similar to the absorbance coefficient (α) spectrum, the σ spectrum aligns with the metallic nature of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). #### 4 Conclusions In summary, we employed DFT calculations to explore four 212 MAB phases, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), and investigated the structural, electronic, mechanical, lattice dynamical, and optical properties to predict their possible applications. The phonon dispersion curves, formation energy, and elastic constants collectively suggest that the M2AB<sup>2</sup> boride maintains dynamic, mechanical, chemical, and thermodynamic stability. The electronic band structure and density of states (DOS) offer evidence supporting the metallic nature of the compounds under investigation. Concurrently, the charge density mapping and atomic Mulliken population both confirm the presence of a strong B-B covalent bond. The stiffness constants, elastic moduli, findex, fracture toughness (KIC), Pugh's ratio (G/B), hardness parameters, and Cauchy Pressure (CP) of M2AB<sup>2</sup> were computed and compared with those of their 211 equivalents. We found that the values of the 212-phase borides are higher than those of the 211-phase carbides or borides. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are identified as brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit ductile characteristics, as indicated by Poisson's ratio (ν) and Pugh's ratio (B/G or G/B). The elastic characteristics display anisotropy due to the distinct atomic configurations along the aand c-directions. Vickers hardness calculations are considered reliable indicators of material hardness. The results suggest that Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> possess more pliable characteristics than Ta2GaB<sup>2</sup> and Ta2GeB2. The high hardness values of Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compared to the Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> due to higher BOP values of M-B bonding in Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> than in Mo2GaB<sup>2</sup> and Mo2GeB2. The value of minimum thermal conductivity (Kmin), thermal expansion coefficient, and melting temperature (Tm) collectively suggest the potential suitability of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) as a material for thermal barrier coating (TBC) applications in high-temperature devices. The optical conductivity and absorption coefficient corroborate the findings of the electronic band structure. Reflectivity is notably high in infrared (IR) regions and remains nearly constant in the visible and moderate ultraviolet (UV) regions, with an average value exceeding 44% for Ta2GaB2. This suggests that Ta2GaB<sup>2</sup> can be effectively utilized as a coating material to reduce solar heating. We expect that the comprehensive analysis of the diverse physical characteristics of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) presented in this study will establish a robust foundation for future theoretical and experimental explorations of these fascinating MAB phases. ## CRediT author statement A. K. M Naim Ishtiaq and Md Nasir Uddin: Data curation, Writing- Original draft preparation. Md. Rasel Rana, Shariful Islam and Noor Afsary: Reviewing and Editing. Md. 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Fig. 7- (a) Real part of dielectric constant, ε<sup>1</sup> (b) Imaginary part of dielectric constant, ε<sup>2</sup> (c) Reflectivity, R and (d) Loss function, (e) The coefficient of absorption, (α) (f) Refractive index, (n) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) for [100] electric field directions.	# Exploration of new 212 MAB phases: M2AB2 (M=Mo, Ta; A=Ga, Ge) via DFT calculations #### Abstract The recently developed MAB phases, an extension of the MAX phase, have sparked interest in research among scientists because of their better thermo-mechanical properties. In this paper, we have explored four new MAB phases M2AB2 (M=Mo, Ta and A=Ga, Ge) and studied the elastic, electronic, thermal, and optical properties to predict the possible applications. The stability of the new phases has been confirmed by calculating formation energy (Ef), formation enthalpy (∆H), phonon dispersion curve (PDC), and elastic constant (Cij). The study reveals that M2AB<sup>2</sup> (M=Mo, Ta and A=Ga, Ge) exhibit significantly higher elastic constants, elastic moduli, and Vickers hardness values than their counterpart 211 borides. Higher Vickers hardness values of Ta2AB2 (A=Ga, Ge) than Mo2AB2 (A=Ga, Ge) have been explained based on the values of the bond overlap population. The analysis of the density of states and electronic band structure revealed the metallic nature of the borides under examination. The thermodynamic characteristics of M2AB2 (M=Mo, Ta and A=Ga, Ge) under high temperatures (0–1000 K) are investigated using the quasi-harmonic Debye model. Critical thermal properties such as melting temperature (Tm), Grüneisen parameter (γ), minimum thermal conductivity (Kmin), Debye temperature (ΘD), and others are also computed. Compared with 211 MAX phases, the 212 phases exhibit higher values of (ΘD) and Tm, along with a lower value of Kmin. These findings suggest that the studied compounds exhibit superior thermal properties that are suitable for practical applications. The optical characteristics have been examined, and the reflectance spectrum indicates that the materials have the potential to mitigate solar heating across various energy regions. #First two authors contributed equally. ### 1. Introduction The MAX phase has garnered significant attention in the present era due to its outstanding mechanical and thermal characteristics at high temperatures, showcasing attributes shared by both metals and ceramics. The increased interest in MAX phase materials can be traced back to Barsoum's noteworthy contributions [1], [2]. The term MAX phase represents a family of multilayer solids where M is an earlier transition metal, A is an element from the IIIA or IVA group of the periodic table, and X is an atom of C/N/B [3]. MAX phase materials showcase metallic behavior due to alternate metallic A-layers and ceramic behavior attributed to the MX layers [4]. Like most metals and alloys, MAX phase materials have excellent thermal shock resistance, superior machinability, and enhanced thermal and electrical conductivities. On the other hand, they have high melting or decomposition temperatures and strong elastic stiffness, similar to many ceramics[5]. The unique combination of metallic and ceramic properties makes them versatile, with applications ranging from high-temperature coatings to nuclear accidenttolerant fuel (ATF), concentrated solar power (CSP), catalysis, and as precursors for MXenes [6], [7], [8], [9]. The diversity of MAX phases was confined to C and N as X elements for years (from 1960 to 2014). However, recent advancements have overcome this limitation by successfully synthesizing MAX phases that contain B [10]. The physical and chemical properties of B and Bcontaining compounds highlight the potential of MAX phase borides, which replace C/N with boron [11]. Due to the presence of B in the composition, they are also called the MAB phase. Furthermore, scientific communities are actively working to broaden the diversity of MAX phases by introducing structural changes, as seen in examples like Cr3AlB<sup>4</sup> (space group Immm), Cr4AlB<sup>6</sup> (space group Cmmm), Cr4AlB<sup>4</sup> [12], [13], [14]. Khazaei et al. reported on the first investigation of the theoretical MAX phase borides M2AlB (where M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta). They evaluated these compounds' electronic structure, mechanical characteristics, and dynamical stability in their investigation [15]. The diversity of MAX phases is a subject of research interest due to their structure variations and the number of atoms in the compounds, resulting in changes to their characteristics. In examples like the 211 MAX phase [16], 312 MAX phase [8][17], 212 MAX phase [3], and i-MAX phase [18], the recent additions to the MAX family include the 314 MAX phase and 212 MAX phase, where the element B has been incorporated as an X element [3] [19]. This introduces a novel aspect to the MAX phase family, contributing to its diversity. The unique structural features of MAB phases include both orthorhombic and hexagonal symmetry observed in crystals. These distinctive symmetries set MAB phases apart from the typical MAX phases. Yinqiao Liu's synthesis of the orthorhombic phase of M2AlB2, particularly the 212-MAB phases with M = Sc, Ti, Zr, Hf, V, Nb, Cr, Mo, W, Mn, Tc, Fe, Co, and Ni, has revealed unique structural stability and notable electrical and mechanical properties [20]. The MAB phase structure slightly differs from the standard MAX phases, which typically crystallize in the hexagonal system with a space group -6<sup>2</sup> (No. 187). Ali et al [3]. investigated the diverse physical properties of Zr2AB<sup>2</sup> (A = In, Tl). Martin Ade et al [13] synthesized ternary borides, namely Cr2AlB2, Cr3AlB4, and Cr4AlB6, and subsequently compared their mechanical properties. Qureshi et al. [19] investigated the 314 Zr3CdB<sup>4</sup> MAX phase boride, calculating its mechanical, thermodynamic, and optical properties. The 314 MAX phase Hf3PB<sup>4</sup> has been thoroughly studied using Density Functional Theory (DFT) and revealed that it was the hardest MAX phase compound discovered until that date [21]. The structure of 212 phases exhibits a slight deviation from 211 MAX phases. In the case of 212 phases, a 2D layer of B is situated between M layers, featuring an additional B atom at the X position, unlike in 211 phases [22] [23], [24]. The B-B bonding in 212 MAX phase borides has improved mechanical and thermal properties. To date, the physical properties of Zr2AB<sup>2</sup> (A = In, Tl) [25] and Hf2AB<sup>2</sup> (A = In, Sn) [26], M2AB (M= Ti, Zr, Hf; A=Al, Ga, In) [27] MAX phases, as well as Nb2AC (A = Ga, Ge, Tl, Zn, P, In, and Cd) MAX phases, have been investigated using density functional theory (DFT). In each instance, the mechanical properties of B-containing compounds show significant improvement compared to their traditional C/N containing 211 MAX phases. The Debye temperature and melting temperature are higher for boron-containing 212 phases than for 211 carbides/nitrides, while the minimum thermal conductivity is lower. The thermal expansion coefficient of borides remains well-suited for use as coating materials. Consequently, the superior thermomechanical properties of B-containing 212 MAX phases demonstrate their suitability for high-temperature technological applications, surpassing the commonly used 211 MAX phase carbides. It should be noted that the 212 MAB phase with a hexagonal structure has already been synthesized [24]. In addition, Ga and Ge-based MAX phases have also been synthesized previously [28]. Thus, the reports on the synthesis of 212 phase and Ga and Gebased MAX phases motivated us to select the 212 MAB phases: M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) for our present study, and we have performed an in-depth investigation of their physical properties through DFT method. Therefore, in this paper, the first-time prediction of the stability and mechanical, electronic, thermal, and optical properties of M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) phases has been presented. The results revealed that M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) compounds are stable and suitable for thermal barrier coating (TBC) and reflection coating applications. Additionally, to provide a comparison, the properties determined for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) are compared with those of other 212 and 211 compounds MAX phase borides. ### 2. Methods of calculations First principles density-functional theory (DFT) computations are conducted utilizing the Cambridge Serial Total Energy Package (CASTEP) module integrated within Materials Studio 2017 [29], [30]. The exchange-correlation function is estimated using the Generalized Gradient Approximation (GGA) method, originally suggested by Perdew, Burke, and Ernzerhof [31]. Pseudo-atomic simulations accounted for electronic orbitals corresponding to B (2s 2 2p 1 ), Ga (3d <sup>10</sup> 4s 2 4p 1 ), Ge (4s 2 4p 2 ), Mo (4s 2 4p 6 4d 5 5s 1 ), and Ta (5d 3 6s 2 ). The energy cutoff and k-point grids were established at 650 eV and 11 × 11 × 4, respectively. The structural relaxation was performed utilizing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique [32], while the electronic structure was computed employing density mixing. The parameters for relaxed structures incorporate the following tolerance thresholds: the self-consistent convergence of the total energy is set at 5 × 10-6 eV/atom, the maximum force exerted on the atom is limited to 0.01 eV/Å, the maximum ionic displacement is constrained to 5 × 10-4 Å, and a maximum stress threshold of 0.02 GPa is imposed. The finite strain method [33], grounded in density functional theory (DFT), is utilized to compute the elastic properties within this framework. All necessary equations for determining various properties are provided in the supplementary document. #### 3 Results and discussion #### 3.1 Structural properties The M2AB<sup>2</sup> compounds (where M=Mo or Ta; A=Ga or Ge) belong to the P6m<sup>2</sup> (No. 187)[23] space group and crystallize in the hexagonal system. Unlike conventional MAX phases, which typically belong to the P63/mmc (194) space group, the 212 MAX phases exhibit distinct characteristics. In Fig. 1, the unit cell structure of Mo2GaB<sup>2</sup> is depicted as a representative of M2AB<sup>2</sup> alongside Mo2GaB, facilitating a comparison to discern their differences easily. The atomic positions are as follows: M (Mo or Ta) at (0.3333, 0.6667, 0.6935), A at (0.6667, 0.3333, 0.0), and two B atoms positioned at (0.6667, 0.3333, 0.5) and (0.0, 0.0, 0.5). The B components are arranged at the corners of the unit cell in typical 211 MAX phases, but in 212 boride MAX phases, they form a 2D layer between the M layers. This structural arrangement results in B-B covalent bonds in the 2D layer, enhancing stability compared to conventional 211 MAX phases. The reflectivity of a material measures the percentage of incident light energy reflected off it. Equation (S24) was employed to compute reflectivity using the dielectric function, as shown in Fig.7(c). The spectra's visible and infrared (IR) portions consistently exhibit values exceeding 55%. Consequently, the materials under examination are expected to appear metallic gray. In the infrared (IR) region, the maximum reflectivity for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> is 65% and 74%, respectively, occurring at 0.33 eV. Conversely, for Ta2GaB2, the maximum reflectivity is 59% at 14.02 eV, while for Ta2GeB2, it is 65% at 14.78 eV in the UV region. After 18 eV, the reflectance decreases significantly. According to reports, substances with an average reflectivity value above 44% in the visible light region can effectively reduce solar heating by reflecting a significant amount of the solar spectrum. Among all the compounds we examined, Ta2GaB<sup>2</sup> has a reflectivity in the visible range of more than 44% [78]. Therefore, Ta2GaB<sup>2</sup> can be utilized as a coating material and should be able to mitigate solar heating. plasma frequencies (ωp). The plasma frequency for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is observed at 23 eV, 25 eV, 21 eV, and 23 eV, respectively. At this specific frequency, the absorption coefficient rapidly decreases, ε<sup>1</sup> crosses zero from the negative side, and the reflectance R(ω) displays a falling tail. Above this distinctive frequency, the materials are transparent to the incident electromagnetic radiation. Fig. 7(e) illustrates the absorbance coefficient (α) of compounds M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) determined using equation (S26). As α begins to rise from zero photon energy, the metallic nature of the substances under study is again indicated. The visible light region experiences a sharp increase in absorption, peaking in the UV region at around 14 eV and then progressively declining. The IR region exhibits negligible absorption. However, the materials mentioned above appear to have a significant absorption band primarily located in the visible and ultraviolet spectrum. This suggests that the materials can be used in UV surface-disinfection devices, medical sterilization equipment, and other optoelectronic device designs. Its refractive index is crucial to a material's potential application in optical devices like waveguides and photonic crystals. Equations (S27) and (S28) were used to derive the refractive index (n) and extinction coefficient (k) for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), as depicted in Fig.7(c) and Fig.S-1(a). The variations of n and k in MAX phase carbides with incident photon energy closely resemble ε1(ω) and ε2(ω). The static refractive index n(0) for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is 9.4,13.07,7 and 9.2, respectively, and decreases gradually with the increase in photon energy. The extinction coefficient for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2 gradually increases in the IR region, reaching their maximum values of 2.56, 4.27, 2.42, and 3.28, respectively. Following this, they slowly decrease in the visible and UV regions. Figure S-1(b) illustrates the photoconductivity (σ) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) across various photon energies. The photoconductivity (σ) parameter quantifies the impact of photon irradiation on a material's electrical conductivity. Similar to the absorbance coefficient (α) spectrum, the σ spectrum aligns with the metallic nature of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). #### 4 Conclusions In summary, we employed DFT calculations to explore four 212 MAB phases, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), and investigated the structural, electronic, mechanical, lattice dynamical, and optical properties to predict their possible applications. The phonon dispersion curves, formation energy, and elastic constants collectively suggest that the M2AB<sup>2</sup> boride maintains dynamic, mechanical, chemical, and thermodynamic stability. The electronic band structure and density of states (DOS) offer evidence supporting the metallic nature of the compounds under investigation. Concurrently, the charge density mapping and atomic Mulliken population both confirm the presence of a strong B-B covalent bond. The stiffness constants, elastic moduli, findex, fracture toughness (KIC), Pugh's ratio (G/B), hardness parameters, and Cauchy Pressure (CP) of M2AB<sup>2</sup> were computed and compared with those of their 211 equivalents. We found that the values of the 212-phase borides are higher than those of the 211-phase carbides or borides. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are identified as brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit ductile characteristics, as indicated by Poisson's ratio (ν) and Pugh's ratio (B/G or G/B). The elastic characteristics display anisotropy due to the distinct atomic configurations along the aand c-directions. Vickers hardness calculations are considered reliable indicators of material hardness. The results suggest that Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> possess more pliable characteristics than Ta2GaB<sup>2</sup> and Ta2GeB2. The high hardness values of Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compared to the Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> due to higher BOP values of M-B bonding in Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> than in Mo2GaB<sup>2</sup> and Mo2GeB2. The value of minimum thermal conductivity (Kmin), thermal expansion coefficient, and melting temperature (Tm) collectively suggest the potential suitability of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) as a material for thermal barrier coating (TBC) applications in high-temperature devices. The optical conductivity and absorption coefficient corroborate the findings of the electronic band structure. Reflectivity is notably high in infrared (IR) regions and remains nearly constant in the visible and moderate ultraviolet (UV) regions, with an average value exceeding 44% for Ta2GaB2. This suggests that Ta2GaB<sup>2</sup> can be effectively utilized as a coating material to reduce solar heating. We expect that the comprehensive analysis of the diverse physical characteristics of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) presented in this study will establish a robust foundation for future theoretical and experimental explorations of these fascinating MAB phases. ## CRediT author statement A. K. M Naim Ishtiaq and Md Nasir Uddin: Data curation, Writing- Original draft preparation. Md. Rasel Rana, Shariful Islam and Noor Afsary: Reviewing and Editing. Md. 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Fig. 3 – Band structure of (a) Mo2GaB2, (b) Mo2GeB2, (c) Ta2GaB2, and (d) Ta2GeB<sup>2</sup> compounds.	# Exploration of new 212 MAB phases: M2AB2 (M=Mo, Ta; A=Ga, Ge) via DFT calculations #### Abstract The recently developed MAB phases, an extension of the MAX phase, have sparked interest in research among scientists because of their better thermo-mechanical properties. In this paper, we have explored four new MAB phases M2AB2 (M=Mo, Ta and A=Ga, Ge) and studied the elastic, electronic, thermal, and optical properties to predict the possible applications. The stability of the new phases has been confirmed by calculating formation energy (Ef), formation enthalpy (∆H), phonon dispersion curve (PDC), and elastic constant (Cij). The study reveals that M2AB<sup>2</sup> (M=Mo, Ta and A=Ga, Ge) exhibit significantly higher elastic constants, elastic moduli, and Vickers hardness values than their counterpart 211 borides. Higher Vickers hardness values of Ta2AB2 (A=Ga, Ge) than Mo2AB2 (A=Ga, Ge) have been explained based on the values of the bond overlap population. The analysis of the density of states and electronic band structure revealed the metallic nature of the borides under examination. The thermodynamic characteristics of M2AB2 (M=Mo, Ta and A=Ga, Ge) under high temperatures (0–1000 K) are investigated using the quasi-harmonic Debye model. Critical thermal properties such as melting temperature (Tm), Grüneisen parameter (γ), minimum thermal conductivity (Kmin), Debye temperature (ΘD), and others are also computed. Compared with 211 MAX phases, the 212 phases exhibit higher values of (ΘD) and Tm, along with a lower value of Kmin. These findings suggest that the studied compounds exhibit superior thermal properties that are suitable for practical applications. The optical characteristics have been examined, and the reflectance spectrum indicates that the materials have the potential to mitigate solar heating across various energy regions. #First two authors contributed equally. ### 1. Introduction The MAX phase has garnered significant attention in the present era due to its outstanding mechanical and thermal characteristics at high temperatures, showcasing attributes shared by both metals and ceramics. The increased interest in MAX phase materials can be traced back to Barsoum's noteworthy contributions [1], [2]. The term MAX phase represents a family of multilayer solids where M is an earlier transition metal, A is an element from the IIIA or IVA group of the periodic table, and X is an atom of C/N/B [3]. MAX phase materials showcase metallic behavior due to alternate metallic A-layers and ceramic behavior attributed to the MX layers [4]. Like most metals and alloys, MAX phase materials have excellent thermal shock resistance, superior machinability, and enhanced thermal and electrical conductivities. On the other hand, they have high melting or decomposition temperatures and strong elastic stiffness, similar to many ceramics[5]. The unique combination of metallic and ceramic properties makes them versatile, with applications ranging from high-temperature coatings to nuclear accidenttolerant fuel (ATF), concentrated solar power (CSP), catalysis, and as precursors for MXenes [6], [7], [8], [9]. The diversity of MAX phases was confined to C and N as X elements for years (from 1960 to 2014). However, recent advancements have overcome this limitation by successfully synthesizing MAX phases that contain B [10]. The physical and chemical properties of B and Bcontaining compounds highlight the potential of MAX phase borides, which replace C/N with boron [11]. Due to the presence of B in the composition, they are also called the MAB phase. Furthermore, scientific communities are actively working to broaden the diversity of MAX phases by introducing structural changes, as seen in examples like Cr3AlB<sup>4</sup> (space group Immm), Cr4AlB<sup>6</sup> (space group Cmmm), Cr4AlB<sup>4</sup> [12], [13], [14]. Khazaei et al. reported on the first investigation of the theoretical MAX phase borides M2AlB (where M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta). They evaluated these compounds' electronic structure, mechanical characteristics, and dynamical stability in their investigation [15]. The diversity of MAX phases is a subject of research interest due to their structure variations and the number of atoms in the compounds, resulting in changes to their characteristics. In examples like the 211 MAX phase [16], 312 MAX phase [8][17], 212 MAX phase [3], and i-MAX phase [18], the recent additions to the MAX family include the 314 MAX phase and 212 MAX phase, where the element B has been incorporated as an X element [3] [19]. This introduces a novel aspect to the MAX phase family, contributing to its diversity. The unique structural features of MAB phases include both orthorhombic and hexagonal symmetry observed in crystals. These distinctive symmetries set MAB phases apart from the typical MAX phases. Yinqiao Liu's synthesis of the orthorhombic phase of M2AlB2, particularly the 212-MAB phases with M = Sc, Ti, Zr, Hf, V, Nb, Cr, Mo, W, Mn, Tc, Fe, Co, and Ni, has revealed unique structural stability and notable electrical and mechanical properties [20]. The MAB phase structure slightly differs from the standard MAX phases, which typically crystallize in the hexagonal system with a space group -6<sup>2</sup> (No. 187). Ali et al [3]. investigated the diverse physical properties of Zr2AB<sup>2</sup> (A = In, Tl). Martin Ade et al [13] synthesized ternary borides, namely Cr2AlB2, Cr3AlB4, and Cr4AlB6, and subsequently compared their mechanical properties. Qureshi et al. [19] investigated the 314 Zr3CdB<sup>4</sup> MAX phase boride, calculating its mechanical, thermodynamic, and optical properties. The 314 MAX phase Hf3PB<sup>4</sup> has been thoroughly studied using Density Functional Theory (DFT) and revealed that it was the hardest MAX phase compound discovered until that date [21]. The structure of 212 phases exhibits a slight deviation from 211 MAX phases. In the case of 212 phases, a 2D layer of B is situated between M layers, featuring an additional B atom at the X position, unlike in 211 phases [22] [23], [24]. The B-B bonding in 212 MAX phase borides has improved mechanical and thermal properties. To date, the physical properties of Zr2AB<sup>2</sup> (A = In, Tl) [25] and Hf2AB<sup>2</sup> (A = In, Sn) [26], M2AB (M= Ti, Zr, Hf; A=Al, Ga, In) [27] MAX phases, as well as Nb2AC (A = Ga, Ge, Tl, Zn, P, In, and Cd) MAX phases, have been investigated using density functional theory (DFT). In each instance, the mechanical properties of B-containing compounds show significant improvement compared to their traditional C/N containing 211 MAX phases. The Debye temperature and melting temperature are higher for boron-containing 212 phases than for 211 carbides/nitrides, while the minimum thermal conductivity is lower. The thermal expansion coefficient of borides remains well-suited for use as coating materials. Consequently, the superior thermomechanical properties of B-containing 212 MAX phases demonstrate their suitability for high-temperature technological applications, surpassing the commonly used 211 MAX phase carbides. It should be noted that the 212 MAB phase with a hexagonal structure has already been synthesized [24]. In addition, Ga and Ge-based MAX phases have also been synthesized previously [28]. Thus, the reports on the synthesis of 212 phase and Ga and Gebased MAX phases motivated us to select the 212 MAB phases: M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) for our present study, and we have performed an in-depth investigation of their physical properties through DFT method. Therefore, in this paper, the first-time prediction of the stability and mechanical, electronic, thermal, and optical properties of M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) phases has been presented. The results revealed that M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) compounds are stable and suitable for thermal barrier coating (TBC) and reflection coating applications. Additionally, to provide a comparison, the properties determined for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) are compared with those of other 212 and 211 compounds MAX phase borides. ### 2. Methods of calculations First principles density-functional theory (DFT) computations are conducted utilizing the Cambridge Serial Total Energy Package (CASTEP) module integrated within Materials Studio 2017 [29], [30]. The exchange-correlation function is estimated using the Generalized Gradient Approximation (GGA) method, originally suggested by Perdew, Burke, and Ernzerhof [31]. Pseudo-atomic simulations accounted for electronic orbitals corresponding to B (2s 2 2p 1 ), Ga (3d <sup>10</sup> 4s 2 4p 1 ), Ge (4s 2 4p 2 ), Mo (4s 2 4p 6 4d 5 5s 1 ), and Ta (5d 3 6s 2 ). The energy cutoff and k-point grids were established at 650 eV and 11 × 11 × 4, respectively. The structural relaxation was performed utilizing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique [32], while the electronic structure was computed employing density mixing. The parameters for relaxed structures incorporate the following tolerance thresholds: the self-consistent convergence of the total energy is set at 5 × 10-6 eV/atom, the maximum force exerted on the atom is limited to 0.01 eV/Å, the maximum ionic displacement is constrained to 5 × 10-4 Å, and a maximum stress threshold of 0.02 GPa is imposed. The finite strain method [33], grounded in density functional theory (DFT), is utilized to compute the elastic properties within this framework. All necessary equations for determining various properties are provided in the supplementary document. #### 3 Results and discussion #### 3.1 Structural properties The M2AB<sup>2</sup> compounds (where M=Mo or Ta; A=Ga or Ge) belong to the P6m<sup>2</sup> (No. 187)[23] space group and crystallize in the hexagonal system. Unlike conventional MAX phases, which typically belong to the P63/mmc (194) space group, the 212 MAX phases exhibit distinct characteristics. In Fig. 1, the unit cell structure of Mo2GaB<sup>2</sup> is depicted as a representative of M2AB<sup>2</sup> alongside Mo2GaB, facilitating a comparison to discern their differences easily. The atomic positions are as follows: M (Mo or Ta) at (0.3333, 0.6667, 0.6935), A at (0.6667, 0.3333, 0.0), and two B atoms positioned at (0.6667, 0.3333, 0.5) and (0.0, 0.0, 0.5). The B components are arranged at the corners of the unit cell in typical 211 MAX phases, but in 212 boride MAX phases, they form a 2D layer between the M layers. This structural arrangement results in B-B covalent bonds in the 2D layer, enhancing stability compared to conventional 211 MAX phases. ![Fig. 1 - The schematic unit cell of (a) Mo2GaB<sup>2</sup> and (b) Mo2GaB compound.](path) The calculated lattice parameters of M2AB<sup>2</sup> are presented in Table 1 alongside those of other 211 and 212 MAX phases for comparison, demonstrating consistency with prior results [16], [34], [35] and affirming the accuracy of the computational methodology used. The primary distinction between 212- and 211-unit cell structures arises from differences in the lattice parameter c, where the c value for 211 exceeds that of 212. Moreover, the volumes of Mo- and Ta-based 211 MAX phase borides surpass those of 212 MAX phase borides. #### 3.2 Stability Examining a compound's stability is significant for multiple purposes, as it yields valuable information regarding the compound's synthesis parameters and aids in assessing the material's resilience across diverse environments, including thermal, compressive, and mechanical pressures. In this section, we delve into a comprehensive theoretical analysis concerning the chemical, dynamic, and mechanical stability of M2AB<sup>2</sup> compounds. The compound's chemical stability is determined by computing its formation energy by the following equation [36]: 2<sup>2</sup> = 22−( + + ) ++ . In the context provided, 22 represents the total energy of the compound after optimizing the unit cell. , , and denote the energies of the individual elements M, A, and B, respectively. The variables x, y, and z correspond to the number of atoms in the unit cell for M, Ga, and B, respectively. For Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, the calculated formation energy (Ef) are -1.8079 eV/atom, -1.8859 eV/atom, -2.1252 eV/atom, and -2.1933 eV/atom, respectively. The negative values signify the chemical stability of all compounds. The order of chemical stability can be expressed as Ta2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Mo2GaB2, indicating that Ta2GeB<sup>2</sup> is the most stable. Additionally, it's observed that MAX phase borides containing Ta are more stable than those containing Mo. Negative formation energy alone may not fully explain the chemical stability of M2AB2 (M=Mo, Ta and A=Ga, Ge). We calculated its formation enthalpy by examining potential pathways to evaluate its thermodynamic stability. For this analysis, we used the experimentally identified stable phases of MoB[37] and TaB[38], Ga4Mo[39], and B2Mo[40]. The potential decomposition pathways for our compounds, as determined from the Open Quantum Materials Database (OQMD), are outlined below. ![We have calculated the reaction energy as follows [41]:](path) Where, M=Mo, Ta and A=Ga, Ge. The following formula can calculate the decomposition energy associated with the reaction energy. = . Where n is the number of participating atoms. The calculated decomposition energies for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> are -0.16, -0.17, -0.26, and -0.28 meV/atom, respectively. So, we can state that the M2AB2 (M=Mo, Ta and A=Ga, Ge) system exhibits thermodynamic stability. Phonon dispersion curves (PDCs) have been calculated at the ground state utilizing the density functional perturbation theory (DFPT) linear-response approach to evaluate the dynamic stability of the MAX phase borides under investigation [42]. The PDCs, depicting the phonon dispersion along the high symmetry directions of the crystal Brillouin zone (BZ), along with the total phonon density of states (PHDOS) of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, are illustrated in Fig. 2(a, b, c and d). Analysis of the PDCs reveals no negative phonon frequencies for any of the compounds, indicating their dynamic stability. The PHDOS of the M2AB<sup>2</sup> compounds are derived from the PDCs and are presented alongside the PDCs in Fig. 2(a, b, c, and d), facilitating band identification through comparison of corresponding peaks. From Fig. 2, it is observed that in Mo2GaB2, the flatness of the bands for the Transverse Optical (TO) modes results in a prominent peak in the PHDOS, whereas non-flat bands for the Longitudinal Optical (LO) modes lead to weaker peaks in the PHDOS. Similar trends are observed in Mo2GeB2, Ta2GaB2, and Ta2GeB2. Notably, a distinct discrepancy arises between the optical and acoustic branches, with the top of the LO and bottom of the TO modes situated at the G point, with separations of 7.49 THz, 6.39 THz, 9.39 THz, and 8.51 THz for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, respectively. ![Fig. 2 – Phonon DOS and Phonon dispersion curves of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) Materials are subjected to various forces and loads in practical applications, necessitating understanding their mechanical stability. The mechanical stability of a compound can be assessed using stiffness constants. For a hexagonal system, the conditions for mechanical stability are as follows [43]: C<sup>11</sup> > 0, C<sup>11</sup> > C12, C<sup>44</sup> > 0, and (C<sup>11</sup> + C12)C<sup>33</sup> - 2(C13)² > 0. As indicated in Table 2, the Cij values of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) satisfy these conditions, thus confirming the mechanical stability of herein predicted phases: M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge). #### 3.3 Electronic properties Analyzing the electronic band structure (EBS) is crucial for gaining insights into the electronic behavior of a compound. The EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) MAX phases are depicted in Fig. 3(a, b, c, and d), with the Fermi energy (EF) level set at 0 eV, represented by a horizontal line. Observing the EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge), it is evident that the conduction band overlaps with the valence band, indicating the absence of a band gap. This observation confirms that the M2AB<sup>2</sup> compounds exhibit metallic behavior, which aligns with conventional MAX phases. The red lines illustrate the overlapping band at the Fermi level. Fig. 3 (a, b) shows that for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> compounds, the maximum band overlap occurs along the A-H path. Conversely, in Fig. 3(c, d), the maximum band overlap is observed along the G-M paths. Utilizing the band structure, we can analyze the electrical anisotropy of M2AB<sup>2</sup> MAX phase compounds. The anisotropic nature can be understood by analyzing the energy dispersion in the basal plane and along the c-axis. The paths G-A, H-K, and M-L show energy dispersion along the c-direction, and A-H, K-G, G-M, and L-H show energy dispersion in the basal plane. In comparison to the paths A-H, K-G, G-M, and L-H (basal plane), there is less energy dispersion along the lines G-A, H–K, and M-L (c-direction), as shown by Fig. 2(a, b, c and d). Lower energy dispersion in the c-direction results from a higher effective mass [44], indicating the strong electronic anisotropy of the M2AB<sup>2</sup> MAX phase compound. Consequently, conductivity along the c-axis is expected to be lower than in the basal planes. These findings are consistent with prior studies [34], [45]. ![Fig. 3 – Band structure of (a) Mo2GaB2, (b) Mo2GeB2, (c) Ta2GaB2, and (d) Ta2GeB<sup>2</sup> compounds.](path) To investigate the bonding nature and electronic conductivity, the total density of states (TDOS) and partial density of states (PDOS) of M2AB<sup>2</sup> compounds were computed. Figure 4 (a, b, c, and d) illustrates the TDOS and PDOS of these compounds, with the Fermi energy (EF) set at zero energy level, indicated by a straight line. These profiles exhibit typical characteristics of MAX phase materials. The Mo or Ta-d electronic states predominantly contribute to the Fermi level, with a minor contribution from the B-p and Ga or Ge-p electronic states. To ascertain the hybridization characteristics of various electronic states within the valence band, the energy spectrum of the valence band has been partitioned into two distinct segments. The first segment encompasses the lower valence band region spanning from -7 eV to -3.5 eV, originating from the hybridization of Mo-p, Mo-d, and B-s orbitals in the case of the Mo2GaB<sup>2</sup> compound. Conversely, for the Mo2GeB2 compound, the lower valence band region arises from the hybridization of Mo-d, Ge-p, and B-s orbitals. Notably, for both the Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compounds, the dominance of the B-p state characterizes the lower valence band region. The second segment pertains to the upper valence band region from -3.5 eV to 0 eV. In the case of the Mo2GaB<sup>2</sup> compound, this region arises from the hybridization of Mo-p and Mo-d orbitals. However, for the Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, the upper valence band region results from the hybridization of Mo-d orbitals and (Ga/Ge)-p orbitals. Notably, the Fermi level of M2AB<sup>2</sup> resides near the pseudogap in the TDOS profile, indicating a high level of electronic stability. Similar trends are observed in other compounds like Zr2AlN, V2AlN, Sc2AlB, Sc2GaB, Ta2GaB, and Hf2GaB<sup>2</sup> [15], [35], [45]. ![Fig. 4 – TDOS and PDOS of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) The electron charge density mapping is helpful in understanding the distribution of electron densities linked to chemical bonds. It delineates areas of positive and negative charge densities, signifying the development and exhaustion of electrical charges, respectively. As depicted in the map, covalent bonds become apparent by accumulating charges between two atoms. Furthermore, the presence of ionic bonds can be inferred from a balance between negative and positive charges at specific atom positions[46]. The valence electronic CDM, denoted in units of eÅ-3 , for M2AB<sup>2</sup> (where M = Mo, Ta; A = Ga, Ge) is showcased in Fig. 5(a, b, c, and d) along the (110) crystallographic plane. The accompanying scale illustrates the intensity of electronic charge density, with red and blue colors indicating low and high electronic charge density, respectively. As depicted in Fig. 5(a, b, c, and d), it is evident that charges accumulate in the regions between the B sites. Consequently, it is anticipated that strong covalent B‒B bonding occurs through the formation of two center-two electron (2c‒2e) bonds in the M2AB<sup>2</sup> (where M=Mo, Ta; A=Ga, Ge) compound, similar to other 212 MAX phase borides like Ti2PB2, Zr2PbB2, Nb2SB2, Zr2GaB<sup>2</sup> and Hf2GaB<sup>2</sup> [34], [45]. Mulliken analysis has corroborated the charge transfer from Mo/Ta atoms to B atoms. The charge received from Mo/Ta atoms is distributed among the B atoms positioned at the transitions and those located at the edges, facilitating the formation of a two center-two electron (2c‒2e) bond between B atoms within the 2D layer of B, as illustrated in Fig. 1(a). The hardness of each bond value presented in Table 5 also aligns with the results obtained from charge density mapping (CDM) and our analysis using moduli and elastic stiffness constants. ![Fig. 5 – The charge density mapping of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) ## 3.4.1 Mechanical properties The mechanical properties of materials play a pivotal role in determining their potential applications, serving as crucial indicators of their behavior and suitability in materials engineering endeavors. These properties are equally applicable to MAX phase materials. Initially, to assess the mechanical properties of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phases, we employed the stress-strain method within the CASTEP code to compute the elastic constants (Cij) [22], [47], [48]. These calculated elastic constants (Cij) are presented in Table 2, alongside those of other 212 and 211 MAX phases. Due to the hexagonal crystal structure of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phase borides, five stiffness constants emerge: C11, C12, C13, C33, and C<sup>44</sup> [49]. The mechanical stability was evaluated using these stiffness constants in the preceding section. For instance, C<sup>11</sup> and C<sup>33</sup> determine stiffness when stress is exerted along the (100) and (001) directions, respectively, whereas C<sup>44</sup> evaluates resistance to shear deformation on the (100) and (001) planes. The stiffness constants C<sup>11</sup> and C<sup>12</sup> directly reflect the strength of atomic bonds along the a- and c-axes. When C<sup>11</sup> exceeds C<sup>33</sup> (or vice versa), it signifies stronger atomic bonding along the a-axis (or c-axis). In Table 2, for Mo2GaB2, Mo2GeB2, and Ta2GeB<sup>2</sup> compounds, C<sup>11</sup> surpasses C33, indicating superior atomic bonding along the a-axis compared to the c-axis. This robust bonding along the a-axis suggests heightened resistance to a-axial deformation. Conversely, in Ta2GeB2, where C<sup>11</sup> < C33, stronger atomic bonding along the c-axis translates to increased resistance against c-axial deformation. Analysis of Table-2 reveals that the values of C<sup>11</sup> and C<sup>22</sup> are notably higher for 212 MAX phase borides than 211 MAX phase borides. Consequently, it can be inferred that 212 MAX phases exhibit stronger resistance to axial deformation when juxtaposed with 211 MAX phases. C<sup>44</sup> is commonly utilized to gauge shear deformation tolerance among the elastic constants. Notably, Ta2GaB exhibits superior shear deformation resistance owing to its highest C<sup>44</sup> value among the studied compounds. Furthermore, the individual elastic constants C12, C13, and C<sup>44</sup> denote shear deformation response under external stress. The observation that C<sup>11</sup> and C<sup>33</sup> possess larger magnitudes than C<sup>44</sup> implies that shear deformation is more facile than axial strain. Another crucial parameter derived from the stiffness constants is the Cauchy pressure (CP), calculated as C<sup>12</sup> - C44, which provides vital insights relevant to the practical applications of solids [51]. Pettifor [51] emphasized the significance of Cauchy pressure (CP) in discerning the chemical bonding and ductile/brittle properties of solids. A negative CP value indicates covalently bonded brittle solids, whereas a positive value signifies isotropic ionic ductile solids. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> fall into covalently bonded brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit metallic ductile characteristics. Similarly, like Ga-containing 211 MAX phase borides, Ga-containing 212 MAX phases also demonstrate negative CP values and brittle behavior [52][53]. The elastic constants obtained are used to produce several bulk elastic parameters that are used to characterize polycrystalline materials, such as Young's modulus (Y), bulk modulus (B), and shear modulus (G). In Table 2, the bulk modulus (B) and shear modulus (G) calculated using Hill's approximation [54] are also presented. Hill's values represent the average of the upper limit (Voigt [55]) and lower limit (Reuss [56]) of B. The necessary equations for these calculations are provided in the supplementary document (S1). Young's modulus (Y) is a crucial indicator of material stiffness. A higher Y value indicates a stiffer material. It can be observed from the table that Mo2GaB<sup>2</sup> exhibits the highest Y value, signifying its greater stiffness compared to others. According to the sequence of Y values, stiffness can be ranked as follows: Mo2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Ta2GeB2. Therefore, Mo2GaB2, with its higher Young's modulus, is anticipated to demonstrate superior mechanical stability and deformation resistance compared to the other compounds under investigation. This quality is essential for aircraft parts or high-performance machinery applications where critical dimensional stability and structural integrity are required [57], [58]. Table 2 shows that 212 MAX phase materials exhibit larger Young's modulus (Y) values than 211 MAX phases. This suggests that 212 MAX phase borides are stiffer than their 211 MAX phase counterparts. Young's modulus (E) also correlates well with thermal shock resistance (R): R ∝ 1/E [59]. Lower Young's modulus values correspond to higher thermal shock resistance. Therefore, materials with higher thermal shock resistance (i.e., lower Young's modulus) are more suitable for use as Thermal Barrier Coating (TBC) materials. Given that Ta2GaB<sup>2</sup> possesses the lowest Y value among the materials studied, it should be considered a superior candidate for TBC material due to its higher thermal shock resistance. The material's ability to withstand shape distortion is elucidated by the shear modulus (G). On the other hand, the bulk modulus (B) indicates the strength of a material's chemical bonds and its ability to withstand uniform compression or volume change. We computed Young's modulus (Y), bulk modulus (B), and shear modulus for our analysis. Based on the bulk modulus values presented in Table 2, the sequence of a material that resists compression when pressure is applied can be outlined as follows: Mo2GeB<sup>2</sup> > Ta2GeB<sup>2</sup> > Mo2GaB<sup>2</sup> > Ta2GeB2. Mo2GeB2, boasting a higher bulk modulus, may exhibit reduced plastic deformation and superior resistance to stress-induced deformation compared to other compounds examined. The ductile and brittle characteristics of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds can also be assessed through Poisson's ratio (ν) and Pugh's ratio (G/B). A compound demonstrates ductile (brittle) behavior if the ν value surpasses (falls below) 0.26 [60]. Furthermore, if the G/B value exceeds (is less than) 0.571, then the compound exhibits brittle (ductile) behavior [19]. According to both criteria, Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are classified as brittle compounds, whereas Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> are categorized as ductile compounds. The ductile nature of Ge-based MAX phases has been reported previously [61]. Fracture toughness (KIC) is a vital property that gauges a material's ability to resist crack propagation. In the case of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, Equation (S2) was employed to determine KIC. The KIC values are documented in Table 2, and notably, these values surpass those of 211 MAX phases [16], [35]. Another parameter, the "f-index," characterizes the isotropic nature and strength of atom-atom bonds within a single hexagonal crystalline lattice along the a- and c-directions. If the f-index is less than 1, chemical bonds exhibit greater rigidity along the c-axis; conversely, if f exceeds 1, bonds are more rigid in the ab-plane. When the value of f is set to 1, atomic bonds exhibit similar strength and uniformity in all directions [15]. The fvalue is computed using Equation (S3) and displayed in Table 2. Table 2 shows that the f-values of Mo2GeB2, Ta2GaB2, and Ta2GeB2, which are close to one, indicate a slight anisotropic bonding strength. Nevertheless, substances with robust bonds in the horizontal plane (ab plane) (f > 1), such as Mo2GaB2, are deemed optimal candidates for the exfoliation process. In engineering applications, the hardness of a solid material serves as a valuable criterion for designing various devices. The elastic properties of polycrystalline materials can be utilized to calculate hardness values, as the ability to resist indentation is closely linked to a material's hardness. Both micro-hardness (Hmicro) and macro-hardness (Hmacro) were computed using Equation (S4) and are presented in Table 2. Based on the values of Hmicro and Hmacro, Mo2GaB<sup>2</sup> emerges as the toughest among the studied phases. The order of hardness is as follows: Mo2GaB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GeB2. Interestingly, Ga-containing MAX phase borides exhibit superior hardness compared to Ge-containing MAX phase borides. In Table 2, we compare our findings with previously reported MAX phases and observe that our data align perfectly with the earlier results. #### 3.4.2 Elastic anisotropy Additionally, anisotropy is linked to other crucial events like anisotropic plastic deformation and the development and spread of micro-cracks within mechanical stress. By providing directiondependent elastic constants, the understanding of anisotropy also offers a framework for improving the mechanical stability of materials in extreme circumstances. The following formulae are used for hexagonal structures to calculate the various anisotropic variables from elastic constants Cij [62]. $A\_{1} = \frac{\frac{1}{6}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{44}};$ $A\_{2} = \frac{2\mathcal{C}\_{44}}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ and $A\_{3} = A\_{1}.$ $A\_{2} = \frac{\frac{1}{3}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ Table 3 lists every anisotropy parameter. Since the value of A<sup>i</sup> should be 1 to be isotropic, the computed value of A<sup>i</sup> (i = 1-3) indicates that all of the compounds under research exhibit anisotropic behavior [63]. An additional method for estimating elastic anisotropy is to use the percentage anisotropy to compressibility and shear (A<sup>B</sup> & AG). This gives polycrystalline materials a helpful way to measure elastic anisotropy. They have been described as [45]; Zero values for AB, AG, and the universal anisotropy factor (AU) show elastic isotropy; the maximum amount of anisotropy is represented by a value of 1. Table 3 shows that, compared to other compounds, the values of AB, AG, and A U for Mo2GeB<sup>2</sup> and Ta2GaB<sup>2</sup> are incredibly close to zero, suggesting that these compounds have nearly isotropic characteristics. #### 3.4.3 Mulliken Populations The Mulliken charge assigned to an atomic species quantifies the effective valence by calculating the absolute difference between the formal ionic charge and the Mulliken charge. Equations (S5) and (S6) are employed to ascertain the Mulliken charge for each atom (α). Table-4 provides the Mulliken atomic population and effective valence charge. Transition metals Mo and Ta in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have pure valence states of 4d<sup>5</sup> and 5d<sup>3</sup> , respectively. The d- orbital electrons of transition metals have been found to influence their effective valence charge significantly. A non-zero positive value indicates a combination of covalent and ionic attributes within chemical bonds. As this value decreases towards zero, it signifies a rise in ionicity. A zero value suggests an ideal ionic character in the bond. Conversely, a progression from zero with a positive value indicates an elevation in the covalency level of the bonds. Based on their effective valence, M atoms move from the left to the right in the periodic table, increasing the covalency of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). Table-4 reveals that the Mulliken atomic charge ascribed to the B atoms is solely negative. Conversely, positive Mulliken atomic charges are associated with transition metals (M) and A. This suggests a charge transfer from M and A to B for each compound within M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge), thereby fostering ionic chemical bonds among these atoms. Bond population serves as another indicator of bond covalency within a crystal, as a high value of bond population essentially signifies a heightened degree of covalency within the chemical bond. The bonding and anti-bonding states influence the populations with positive and negative bond overlap. As demonstrated in Table-5, the B-B bond exhibits greater covalency compared to any other bond in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The presence of an antibonding state between two relevant atoms, which subsequently decreases their chemical bonding, is indicated by a hostile bond overlap population. In Mo2GeB<sup>2</sup> and Ta2GeB2, a hostile bond overlap population is observed in the Ge-Mo and Ge-Ta bonds, indicating the presence of an antibonding state. Therefore, the existence of ionic bonding is guaranteed by electronic charge transfer. On the other hand, the high positive value of the bond overlap population (BOP) denotes the presence of covalent bonding, a feature shared by materials in the MAX phase. ## 3.4.4 Theoretical Vickers Hardness Vickers hardness, derived from the atomic bonds found in solids, indicates how resistant a material is to deformation in extreme circumstances. Several variables influence this feature, such as the crystal flaws, solid structure, atomic arrangement, and bond strength. The Vickers hardness of the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) MAX phases is determined using the Mulliken bond population method, as described by Gou et al. [66] using the formula (S7-S10). This method is particularly suitable for partial metallic systems such as MAX phases. Table-5 lists the computed Vickers hardness values for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The calculated values are 3.35 GPa, 4.76 GPa, 8.21 GPa, and 8.74 GPa for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, respectively. We observed that Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> have much higher hardness values than Mo2GaB<sup>2</sup> and Mo2GeB2. These values are also higher than that of other 212 phases, like, Ti2InB<sup>2</sup> (4.05 GPa) [50], Hf2InB<sup>2</sup> (3.94 GPa), and Hf2SnB<sup>2</sup> (4.41 GPa) [26], Zr2InB<sup>2</sup> (2.92 GPa) and Zr2TlB<sup>2</sup> (2.19 GPa) [25], Zr2GaB<sup>2</sup> (2.53GPa), Zr2GeB<sup>2</sup> (3.31GPa), Hf2GaB<sup>2</sup> (4.73GPa) and Hf2GeB2(4.83GPa) [67]The Hv calculated by the geometrical average of the individual bonding, where the bonding strength mainly determined by the BOP values. In the case of Ta2GaB<sup>2</sup> and Ta2GeB2, the BOP of M-B bonding is much higher compared to Mo2GaB<sup>2</sup> and Mo2GeB2. Even though the BOP of M-B for Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> is higher than that of the other 212 phases mentioned earlier. Therefore, higher values of Hv are expected for Ta2GaB<sup>2</sup> and Ta2GeB2. Additionally, we looked at the Vickers hardness value between 211 and 212 compounds and discovered that the 212 MAX phase compounds had a higher H<sup>V</sup> value. This is because a 2D layer of B atoms is positioned between the M atoms. The B atoms share two center-two electrons to form an extremely strong B-B bond [3]. Table 5 Calculated data for Mulliken bond number (n μ ), bond length (d μ ), bond overlap populations BOP, (P μ ), metallic populations (P μˊ), Vickers hardness (HV) M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. # 3.5 Thermal Properties MAX phases are ideal for high-temperature applications due to their exceptional mechanical qualities at elevated temperatures. As a result, researching the fundamental parameters necessary for predicting their application is of great interest, and these can be obtained from the vibrations of atoms or phonons. The Debye temperature (ΘD) of a solid is directly connected to its bonding strength, melting temperature, thermal expansion, and conductivity. Using the sound velocity and Anderson's technique [68], the Θ<sup>D</sup> of the phases under study has been computed using the formula (S11). Equation (S12) can be used to get the average sound velocity (Vm) from the longitudinal and transverse sound velocities. Equations (S13–14) were used to determine v<sup>l</sup> and vt. The calculated values of Debye's temperature are shown in Table 6, where Mo2GaB<sup>2</sup> has the highest Θ<sup>D</sup> and Ta2GeB<sup>2</sup> has the lowest. If we rank them, it is as follows: Mo2GaB<sup>2</sup> < Mo2GeB<sup>2</sup> < Ta2GaB2 < Ta2GeB2. Hadi et al. recently reported a MAX phase (V2SnC) as a TBC material with a Θ<sup>D</sup> value of 472 K [69]. Thus, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) exhibit encouraging potential as TBC materials, as shown in Table 6. Table 6-Data for density (ρ), longitudinal, transverse, and average sound velocities (vl, vt, and vm), Debye temperature (ΘD), minimum thermal conductivity (Kmin), Grüneisen parameter (γ), thermal expansion coefficient (TEC) at 300K and melting temperature (Tm) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The constant thermal conductivity value at high temperatures is the minimum thermal conductivity (Kmin). As the name implies, this conductivity is minimal because, at high temperatures, phonon coupling breaks. The formula (S15) for the minimum thermal conductivity of solids was derived using the Clarke model [70]. Table 6 lists the calculated value of kmin, with Ta2GaB<sup>2</sup> having the lowest value and Mo2GaB<sup>2</sup> having the highest. When selecting suitable materials for TBC applications, a minimum thermal conductivity of 1.25 W/mK is used as a screening criterion [71]. Our compounds exhibit lower minimum thermal conductivity values, holding promising potential as TBC materials. Gd2Zr2O<sup>7</sup> and Y2SiO5, two recently developed thermal barrier coating (TBC) materials, have minimal thermal conductivities (Kmin) of 1.22 W/m.K. and 1.3 W/m.K. [72], respectively, as confirmed by experiment. These numbers roughly match the values we computed for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The Grüneisen parameter (γ) is a crucial thermal parameter that helps explain the anharmonic effects of lattice dynamics; solids utilized at high temperatures are expected to have lower anharmonic effects. The Grüneisen parameter (γ) can be determined with the help of Poisson's ratio using equation (S16) [73]. The computed γ values, as shown in Table 6, suggest that the compounds under investigation exhibit a weak anharmonic effect. Additionally, for solids with a Poisson's ratio between 0.05 and 0.46, the values similarly fall within the range of 0.85 and 3.53 [74]. The melting temperature (Tm) of the compounds under investigation has been calculated using the following formula (S17). The strength of atomic bonding is the primary factor determining the melting temperature of solids; the higher the Tm, the stronger the atomic bonding. The order of T<sup>m</sup> for the titled phases is found to mirror the Y-based (Young's modulus) order, indicating a close link between T<sup>m</sup> and Y [75]. As observed in Table 6, our compounds roughly follow the Ybased ranking with Mo2GeB<sup>2</sup> < Mo2GaB<sup>2</sup> < Ta2GeB<sup>2</sup> < Ta2GaB2. The T<sup>m</sup> value for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) is also comparable with the TBC material Y4Al2O<sup>9</sup> (2000 K)[75]. ![Fig. 6: Temperature dependence of the calculated thermodynamic parameters. (a) Specific heat at constant pressure, C<sup>p</sup> (b) Specific heat at constant volume, C<sup>v</sup> (c) Free energy, F (d) Entropy, S (e) Internal energy, E and (f) Thermal expansion coefficient, TEC of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge).](path) The temperature dependence of specific heats, Cv, Cp, for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds was obtained using the formulas (S19-S20), as shown in Fig.6(a,b). Assuming that the quasi-harmonic model is accurate and that phase transitions are not anticipated for the compounds under study, these properties are approximated across a temperature range of 0 to 1000 K. Because phonon thermal softening occurs at higher temperatures, the heat capacity rises with temperature. Heat capacities increase quickly and follow the Debye-T 3 power law at lower temperatures. At higher temperature regimes, where C<sup>v</sup> and C<sup>p</sup> do not greatly depend on temperature, they approach the Dulong-Petit (3) limit.[76]. Using the quasi-harmonic approximation, various temperature-dependent thermodynamic potential functions for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have been estimated at zero pressure and displayed in Fig.6 [77]. These functions include the Helmholtz free energy (F), internal energy (E), and entropy (S) within 0-1000 K temperature using equation (S21-S23). The free energy progressively decreases as the temperature rises, as shown in Fig.6(c). Free energy typically declines, and this trend becomes increasingly negative as a natural process proceeds. As demonstrated in Fig. 6(d), the internal energy (E) shows a rising trend with temperature, in contrast to the free energy. Since thermal agitation creates disorder, a system's entropy rises as temperature rises. This is illustrated in Fig.6(d). A material's thermal expansion coefficient (TEC) is derived from the anharmonicity in the lattice dynamics and can be found using equation (S18). The measure of a material's capacity to expand or contract with heat or cold is called the Thermal Expansion Coefficient, or TEC. As observed in Fig. 5(f), the Thermal Expansion Coefficient (TEC) increases rapidly up to 365 K. Then it approaches a constant value, which indicates lower saturations in the materials with temperature changes. The materials under study have a very low TEC value, a crucial characteristic of materials intended for application in high-temperature technology. To be effective as thermal barrier coating (TBC) materials, compounds must exhibit a low thermal conductivity (Kmin) to impede heat transfer, a high melting temperature to withstand extreme heat, and a low thermal expansion coefficient (TEC) to maintain dimensional stability under thermal stress. The compounds M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge) possess these properties, making them suitable candidates for use as TBC materials #### 3.6 Optical Properties Different materials exhibit unique behaviors when exposed to electromagnetic radiation. The optical constants determine the overall response of the sample to the incident radiation. The complex dielectric function, defined as ε(ω)=ε1(ω)+iε2(ω), is one of the main optical characteristics of solids. The following formula determines the imaginary part of the dielectric function ε2(ω) from the momentum matrix element between the occupied and unoccupied electronic states. In this formula, e stands for an electronic charge, ω for light angular frequency, u for the polarization vector of the incident electric field, and and for the conduction and valence band wave functions, respectively, at k. The Kramers-Kronig equation can estimate the real part of the dielectric function, ε1(ω). In contrast, ε2(ω) and ε1(ω) are utilized to evaluate all other optical parameters, such as the absorption coefficient, photoconductivity, reflectivity, and loss function [78]. In this part, several energy-dependent optical properties of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) 212 MAX phases are calculated and analyzed in detail for the photon energy range of 0 to 30 eV, for [100] plane enabling the first assessment of the compounds' practical applicability. Given the metallic conductivity of the MAX compounds' electronic structure, additional parameters were chosen to analyze the optical properties. These include a plasma frequency of 3 eV, damping of 0.05 eV, and Gaussian smearing of 0.5 eV [79]. Figure 7(a) shows the real component of the dielectric function, ε1, which exhibits metallic behavior. In metallic systems, ε<sup>1</sup> has a considerably high negative value in the low-energy range, with the real component reaching negative, which aligns with the band structure finding. Fig. 7(b) depicts the imaginary part of the dielectric function, ε2(ω), representing dielectric losses about frequency. Mo2GeB<sup>2</sup> demonstrates the highest peak in the low-energy region, with all compounds approaching zero from above at around 17 eV. This observation confirms Drude's behavior. ![Fig. 7- (a) Real part of dielectric constant, ε<sup>1</sup> (b) Imaginary part of dielectric constant, ε<sup>2</sup> (c) Reflectivity, R and (d) Loss function, (e) The coefficient of absorption, (α) (f) Refractive index, (n) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) for [100] electric field directions.](path) The reflectivity of a material measures the percentage of incident light energy reflected off it. Equation (S24) was employed to compute reflectivity using the dielectric function, as shown in Fig.7(c). The spectra's visible and infrared (IR) portions consistently exhibit values exceeding 55%. Consequently, the materials under examination are expected to appear metallic gray. In the infrared (IR) region, the maximum reflectivity for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> is 65% and 74%, respectively, occurring at 0.33 eV. Conversely, for Ta2GaB2, the maximum reflectivity is 59% at 14.02 eV, while for Ta2GeB2, it is 65% at 14.78 eV in the UV region. After 18 eV, the reflectance decreases significantly. According to reports, substances with an average reflectivity value above 44% in the visible light region can effectively reduce solar heating by reflecting a significant amount of the solar spectrum. Among all the compounds we examined, Ta2GaB<sup>2</sup> has a reflectivity in the visible range of more than 44% [78]. Therefore, Ta2GaB<sup>2</sup> can be utilized as a coating material and should be able to mitigate solar heating. plasma frequencies (ωp). The plasma frequency for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is observed at 23 eV, 25 eV, 21 eV, and 23 eV, respectively. At this specific frequency, the absorption coefficient rapidly decreases, ε<sup>1</sup> crosses zero from the negative side, and the reflectance R(ω) displays a falling tail. Above this distinctive frequency, the materials are transparent to the incident electromagnetic radiation. Fig. 7(e) illustrates the absorbance coefficient (α) of compounds M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) determined using equation (S26). As α begins to rise from zero photon energy, the metallic nature of the substances under study is again indicated. The visible light region experiences a sharp increase in absorption, peaking in the UV region at around 14 eV and then progressively declining. The IR region exhibits negligible absorption. However, the materials mentioned above appear to have a significant absorption band primarily located in the visible and ultraviolet spectrum. This suggests that the materials can be used in UV surface-disinfection devices, medical sterilization equipment, and other optoelectronic device designs. Its refractive index is crucial to a material's potential application in optical devices like waveguides and photonic crystals. Equations (S27) and (S28) were used to derive the refractive index (n) and extinction coefficient (k) for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), as depicted in Fig.7(c) and Fig.S-1(a). The variations of n and k in MAX phase carbides with incident photon energy closely resemble ε1(ω) and ε2(ω). The static refractive index n(0) for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is 9.4,13.07,7 and 9.2, respectively, and decreases gradually with the increase in photon energy. The extinction coefficient for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2 gradually increases in the IR region, reaching their maximum values of 2.56, 4.27, 2.42, and 3.28, respectively. Following this, they slowly decrease in the visible and UV regions. Figure S-1(b) illustrates the photoconductivity (σ) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) across various photon energies. The photoconductivity (σ) parameter quantifies the impact of photon irradiation on a material's electrical conductivity. Similar to the absorbance coefficient (α) spectrum, the σ spectrum aligns with the metallic nature of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). #### 4 Conclusions In summary, we employed DFT calculations to explore four 212 MAB phases, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), and investigated the structural, electronic, mechanical, lattice dynamical, and optical properties to predict their possible applications. The phonon dispersion curves, formation energy, and elastic constants collectively suggest that the M2AB<sup>2</sup> boride maintains dynamic, mechanical, chemical, and thermodynamic stability. The electronic band structure and density of states (DOS) offer evidence supporting the metallic nature of the compounds under investigation. Concurrently, the charge density mapping and atomic Mulliken population both confirm the presence of a strong B-B covalent bond. The stiffness constants, elastic moduli, findex, fracture toughness (KIC), Pugh's ratio (G/B), hardness parameters, and Cauchy Pressure (CP) of M2AB<sup>2</sup> were computed and compared with those of their 211 equivalents. We found that the values of the 212-phase borides are higher than those of the 211-phase carbides or borides. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are identified as brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit ductile characteristics, as indicated by Poisson's ratio (ν) and Pugh's ratio (B/G or G/B). The elastic characteristics display anisotropy due to the distinct atomic configurations along the aand c-directions. Vickers hardness calculations are considered reliable indicators of material hardness. The results suggest that Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> possess more pliable characteristics than Ta2GaB<sup>2</sup> and Ta2GeB2. The high hardness values of Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compared to the Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> due to higher BOP values of M-B bonding in Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> than in Mo2GaB<sup>2</sup> and Mo2GeB2. The value of minimum thermal conductivity (Kmin), thermal expansion coefficient, and melting temperature (Tm) collectively suggest the potential suitability of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) as a material for thermal barrier coating (TBC) applications in high-temperature devices. The optical conductivity and absorption coefficient corroborate the findings of the electronic band structure. Reflectivity is notably high in infrared (IR) regions and remains nearly constant in the visible and moderate ultraviolet (UV) regions, with an average value exceeding 44% for Ta2GaB2. This suggests that Ta2GaB<sup>2</sup> can be effectively utilized as a coating material to reduce solar heating. We expect that the comprehensive analysis of the diverse physical characteristics of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) presented in this study will establish a robust foundation for future theoretical and experimental explorations of these fascinating MAB phases. ## CRediT author statement A. K. M Naim Ishtiaq and Md Nasir Uddin: Data curation, Writing- Original draft preparation. Md. Rasel Rana, Shariful Islam and Noor Afsary: Reviewing and Editing. Md. 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Fig. 5 – The charge density mapping of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.	# Exploration of new 212 MAB phases: M2AB2 (M=Mo, Ta; A=Ga, Ge) via DFT calculations #### Abstract The recently developed MAB phases, an extension of the MAX phase, have sparked interest in research among scientists because of their better thermo-mechanical properties. In this paper, we have explored four new MAB phases M2AB2 (M=Mo, Ta and A=Ga, Ge) and studied the elastic, electronic, thermal, and optical properties to predict the possible applications. The stability of the new phases has been confirmed by calculating formation energy (Ef), formation enthalpy (∆H), phonon dispersion curve (PDC), and elastic constant (Cij). The study reveals that M2AB<sup>2</sup> (M=Mo, Ta and A=Ga, Ge) exhibit significantly higher elastic constants, elastic moduli, and Vickers hardness values than their counterpart 211 borides. Higher Vickers hardness values of Ta2AB2 (A=Ga, Ge) than Mo2AB2 (A=Ga, Ge) have been explained based on the values of the bond overlap population. The analysis of the density of states and electronic band structure revealed the metallic nature of the borides under examination. The thermodynamic characteristics of M2AB2 (M=Mo, Ta and A=Ga, Ge) under high temperatures (0–1000 K) are investigated using the quasi-harmonic Debye model. Critical thermal properties such as melting temperature (Tm), Grüneisen parameter (γ), minimum thermal conductivity (Kmin), Debye temperature (ΘD), and others are also computed. Compared with 211 MAX phases, the 212 phases exhibit higher values of (ΘD) and Tm, along with a lower value of Kmin. These findings suggest that the studied compounds exhibit superior thermal properties that are suitable for practical applications. The optical characteristics have been examined, and the reflectance spectrum indicates that the materials have the potential to mitigate solar heating across various energy regions. #First two authors contributed equally. ### 1. Introduction The MAX phase has garnered significant attention in the present era due to its outstanding mechanical and thermal characteristics at high temperatures, showcasing attributes shared by both metals and ceramics. The increased interest in MAX phase materials can be traced back to Barsoum's noteworthy contributions [1], [2]. The term MAX phase represents a family of multilayer solids where M is an earlier transition metal, A is an element from the IIIA or IVA group of the periodic table, and X is an atom of C/N/B [3]. MAX phase materials showcase metallic behavior due to alternate metallic A-layers and ceramic behavior attributed to the MX layers [4]. Like most metals and alloys, MAX phase materials have excellent thermal shock resistance, superior machinability, and enhanced thermal and electrical conductivities. On the other hand, they have high melting or decomposition temperatures and strong elastic stiffness, similar to many ceramics[5]. The unique combination of metallic and ceramic properties makes them versatile, with applications ranging from high-temperature coatings to nuclear accidenttolerant fuel (ATF), concentrated solar power (CSP), catalysis, and as precursors for MXenes [6], [7], [8], [9]. The diversity of MAX phases was confined to C and N as X elements for years (from 1960 to 2014). However, recent advancements have overcome this limitation by successfully synthesizing MAX phases that contain B [10]. The physical and chemical properties of B and Bcontaining compounds highlight the potential of MAX phase borides, which replace C/N with boron [11]. Due to the presence of B in the composition, they are also called the MAB phase. Furthermore, scientific communities are actively working to broaden the diversity of MAX phases by introducing structural changes, as seen in examples like Cr3AlB<sup>4</sup> (space group Immm), Cr4AlB<sup>6</sup> (space group Cmmm), Cr4AlB<sup>4</sup> [12], [13], [14]. Khazaei et al. reported on the first investigation of the theoretical MAX phase borides M2AlB (where M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta). They evaluated these compounds' electronic structure, mechanical characteristics, and dynamical stability in their investigation [15]. The diversity of MAX phases is a subject of research interest due to their structure variations and the number of atoms in the compounds, resulting in changes to their characteristics. In examples like the 211 MAX phase [16], 312 MAX phase [8][17], 212 MAX phase [3], and i-MAX phase [18], the recent additions to the MAX family include the 314 MAX phase and 212 MAX phase, where the element B has been incorporated as an X element [3] [19]. This introduces a novel aspect to the MAX phase family, contributing to its diversity. The unique structural features of MAB phases include both orthorhombic and hexagonal symmetry observed in crystals. These distinctive symmetries set MAB phases apart from the typical MAX phases. Yinqiao Liu's synthesis of the orthorhombic phase of M2AlB2, particularly the 212-MAB phases with M = Sc, Ti, Zr, Hf, V, Nb, Cr, Mo, W, Mn, Tc, Fe, Co, and Ni, has revealed unique structural stability and notable electrical and mechanical properties [20]. The MAB phase structure slightly differs from the standard MAX phases, which typically crystallize in the hexagonal system with a space group -6<sup>2</sup> (No. 187). Ali et al [3]. investigated the diverse physical properties of Zr2AB<sup>2</sup> (A = In, Tl). Martin Ade et al [13] synthesized ternary borides, namely Cr2AlB2, Cr3AlB4, and Cr4AlB6, and subsequently compared their mechanical properties. Qureshi et al. [19] investigated the 314 Zr3CdB<sup>4</sup> MAX phase boride, calculating its mechanical, thermodynamic, and optical properties. The 314 MAX phase Hf3PB<sup>4</sup> has been thoroughly studied using Density Functional Theory (DFT) and revealed that it was the hardest MAX phase compound discovered until that date [21]. The structure of 212 phases exhibits a slight deviation from 211 MAX phases. In the case of 212 phases, a 2D layer of B is situated between M layers, featuring an additional B atom at the X position, unlike in 211 phases [22] [23], [24]. The B-B bonding in 212 MAX phase borides has improved mechanical and thermal properties. To date, the physical properties of Zr2AB<sup>2</sup> (A = In, Tl) [25] and Hf2AB<sup>2</sup> (A = In, Sn) [26], M2AB (M= Ti, Zr, Hf; A=Al, Ga, In) [27] MAX phases, as well as Nb2AC (A = Ga, Ge, Tl, Zn, P, In, and Cd) MAX phases, have been investigated using density functional theory (DFT). In each instance, the mechanical properties of B-containing compounds show significant improvement compared to their traditional C/N containing 211 MAX phases. The Debye temperature and melting temperature are higher for boron-containing 212 phases than for 211 carbides/nitrides, while the minimum thermal conductivity is lower. The thermal expansion coefficient of borides remains well-suited for use as coating materials. Consequently, the superior thermomechanical properties of B-containing 212 MAX phases demonstrate their suitability for high-temperature technological applications, surpassing the commonly used 211 MAX phase carbides. It should be noted that the 212 MAB phase with a hexagonal structure has already been synthesized [24]. In addition, Ga and Ge-based MAX phases have also been synthesized previously [28]. Thus, the reports on the synthesis of 212 phase and Ga and Gebased MAX phases motivated us to select the 212 MAB phases: M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) for our present study, and we have performed an in-depth investigation of their physical properties through DFT method. Therefore, in this paper, the first-time prediction of the stability and mechanical, electronic, thermal, and optical properties of M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) phases has been presented. The results revealed that M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) compounds are stable and suitable for thermal barrier coating (TBC) and reflection coating applications. Additionally, to provide a comparison, the properties determined for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) are compared with those of other 212 and 211 compounds MAX phase borides. ### 2. Methods of calculations First principles density-functional theory (DFT) computations are conducted utilizing the Cambridge Serial Total Energy Package (CASTEP) module integrated within Materials Studio 2017 [29], [30]. The exchange-correlation function is estimated using the Generalized Gradient Approximation (GGA) method, originally suggested by Perdew, Burke, and Ernzerhof [31]. Pseudo-atomic simulations accounted for electronic orbitals corresponding to B (2s 2 2p 1 ), Ga (3d <sup>10</sup> 4s 2 4p 1 ), Ge (4s 2 4p 2 ), Mo (4s 2 4p 6 4d 5 5s 1 ), and Ta (5d 3 6s 2 ). The energy cutoff and k-point grids were established at 650 eV and 11 × 11 × 4, respectively. The structural relaxation was performed utilizing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique [32], while the electronic structure was computed employing density mixing. The parameters for relaxed structures incorporate the following tolerance thresholds: the self-consistent convergence of the total energy is set at 5 × 10-6 eV/atom, the maximum force exerted on the atom is limited to 0.01 eV/Å, the maximum ionic displacement is constrained to 5 × 10-4 Å, and a maximum stress threshold of 0.02 GPa is imposed. The finite strain method [33], grounded in density functional theory (DFT), is utilized to compute the elastic properties within this framework. All necessary equations for determining various properties are provided in the supplementary document. #### 3 Results and discussion #### 3.1 Structural properties The M2AB<sup>2</sup> compounds (where M=Mo or Ta; A=Ga or Ge) belong to the P6m<sup>2</sup> (No. 187)[23] space group and crystallize in the hexagonal system. Unlike conventional MAX phases, which typically belong to the P63/mmc (194) space group, the 212 MAX phases exhibit distinct characteristics. In Fig. 1, the unit cell structure of Mo2GaB<sup>2</sup> is depicted as a representative of M2AB<sup>2</sup> alongside Mo2GaB, facilitating a comparison to discern their differences easily. The atomic positions are as follows: M (Mo or Ta) at (0.3333, 0.6667, 0.6935), A at (0.6667, 0.3333, 0.0), and two B atoms positioned at (0.6667, 0.3333, 0.5) and (0.0, 0.0, 0.5). The B components are arranged at the corners of the unit cell in typical 211 MAX phases, but in 212 boride MAX phases, they form a 2D layer between the M layers. This structural arrangement results in B-B covalent bonds in the 2D layer, enhancing stability compared to conventional 211 MAX phases. ## 3.4.1 Mechanical properties The mechanical properties of materials play a pivotal role in determining their potential applications, serving as crucial indicators of their behavior and suitability in materials engineering endeavors. These properties are equally applicable to MAX phase materials. Initially, to assess the mechanical properties of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phases, we employed the stress-strain method within the CASTEP code to compute the elastic constants (Cij) [22], [47], [48]. These calculated elastic constants (Cij) are presented in Table 2, alongside those of other 212 and 211 MAX phases. Due to the hexagonal crystal structure of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phase borides, five stiffness constants emerge: C11, C12, C13, C33, and C<sup>44</sup> [49]. The mechanical stability was evaluated using these stiffness constants in the preceding section. For instance, C<sup>11</sup> and C<sup>33</sup> determine stiffness when stress is exerted along the (100) and (001) directions, respectively, whereas C<sup>44</sup> evaluates resistance to shear deformation on the (100) and (001) planes. The stiffness constants C<sup>11</sup> and C<sup>12</sup> directly reflect the strength of atomic bonds along the a- and c-axes. When C<sup>11</sup> exceeds C<sup>33</sup> (or vice versa), it signifies stronger atomic bonding along the a-axis (or c-axis). In Table 2, for Mo2GaB2, Mo2GeB2, and Ta2GeB<sup>2</sup> compounds, C<sup>11</sup> surpasses C33, indicating superior atomic bonding along the a-axis compared to the c-axis. This robust bonding along the a-axis suggests heightened resistance to a-axial deformation. Conversely, in Ta2GeB2, where C<sup>11</sup> < C33, stronger atomic bonding along the c-axis translates to increased resistance against c-axial deformation. Analysis of Table-2 reveals that the values of C<sup>11</sup> and C<sup>22</sup> are notably higher for 212 MAX phase borides than 211 MAX phase borides. Consequently, it can be inferred that 212 MAX phases exhibit stronger resistance to axial deformation when juxtaposed with 211 MAX phases. C<sup>44</sup> is commonly utilized to gauge shear deformation tolerance among the elastic constants. Notably, Ta2GaB exhibits superior shear deformation resistance owing to its highest C<sup>44</sup> value among the studied compounds. Furthermore, the individual elastic constants C12, C13, and C<sup>44</sup> denote shear deformation response under external stress. The observation that C<sup>11</sup> and C<sup>33</sup> possess larger magnitudes than C<sup>44</sup> implies that shear deformation is more facile than axial strain. Another crucial parameter derived from the stiffness constants is the Cauchy pressure (CP), calculated as C<sup>12</sup> - C44, which provides vital insights relevant to the practical applications of solids [51]. Pettifor [51] emphasized the significance of Cauchy pressure (CP) in discerning the chemical bonding and ductile/brittle properties of solids. A negative CP value indicates covalently bonded brittle solids, whereas a positive value signifies isotropic ionic ductile solids. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> fall into covalently bonded brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit metallic ductile characteristics. Similarly, like Ga-containing 211 MAX phase borides, Ga-containing 212 MAX phases also demonstrate negative CP values and brittle behavior [52][53]. The elastic constants obtained are used to produce several bulk elastic parameters that are used to characterize polycrystalline materials, such as Young's modulus (Y), bulk modulus (B), and shear modulus (G). In Table 2, the bulk modulus (B) and shear modulus (G) calculated using Hill's approximation [54] are also presented. Hill's values represent the average of the upper limit (Voigt [55]) and lower limit (Reuss [56]) of B. The necessary equations for these calculations are provided in the supplementary document (S1). Young's modulus (Y) is a crucial indicator of material stiffness. A higher Y value indicates a stiffer material. It can be observed from the table that Mo2GaB<sup>2</sup> exhibits the highest Y value, signifying its greater stiffness compared to others. According to the sequence of Y values, stiffness can be ranked as follows: Mo2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Ta2GeB2. Therefore, Mo2GaB2, with its higher Young's modulus, is anticipated to demonstrate superior mechanical stability and deformation resistance compared to the other compounds under investigation. This quality is essential for aircraft parts or high-performance machinery applications where critical dimensional stability and structural integrity are required [57], [58]. Table 2 shows that 212 MAX phase materials exhibit larger Young's modulus (Y) values than 211 MAX phases. This suggests that 212 MAX phase borides are stiffer than their 211 MAX phase counterparts. Young's modulus (E) also correlates well with thermal shock resistance (R): R ∝ 1/E [59]. Lower Young's modulus values correspond to higher thermal shock resistance. Therefore, materials with higher thermal shock resistance (i.e., lower Young's modulus) are more suitable for use as Thermal Barrier Coating (TBC) materials. Given that Ta2GaB<sup>2</sup> possesses the lowest Y value among the materials studied, it should be considered a superior candidate for TBC material due to its higher thermal shock resistance. The material's ability to withstand shape distortion is elucidated by the shear modulus (G). On the other hand, the bulk modulus (B) indicates the strength of a material's chemical bonds and its ability to withstand uniform compression or volume change. We computed Young's modulus (Y), bulk modulus (B), and shear modulus for our analysis. Based on the bulk modulus values presented in Table 2, the sequence of a material that resists compression when pressure is applied can be outlined as follows: Mo2GeB<sup>2</sup> > Ta2GeB<sup>2</sup> > Mo2GaB<sup>2</sup> > Ta2GeB2. Mo2GeB2, boasting a higher bulk modulus, may exhibit reduced plastic deformation and superior resistance to stress-induced deformation compared to other compounds examined. The ductile and brittle characteristics of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds can also be assessed through Poisson's ratio (ν) and Pugh's ratio (G/B). A compound demonstrates ductile (brittle) behavior if the ν value surpasses (falls below) 0.26 [60]. Furthermore, if the G/B value exceeds (is less than) 0.571, then the compound exhibits brittle (ductile) behavior [19]. According to both criteria, Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are classified as brittle compounds, whereas Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> are categorized as ductile compounds. The ductile nature of Ge-based MAX phases has been reported previously [61]. Fracture toughness (KIC) is a vital property that gauges a material's ability to resist crack propagation. In the case of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, Equation (S2) was employed to determine KIC. The KIC values are documented in Table 2, and notably, these values surpass those of 211 MAX phases [16], [35]. Another parameter, the "f-index," characterizes the isotropic nature and strength of atom-atom bonds within a single hexagonal crystalline lattice along the a- and c-directions. If the f-index is less than 1, chemical bonds exhibit greater rigidity along the c-axis; conversely, if f exceeds 1, bonds are more rigid in the ab-plane. When the value of f is set to 1, atomic bonds exhibit similar strength and uniformity in all directions [15]. The fvalue is computed using Equation (S3) and displayed in Table 2. Table 2 shows that the f-values of Mo2GeB2, Ta2GaB2, and Ta2GeB2, which are close to one, indicate a slight anisotropic bonding strength. Nevertheless, substances with robust bonds in the horizontal plane (ab plane) (f > 1), such as Mo2GaB2, are deemed optimal candidates for the exfoliation process. In engineering applications, the hardness of a solid material serves as a valuable criterion for designing various devices. The elastic properties of polycrystalline materials can be utilized to calculate hardness values, as the ability to resist indentation is closely linked to a material's hardness. Both micro-hardness (Hmicro) and macro-hardness (Hmacro) were computed using Equation (S4) and are presented in Table 2. Based on the values of Hmicro and Hmacro, Mo2GaB<sup>2</sup> emerges as the toughest among the studied phases. The order of hardness is as follows: Mo2GaB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GeB2. Interestingly, Ga-containing MAX phase borides exhibit superior hardness compared to Ge-containing MAX phase borides. In Table 2, we compare our findings with previously reported MAX phases and observe that our data align perfectly with the earlier results. #### 3.4.2 Elastic anisotropy Additionally, anisotropy is linked to other crucial events like anisotropic plastic deformation and the development and spread of micro-cracks within mechanical stress. By providing directiondependent elastic constants, the understanding of anisotropy also offers a framework for improving the mechanical stability of materials in extreme circumstances. The following formulae are used for hexagonal structures to calculate the various anisotropic variables from elastic constants Cij [62]. $A\_{1} = \frac{\frac{1}{6}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{44}};$ $A\_{2} = \frac{2\mathcal{C}\_{44}}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ and $A\_{3} = A\_{1}.$ $A\_{2} = \frac{\frac{1}{3}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ Table 3 lists every anisotropy parameter. Since the value of A<sup>i</sup> should be 1 to be isotropic, the computed value of A<sup>i</sup> (i = 1-3) indicates that all of the compounds under research exhibit anisotropic behavior [63]. An additional method for estimating elastic anisotropy is to use the percentage anisotropy to compressibility and shear (A<sup>B</sup> & AG). This gives polycrystalline materials a helpful way to measure elastic anisotropy. They have been described as [45]; Zero values for AB, AG, and the universal anisotropy factor (AU) show elastic isotropy; the maximum amount of anisotropy is represented by a value of 1. Table 3 shows that, compared to other compounds, the values of AB, AG, and A U for Mo2GeB<sup>2</sup> and Ta2GaB<sup>2</sup> are incredibly close to zero, suggesting that these compounds have nearly isotropic characteristics. #### 3.4.3 Mulliken Populations The Mulliken charge assigned to an atomic species quantifies the effective valence by calculating the absolute difference between the formal ionic charge and the Mulliken charge. Equations (S5) and (S6) are employed to ascertain the Mulliken charge for each atom (α). Table-4 provides the Mulliken atomic population and effective valence charge. Transition metals Mo and Ta in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have pure valence states of 4d<sup>5</sup> and 5d<sup>3</sup> , respectively. The d- orbital electrons of transition metals have been found to influence their effective valence charge significantly. A non-zero positive value indicates a combination of covalent and ionic attributes within chemical bonds. As this value decreases towards zero, it signifies a rise in ionicity. A zero value suggests an ideal ionic character in the bond. Conversely, a progression from zero with a positive value indicates an elevation in the covalency level of the bonds. Based on their effective valence, M atoms move from the left to the right in the periodic table, increasing the covalency of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). Table-4 reveals that the Mulliken atomic charge ascribed to the B atoms is solely negative. Conversely, positive Mulliken atomic charges are associated with transition metals (M) and A. This suggests a charge transfer from M and A to B for each compound within M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge), thereby fostering ionic chemical bonds among these atoms. Bond population serves as another indicator of bond covalency within a crystal, as a high value of bond population essentially signifies a heightened degree of covalency within the chemical bond. The bonding and anti-bonding states influence the populations with positive and negative bond overlap. As demonstrated in Table-5, the B-B bond exhibits greater covalency compared to any other bond in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The presence of an antibonding state between two relevant atoms, which subsequently decreases their chemical bonding, is indicated by a hostile bond overlap population. In Mo2GeB<sup>2</sup> and Ta2GeB2, a hostile bond overlap population is observed in the Ge-Mo and Ge-Ta bonds, indicating the presence of an antibonding state. Therefore, the existence of ionic bonding is guaranteed by electronic charge transfer. On the other hand, the high positive value of the bond overlap population (BOP) denotes the presence of covalent bonding, a feature shared by materials in the MAX phase. ## 3.4.4 Theoretical Vickers Hardness Vickers hardness, derived from the atomic bonds found in solids, indicates how resistant a material is to deformation in extreme circumstances. Several variables influence this feature, such as the crystal flaws, solid structure, atomic arrangement, and bond strength. The Vickers hardness of the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) MAX phases is determined using the Mulliken bond population method, as described by Gou et al. [66] using the formula (S7-S10). This method is particularly suitable for partial metallic systems such as MAX phases. Table-5 lists the computed Vickers hardness values for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The calculated values are 3.35 GPa, 4.76 GPa, 8.21 GPa, and 8.74 GPa for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, respectively. We observed that Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> have much higher hardness values than Mo2GaB<sup>2</sup> and Mo2GeB2. These values are also higher than that of other 212 phases, like, Ti2InB<sup>2</sup> (4.05 GPa) [50], Hf2InB<sup>2</sup> (3.94 GPa), and Hf2SnB<sup>2</sup> (4.41 GPa) [26], Zr2InB<sup>2</sup> (2.92 GPa) and Zr2TlB<sup>2</sup> (2.19 GPa) [25], Zr2GaB<sup>2</sup> (2.53GPa), Zr2GeB<sup>2</sup> (3.31GPa), Hf2GaB<sup>2</sup> (4.73GPa) and Hf2GeB2(4.83GPa) [67]The Hv calculated by the geometrical average of the individual bonding, where the bonding strength mainly determined by the BOP values. In the case of Ta2GaB<sup>2</sup> and Ta2GeB2, the BOP of M-B bonding is much higher compared to Mo2GaB<sup>2</sup> and Mo2GeB2. Even though the BOP of M-B for Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> is higher than that of the other 212 phases mentioned earlier. Therefore, higher values of Hv are expected for Ta2GaB<sup>2</sup> and Ta2GeB2. Additionally, we looked at the Vickers hardness value between 211 and 212 compounds and discovered that the 212 MAX phase compounds had a higher H<sup>V</sup> value. This is because a 2D layer of B atoms is positioned between the M atoms. The B atoms share two center-two electrons to form an extremely strong B-B bond [3]. Table 5 Calculated data for Mulliken bond number (n μ ), bond length (d μ ), bond overlap populations BOP, (P μ ), metallic populations (P μˊ), Vickers hardness (HV) M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. # 3.5 Thermal Properties MAX phases are ideal for high-temperature applications due to their exceptional mechanical qualities at elevated temperatures. As a result, researching the fundamental parameters necessary for predicting their application is of great interest, and these can be obtained from the vibrations of atoms or phonons. The Debye temperature (ΘD) of a solid is directly connected to its bonding strength, melting temperature, thermal expansion, and conductivity. Using the sound velocity and Anderson's technique [68], the Θ<sup>D</sup> of the phases under study has been computed using the formula (S11). Equation (S12) can be used to get the average sound velocity (Vm) from the longitudinal and transverse sound velocities. Equations (S13–14) were used to determine v<sup>l</sup> and vt. The calculated values of Debye's temperature are shown in Table 6, where Mo2GaB<sup>2</sup> has the highest Θ<sup>D</sup> and Ta2GeB<sup>2</sup> has the lowest. If we rank them, it is as follows: Mo2GaB<sup>2</sup> < Mo2GeB<sup>2</sup> < Ta2GaB2 < Ta2GeB2. Hadi et al. recently reported a MAX phase (V2SnC) as a TBC material with a Θ<sup>D</sup> value of 472 K [69]. Thus, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) exhibit encouraging potential as TBC materials, as shown in Table 6. Table 6-Data for density (ρ), longitudinal, transverse, and average sound velocities (vl, vt, and vm), Debye temperature (ΘD), minimum thermal conductivity (Kmin), Grüneisen parameter (γ), thermal expansion coefficient (TEC) at 300K and melting temperature (Tm) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The constant thermal conductivity value at high temperatures is the minimum thermal conductivity (Kmin). As the name implies, this conductivity is minimal because, at high temperatures, phonon coupling breaks. The formula (S15) for the minimum thermal conductivity of solids was derived using the Clarke model [70]. Table 6 lists the calculated value of kmin, with Ta2GaB<sup>2</sup> having the lowest value and Mo2GaB<sup>2</sup> having the highest. When selecting suitable materials for TBC applications, a minimum thermal conductivity of 1.25 W/mK is used as a screening criterion [71]. Our compounds exhibit lower minimum thermal conductivity values, holding promising potential as TBC materials. Gd2Zr2O<sup>7</sup> and Y2SiO5, two recently developed thermal barrier coating (TBC) materials, have minimal thermal conductivities (Kmin) of 1.22 W/m.K. and 1.3 W/m.K. [72], respectively, as confirmed by experiment. These numbers roughly match the values we computed for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The Grüneisen parameter (γ) is a crucial thermal parameter that helps explain the anharmonic effects of lattice dynamics; solids utilized at high temperatures are expected to have lower anharmonic effects. The Grüneisen parameter (γ) can be determined with the help of Poisson's ratio using equation (S16) [73]. The computed γ values, as shown in Table 6, suggest that the compounds under investigation exhibit a weak anharmonic effect. Additionally, for solids with a Poisson's ratio between 0.05 and 0.46, the values similarly fall within the range of 0.85 and 3.53 [74]. The melting temperature (Tm) of the compounds under investigation has been calculated using the following formula (S17). The strength of atomic bonding is the primary factor determining the melting temperature of solids; the higher the Tm, the stronger the atomic bonding. The order of T<sup>m</sup> for the titled phases is found to mirror the Y-based (Young's modulus) order, indicating a close link between T<sup>m</sup> and Y [75]. As observed in Table 6, our compounds roughly follow the Ybased ranking with Mo2GeB<sup>2</sup> < Mo2GaB<sup>2</sup> < Ta2GeB<sup>2</sup> < Ta2GaB2. The T<sup>m</sup> value for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) is also comparable with the TBC material Y4Al2O<sup>9</sup> (2000 K)[75]. ![Fig. 6: Temperature dependence of the calculated thermodynamic parameters. (a) Specific heat at constant pressure, C<sup>p</sup> (b) Specific heat at constant volume, C<sup>v</sup> (c) Free energy, F (d) Entropy, S (e) Internal energy, E and (f) Thermal expansion coefficient, TEC of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge).](path) The temperature dependence of specific heats, Cv, Cp, for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds was obtained using the formulas (S19-S20), as shown in Fig.6(a,b). Assuming that the quasi-harmonic model is accurate and that phase transitions are not anticipated for the compounds under study, these properties are approximated across a temperature range of 0 to 1000 K. Because phonon thermal softening occurs at higher temperatures, the heat capacity rises with temperature. Heat capacities increase quickly and follow the Debye-T 3 power law at lower temperatures. At higher temperature regimes, where C<sup>v</sup> and C<sup>p</sup> do not greatly depend on temperature, they approach the Dulong-Petit (3) limit.[76]. Using the quasi-harmonic approximation, various temperature-dependent thermodynamic potential functions for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have been estimated at zero pressure and displayed in Fig.6 [77]. These functions include the Helmholtz free energy (F), internal energy (E), and entropy (S) within 0-1000 K temperature using equation (S21-S23). The free energy progressively decreases as the temperature rises, as shown in Fig.6(c). Free energy typically declines, and this trend becomes increasingly negative as a natural process proceeds. As demonstrated in Fig. 6(d), the internal energy (E) shows a rising trend with temperature, in contrast to the free energy. Since thermal agitation creates disorder, a system's entropy rises as temperature rises. This is illustrated in Fig.6(d). A material's thermal expansion coefficient (TEC) is derived from the anharmonicity in the lattice dynamics and can be found using equation (S18). The measure of a material's capacity to expand or contract with heat or cold is called the Thermal Expansion Coefficient, or TEC. As observed in Fig. 5(f), the Thermal Expansion Coefficient (TEC) increases rapidly up to 365 K. Then it approaches a constant value, which indicates lower saturations in the materials with temperature changes. The materials under study have a very low TEC value, a crucial characteristic of materials intended for application in high-temperature technology. To be effective as thermal barrier coating (TBC) materials, compounds must exhibit a low thermal conductivity (Kmin) to impede heat transfer, a high melting temperature to withstand extreme heat, and a low thermal expansion coefficient (TEC) to maintain dimensional stability under thermal stress. The compounds M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge) possess these properties, making them suitable candidates for use as TBC materials #### 3.6 Optical Properties Different materials exhibit unique behaviors when exposed to electromagnetic radiation. The optical constants determine the overall response of the sample to the incident radiation. The complex dielectric function, defined as ε(ω)=ε1(ω)+iε2(ω), is one of the main optical characteristics of solids. The following formula determines the imaginary part of the dielectric function ε2(ω) from the momentum matrix element between the occupied and unoccupied electronic states. In this formula, e stands for an electronic charge, ω for light angular frequency, u for the polarization vector of the incident electric field, and and for the conduction and valence band wave functions, respectively, at k. The Kramers-Kronig equation can estimate the real part of the dielectric function, ε1(ω). In contrast, ε2(ω) and ε1(ω) are utilized to evaluate all other optical parameters, such as the absorption coefficient, photoconductivity, reflectivity, and loss function [78]. In this part, several energy-dependent optical properties of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) 212 MAX phases are calculated and analyzed in detail for the photon energy range of 0 to 30 eV, for [100] plane enabling the first assessment of the compounds' practical applicability. Given the metallic conductivity of the MAX compounds' electronic structure, additional parameters were chosen to analyze the optical properties. These include a plasma frequency of 3 eV, damping of 0.05 eV, and Gaussian smearing of 0.5 eV [79]. Figure 7(a) shows the real component of the dielectric function, ε1, which exhibits metallic behavior. In metallic systems, ε<sup>1</sup> has a considerably high negative value in the low-energy range, with the real component reaching negative, which aligns with the band structure finding. Fig. 7(b) depicts the imaginary part of the dielectric function, ε2(ω), representing dielectric losses about frequency. Mo2GeB<sup>2</sup> demonstrates the highest peak in the low-energy region, with all compounds approaching zero from above at around 17 eV. This observation confirms Drude's behavior. ![Fig. 7- (a) Real part of dielectric constant, ε<sup>1</sup> (b) Imaginary part of dielectric constant, ε<sup>2</sup> (c) Reflectivity, R and (d) Loss function, (e) The coefficient of absorption, (α) (f) Refractive index, (n) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) for [100] electric field directions.](path) The reflectivity of a material measures the percentage of incident light energy reflected off it. Equation (S24) was employed to compute reflectivity using the dielectric function, as shown in Fig.7(c). The spectra's visible and infrared (IR) portions consistently exhibit values exceeding 55%. Consequently, the materials under examination are expected to appear metallic gray. In the infrared (IR) region, the maximum reflectivity for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> is 65% and 74%, respectively, occurring at 0.33 eV. Conversely, for Ta2GaB2, the maximum reflectivity is 59% at 14.02 eV, while for Ta2GeB2, it is 65% at 14.78 eV in the UV region. After 18 eV, the reflectance decreases significantly. According to reports, substances with an average reflectivity value above 44% in the visible light region can effectively reduce solar heating by reflecting a significant amount of the solar spectrum. Among all the compounds we examined, Ta2GaB<sup>2</sup> has a reflectivity in the visible range of more than 44% [78]. Therefore, Ta2GaB<sup>2</sup> can be utilized as a coating material and should be able to mitigate solar heating. plasma frequencies (ωp). The plasma frequency for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is observed at 23 eV, 25 eV, 21 eV, and 23 eV, respectively. At this specific frequency, the absorption coefficient rapidly decreases, ε<sup>1</sup> crosses zero from the negative side, and the reflectance R(ω) displays a falling tail. Above this distinctive frequency, the materials are transparent to the incident electromagnetic radiation. Fig. 7(e) illustrates the absorbance coefficient (α) of compounds M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) determined using equation (S26). As α begins to rise from zero photon energy, the metallic nature of the substances under study is again indicated. The visible light region experiences a sharp increase in absorption, peaking in the UV region at around 14 eV and then progressively declining. The IR region exhibits negligible absorption. However, the materials mentioned above appear to have a significant absorption band primarily located in the visible and ultraviolet spectrum. This suggests that the materials can be used in UV surface-disinfection devices, medical sterilization equipment, and other optoelectronic device designs. Its refractive index is crucial to a material's potential application in optical devices like waveguides and photonic crystals. Equations (S27) and (S28) were used to derive the refractive index (n) and extinction coefficient (k) for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), as depicted in Fig.7(c) and Fig.S-1(a). The variations of n and k in MAX phase carbides with incident photon energy closely resemble ε1(ω) and ε2(ω). The static refractive index n(0) for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is 9.4,13.07,7 and 9.2, respectively, and decreases gradually with the increase in photon energy. The extinction coefficient for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2 gradually increases in the IR region, reaching their maximum values of 2.56, 4.27, 2.42, and 3.28, respectively. Following this, they slowly decrease in the visible and UV regions. Figure S-1(b) illustrates the photoconductivity (σ) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) across various photon energies. The photoconductivity (σ) parameter quantifies the impact of photon irradiation on a material's electrical conductivity. Similar to the absorbance coefficient (α) spectrum, the σ spectrum aligns with the metallic nature of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). #### 4 Conclusions In summary, we employed DFT calculations to explore four 212 MAB phases, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), and investigated the structural, electronic, mechanical, lattice dynamical, and optical properties to predict their possible applications. The phonon dispersion curves, formation energy, and elastic constants collectively suggest that the M2AB<sup>2</sup> boride maintains dynamic, mechanical, chemical, and thermodynamic stability. The electronic band structure and density of states (DOS) offer evidence supporting the metallic nature of the compounds under investigation. Concurrently, the charge density mapping and atomic Mulliken population both confirm the presence of a strong B-B covalent bond. The stiffness constants, elastic moduli, findex, fracture toughness (KIC), Pugh's ratio (G/B), hardness parameters, and Cauchy Pressure (CP) of M2AB<sup>2</sup> were computed and compared with those of their 211 equivalents. We found that the values of the 212-phase borides are higher than those of the 211-phase carbides or borides. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are identified as brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit ductile characteristics, as indicated by Poisson's ratio (ν) and Pugh's ratio (B/G or G/B). The elastic characteristics display anisotropy due to the distinct atomic configurations along the aand c-directions. Vickers hardness calculations are considered reliable indicators of material hardness. The results suggest that Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> possess more pliable characteristics than Ta2GaB<sup>2</sup> and Ta2GeB2. The high hardness values of Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compared to the Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> due to higher BOP values of M-B bonding in Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> than in Mo2GaB<sup>2</sup> and Mo2GeB2. The value of minimum thermal conductivity (Kmin), thermal expansion coefficient, and melting temperature (Tm) collectively suggest the potential suitability of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) as a material for thermal barrier coating (TBC) applications in high-temperature devices. The optical conductivity and absorption coefficient corroborate the findings of the electronic band structure. Reflectivity is notably high in infrared (IR) regions and remains nearly constant in the visible and moderate ultraviolet (UV) regions, with an average value exceeding 44% for Ta2GaB2. This suggests that Ta2GaB<sup>2</sup> can be effectively utilized as a coating material to reduce solar heating. We expect that the comprehensive analysis of the diverse physical characteristics of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) presented in this study will establish a robust foundation for future theoretical and experimental explorations of these fascinating MAB phases. ## CRediT author statement A. K. M Naim Ishtiaq and Md Nasir Uddin: Data curation, Writing- Original draft preparation. Md. Rasel Rana, Shariful Islam and Noor Afsary: Reviewing and Editing. Md. 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Fig. 4 – TDOS and PDOS of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB2 compounds.	# Exploration of new 212 MAB phases: M2AB2 (M=Mo, Ta; A=Ga, Ge) via DFT calculations #### Abstract The recently developed MAB phases, an extension of the MAX phase, have sparked interest in research among scientists because of their better thermo-mechanical properties. In this paper, we have explored four new MAB phases M2AB2 (M=Mo, Ta and A=Ga, Ge) and studied the elastic, electronic, thermal, and optical properties to predict the possible applications. The stability of the new phases has been confirmed by calculating formation energy (Ef), formation enthalpy (∆H), phonon dispersion curve (PDC), and elastic constant (Cij). The study reveals that M2AB<sup>2</sup> (M=Mo, Ta and A=Ga, Ge) exhibit significantly higher elastic constants, elastic moduli, and Vickers hardness values than their counterpart 211 borides. Higher Vickers hardness values of Ta2AB2 (A=Ga, Ge) than Mo2AB2 (A=Ga, Ge) have been explained based on the values of the bond overlap population. The analysis of the density of states and electronic band structure revealed the metallic nature of the borides under examination. The thermodynamic characteristics of M2AB2 (M=Mo, Ta and A=Ga, Ge) under high temperatures (0–1000 K) are investigated using the quasi-harmonic Debye model. Critical thermal properties such as melting temperature (Tm), Grüneisen parameter (γ), minimum thermal conductivity (Kmin), Debye temperature (ΘD), and others are also computed. Compared with 211 MAX phases, the 212 phases exhibit higher values of (ΘD) and Tm, along with a lower value of Kmin. These findings suggest that the studied compounds exhibit superior thermal properties that are suitable for practical applications. The optical characteristics have been examined, and the reflectance spectrum indicates that the materials have the potential to mitigate solar heating across various energy regions. #First two authors contributed equally. ### 1. Introduction The MAX phase has garnered significant attention in the present era due to its outstanding mechanical and thermal characteristics at high temperatures, showcasing attributes shared by both metals and ceramics. The increased interest in MAX phase materials can be traced back to Barsoum's noteworthy contributions [1], [2]. The term MAX phase represents a family of multilayer solids where M is an earlier transition metal, A is an element from the IIIA or IVA group of the periodic table, and X is an atom of C/N/B [3]. MAX phase materials showcase metallic behavior due to alternate metallic A-layers and ceramic behavior attributed to the MX layers [4]. Like most metals and alloys, MAX phase materials have excellent thermal shock resistance, superior machinability, and enhanced thermal and electrical conductivities. On the other hand, they have high melting or decomposition temperatures and strong elastic stiffness, similar to many ceramics[5]. The unique combination of metallic and ceramic properties makes them versatile, with applications ranging from high-temperature coatings to nuclear accidenttolerant fuel (ATF), concentrated solar power (CSP), catalysis, and as precursors for MXenes [6], [7], [8], [9]. The diversity of MAX phases was confined to C and N as X elements for years (from 1960 to 2014). However, recent advancements have overcome this limitation by successfully synthesizing MAX phases that contain B [10]. The physical and chemical properties of B and Bcontaining compounds highlight the potential of MAX phase borides, which replace C/N with boron [11]. Due to the presence of B in the composition, they are also called the MAB phase. Furthermore, scientific communities are actively working to broaden the diversity of MAX phases by introducing structural changes, as seen in examples like Cr3AlB<sup>4</sup> (space group Immm), Cr4AlB<sup>6</sup> (space group Cmmm), Cr4AlB<sup>4</sup> [12], [13], [14]. Khazaei et al. reported on the first investigation of the theoretical MAX phase borides M2AlB (where M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta). They evaluated these compounds' electronic structure, mechanical characteristics, and dynamical stability in their investigation [15]. The diversity of MAX phases is a subject of research interest due to their structure variations and the number of atoms in the compounds, resulting in changes to their characteristics. In examples like the 211 MAX phase [16], 312 MAX phase [8][17], 212 MAX phase [3], and i-MAX phase [18], the recent additions to the MAX family include the 314 MAX phase and 212 MAX phase, where the element B has been incorporated as an X element [3] [19]. This introduces a novel aspect to the MAX phase family, contributing to its diversity. The unique structural features of MAB phases include both orthorhombic and hexagonal symmetry observed in crystals. These distinctive symmetries set MAB phases apart from the typical MAX phases. Yinqiao Liu's synthesis of the orthorhombic phase of M2AlB2, particularly the 212-MAB phases with M = Sc, Ti, Zr, Hf, V, Nb, Cr, Mo, W, Mn, Tc, Fe, Co, and Ni, has revealed unique structural stability and notable electrical and mechanical properties [20]. The MAB phase structure slightly differs from the standard MAX phases, which typically crystallize in the hexagonal system with a space group -6<sup>2</sup> (No. 187). Ali et al [3]. investigated the diverse physical properties of Zr2AB<sup>2</sup> (A = In, Tl). Martin Ade et al [13] synthesized ternary borides, namely Cr2AlB2, Cr3AlB4, and Cr4AlB6, and subsequently compared their mechanical properties. Qureshi et al. [19] investigated the 314 Zr3CdB<sup>4</sup> MAX phase boride, calculating its mechanical, thermodynamic, and optical properties. The 314 MAX phase Hf3PB<sup>4</sup> has been thoroughly studied using Density Functional Theory (DFT) and revealed that it was the hardest MAX phase compound discovered until that date [21]. The structure of 212 phases exhibits a slight deviation from 211 MAX phases. In the case of 212 phases, a 2D layer of B is situated between M layers, featuring an additional B atom at the X position, unlike in 211 phases [22] [23], [24]. The B-B bonding in 212 MAX phase borides has improved mechanical and thermal properties. To date, the physical properties of Zr2AB<sup>2</sup> (A = In, Tl) [25] and Hf2AB<sup>2</sup> (A = In, Sn) [26], M2AB (M= Ti, Zr, Hf; A=Al, Ga, In) [27] MAX phases, as well as Nb2AC (A = Ga, Ge, Tl, Zn, P, In, and Cd) MAX phases, have been investigated using density functional theory (DFT). In each instance, the mechanical properties of B-containing compounds show significant improvement compared to their traditional C/N containing 211 MAX phases. The Debye temperature and melting temperature are higher for boron-containing 212 phases than for 211 carbides/nitrides, while the minimum thermal conductivity is lower. The thermal expansion coefficient of borides remains well-suited for use as coating materials. Consequently, the superior thermomechanical properties of B-containing 212 MAX phases demonstrate their suitability for high-temperature technological applications, surpassing the commonly used 211 MAX phase carbides. It should be noted that the 212 MAB phase with a hexagonal structure has already been synthesized [24]. In addition, Ga and Ge-based MAX phases have also been synthesized previously [28]. Thus, the reports on the synthesis of 212 phase and Ga and Gebased MAX phases motivated us to select the 212 MAB phases: M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) for our present study, and we have performed an in-depth investigation of their physical properties through DFT method. Therefore, in this paper, the first-time prediction of the stability and mechanical, electronic, thermal, and optical properties of M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) phases has been presented. The results revealed that M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) compounds are stable and suitable for thermal barrier coating (TBC) and reflection coating applications. Additionally, to provide a comparison, the properties determined for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) are compared with those of other 212 and 211 compounds MAX phase borides. ### 2. Methods of calculations First principles density-functional theory (DFT) computations are conducted utilizing the Cambridge Serial Total Energy Package (CASTEP) module integrated within Materials Studio 2017 [29], [30]. The exchange-correlation function is estimated using the Generalized Gradient Approximation (GGA) method, originally suggested by Perdew, Burke, and Ernzerhof [31]. Pseudo-atomic simulations accounted for electronic orbitals corresponding to B (2s 2 2p 1 ), Ga (3d <sup>10</sup> 4s 2 4p 1 ), Ge (4s 2 4p 2 ), Mo (4s 2 4p 6 4d 5 5s 1 ), and Ta (5d 3 6s 2 ). The energy cutoff and k-point grids were established at 650 eV and 11 × 11 × 4, respectively. The structural relaxation was performed utilizing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique [32], while the electronic structure was computed employing density mixing. The parameters for relaxed structures incorporate the following tolerance thresholds: the self-consistent convergence of the total energy is set at 5 × 10-6 eV/atom, the maximum force exerted on the atom is limited to 0.01 eV/Å, the maximum ionic displacement is constrained to 5 × 10-4 Å, and a maximum stress threshold of 0.02 GPa is imposed. The finite strain method [33], grounded in density functional theory (DFT), is utilized to compute the elastic properties within this framework. All necessary equations for determining various properties are provided in the supplementary document. #### 3 Results and discussion #### 3.1 Structural properties The M2AB<sup>2</sup> compounds (where M=Mo or Ta; A=Ga or Ge) belong to the P6m<sup>2</sup> (No. 187)[23] space group and crystallize in the hexagonal system. Unlike conventional MAX phases, which typically belong to the P63/mmc (194) space group, the 212 MAX phases exhibit distinct characteristics. In Fig. 1, the unit cell structure of Mo2GaB<sup>2</sup> is depicted as a representative of M2AB<sup>2</sup> alongside Mo2GaB, facilitating a comparison to discern their differences easily. The atomic positions are as follows: M (Mo or Ta) at (0.3333, 0.6667, 0.6935), A at (0.6667, 0.3333, 0.0), and two B atoms positioned at (0.6667, 0.3333, 0.5) and (0.0, 0.0, 0.5). The B components are arranged at the corners of the unit cell in typical 211 MAX phases, but in 212 boride MAX phases, they form a 2D layer between the M layers. This structural arrangement results in B-B covalent bonds in the 2D layer, enhancing stability compared to conventional 211 MAX phases. ![Fig. 1 - The schematic unit cell of (a) Mo2GaB<sup>2</sup> and (b) Mo2GaB compound.](path) The calculated lattice parameters of M2AB<sup>2</sup> are presented in Table 1 alongside those of other 211 and 212 MAX phases for comparison, demonstrating consistency with prior results [16], [34], [35] and affirming the accuracy of the computational methodology used. The primary distinction between 212- and 211-unit cell structures arises from differences in the lattice parameter c, where the c value for 211 exceeds that of 212. Moreover, the volumes of Mo- and Ta-based 211 MAX phase borides surpass those of 212 MAX phase borides. #### 3.2 Stability Examining a compound's stability is significant for multiple purposes, as it yields valuable information regarding the compound's synthesis parameters and aids in assessing the material's resilience across diverse environments, including thermal, compressive, and mechanical pressures. In this section, we delve into a comprehensive theoretical analysis concerning the chemical, dynamic, and mechanical stability of M2AB<sup>2</sup> compounds. The compound's chemical stability is determined by computing its formation energy by the following equation [36]: 2<sup>2</sup> = 22−( + + ) ++ . In the context provided, 22 represents the total energy of the compound after optimizing the unit cell. , , and denote the energies of the individual elements M, A, and B, respectively. The variables x, y, and z correspond to the number of atoms in the unit cell for M, Ga, and B, respectively. For Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, the calculated formation energy (Ef) are -1.8079 eV/atom, -1.8859 eV/atom, -2.1252 eV/atom, and -2.1933 eV/atom, respectively. The negative values signify the chemical stability of all compounds. The order of chemical stability can be expressed as Ta2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Mo2GaB2, indicating that Ta2GeB<sup>2</sup> is the most stable. Additionally, it's observed that MAX phase borides containing Ta are more stable than those containing Mo. Negative formation energy alone may not fully explain the chemical stability of M2AB2 (M=Mo, Ta and A=Ga, Ge). We calculated its formation enthalpy by examining potential pathways to evaluate its thermodynamic stability. For this analysis, we used the experimentally identified stable phases of MoB[37] and TaB[38], Ga4Mo[39], and B2Mo[40]. The potential decomposition pathways for our compounds, as determined from the Open Quantum Materials Database (OQMD), are outlined below. ![We have calculated the reaction energy as follows [41]:](path) Where, M=Mo, Ta and A=Ga, Ge. The following formula can calculate the decomposition energy associated with the reaction energy. = . Where n is the number of participating atoms. The calculated decomposition energies for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> are -0.16, -0.17, -0.26, and -0.28 meV/atom, respectively. So, we can state that the M2AB2 (M=Mo, Ta and A=Ga, Ge) system exhibits thermodynamic stability. Phonon dispersion curves (PDCs) have been calculated at the ground state utilizing the density functional perturbation theory (DFPT) linear-response approach to evaluate the dynamic stability of the MAX phase borides under investigation [42]. The PDCs, depicting the phonon dispersion along the high symmetry directions of the crystal Brillouin zone (BZ), along with the total phonon density of states (PHDOS) of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, are illustrated in Fig. 2(a, b, c and d). Analysis of the PDCs reveals no negative phonon frequencies for any of the compounds, indicating their dynamic stability. The PHDOS of the M2AB<sup>2</sup> compounds are derived from the PDCs and are presented alongside the PDCs in Fig. 2(a, b, c, and d), facilitating band identification through comparison of corresponding peaks. From Fig. 2, it is observed that in Mo2GaB2, the flatness of the bands for the Transverse Optical (TO) modes results in a prominent peak in the PHDOS, whereas non-flat bands for the Longitudinal Optical (LO) modes lead to weaker peaks in the PHDOS. Similar trends are observed in Mo2GeB2, Ta2GaB2, and Ta2GeB2. Notably, a distinct discrepancy arises between the optical and acoustic branches, with the top of the LO and bottom of the TO modes situated at the G point, with separations of 7.49 THz, 6.39 THz, 9.39 THz, and 8.51 THz for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, respectively. ![Fig. 2 – Phonon DOS and Phonon dispersion curves of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) Materials are subjected to various forces and loads in practical applications, necessitating understanding their mechanical stability. The mechanical stability of a compound can be assessed using stiffness constants. For a hexagonal system, the conditions for mechanical stability are as follows [43]: C<sup>11</sup> > 0, C<sup>11</sup> > C12, C<sup>44</sup> > 0, and (C<sup>11</sup> + C12)C<sup>33</sup> - 2(C13)² > 0. As indicated in Table 2, the Cij values of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) satisfy these conditions, thus confirming the mechanical stability of herein predicted phases: M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge). #### 3.3 Electronic properties Analyzing the electronic band structure (EBS) is crucial for gaining insights into the electronic behavior of a compound. The EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) MAX phases are depicted in Fig. 3(a, b, c, and d), with the Fermi energy (EF) level set at 0 eV, represented by a horizontal line. Observing the EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge), it is evident that the conduction band overlaps with the valence band, indicating the absence of a band gap. This observation confirms that the M2AB<sup>2</sup> compounds exhibit metallic behavior, which aligns with conventional MAX phases. The red lines illustrate the overlapping band at the Fermi level. Fig. 3 (a, b) shows that for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> compounds, the maximum band overlap occurs along the A-H path. Conversely, in Fig. 3(c, d), the maximum band overlap is observed along the G-M paths. Utilizing the band structure, we can analyze the electrical anisotropy of M2AB<sup>2</sup> MAX phase compounds. The anisotropic nature can be understood by analyzing the energy dispersion in the basal plane and along the c-axis. The paths G-A, H-K, and M-L show energy dispersion along the c-direction, and A-H, K-G, G-M, and L-H show energy dispersion in the basal plane. In comparison to the paths A-H, K-G, G-M, and L-H (basal plane), there is less energy dispersion along the lines G-A, H–K, and M-L (c-direction), as shown by Fig. 2(a, b, c and d). Lower energy dispersion in the c-direction results from a higher effective mass [44], indicating the strong electronic anisotropy of the M2AB<sup>2</sup> MAX phase compound. Consequently, conductivity along the c-axis is expected to be lower than in the basal planes. These findings are consistent with prior studies [34], [45]. ![Fig. 3 – Band structure of (a) Mo2GaB2, (b) Mo2GeB2, (c) Ta2GaB2, and (d) Ta2GeB<sup>2</sup> compounds.](path) To investigate the bonding nature and electronic conductivity, the total density of states (TDOS) and partial density of states (PDOS) of M2AB<sup>2</sup> compounds were computed. Figure 4 (a, b, c, and d) illustrates the TDOS and PDOS of these compounds, with the Fermi energy (EF) set at zero energy level, indicated by a straight line. These profiles exhibit typical characteristics of MAX phase materials. The Mo or Ta-d electronic states predominantly contribute to the Fermi level, with a minor contribution from the B-p and Ga or Ge-p electronic states. To ascertain the hybridization characteristics of various electronic states within the valence band, the energy spectrum of the valence band has been partitioned into two distinct segments. The first segment encompasses the lower valence band region spanning from -7 eV to -3.5 eV, originating from the hybridization of Mo-p, Mo-d, and B-s orbitals in the case of the Mo2GaB<sup>2</sup> compound. Conversely, for the Mo2GeB2 compound, the lower valence band region arises from the hybridization of Mo-d, Ge-p, and B-s orbitals. Notably, for both the Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compounds, the dominance of the B-p state characterizes the lower valence band region. The second segment pertains to the upper valence band region from -3.5 eV to 0 eV. In the case of the Mo2GaB<sup>2</sup> compound, this region arises from the hybridization of Mo-p and Mo-d orbitals. However, for the Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, the upper valence band region results from the hybridization of Mo-d orbitals and (Ga/Ge)-p orbitals. Notably, the Fermi level of M2AB<sup>2</sup> resides near the pseudogap in the TDOS profile, indicating a high level of electronic stability. Similar trends are observed in other compounds like Zr2AlN, V2AlN, Sc2AlB, Sc2GaB, Ta2GaB, and Hf2GaB<sup>2</sup> [15], [35], [45]. ![Fig. 4 – TDOS and PDOS of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) The electron charge density mapping is helpful in understanding the distribution of electron densities linked to chemical bonds. It delineates areas of positive and negative charge densities, signifying the development and exhaustion of electrical charges, respectively. As depicted in the map, covalent bonds become apparent by accumulating charges between two atoms. Furthermore, the presence of ionic bonds can be inferred from a balance between negative and positive charges at specific atom positions[46]. The valence electronic CDM, denoted in units of eÅ-3 , for M2AB<sup>2</sup> (where M = Mo, Ta; A = Ga, Ge) is showcased in Fig. 5(a, b, c, and d) along the (110) crystallographic plane. The accompanying scale illustrates the intensity of electronic charge density, with red and blue colors indicating low and high electronic charge density, respectively. As depicted in Fig. 5(a, b, c, and d), it is evident that charges accumulate in the regions between the B sites. Consequently, it is anticipated that strong covalent B‒B bonding occurs through the formation of two center-two electron (2c‒2e) bonds in the M2AB<sup>2</sup> (where M=Mo, Ta; A=Ga, Ge) compound, similar to other 212 MAX phase borides like Ti2PB2, Zr2PbB2, Nb2SB2, Zr2GaB<sup>2</sup> and Hf2GaB<sup>2</sup> [34], [45]. Mulliken analysis has corroborated the charge transfer from Mo/Ta atoms to B atoms. The charge received from Mo/Ta atoms is distributed among the B atoms positioned at the transitions and those located at the edges, facilitating the formation of a two center-two electron (2c‒2e) bond between B atoms within the 2D layer of B, as illustrated in Fig. 1(a). The hardness of each bond value presented in Table 5 also aligns with the results obtained from charge density mapping (CDM) and our analysis using moduli and elastic stiffness constants. ![Fig. 5 – The charge density mapping of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) ## 3.4.1 Mechanical properties The mechanical properties of materials play a pivotal role in determining their potential applications, serving as crucial indicators of their behavior and suitability in materials engineering endeavors. These properties are equally applicable to MAX phase materials. Initially, to assess the mechanical properties of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phases, we employed the stress-strain method within the CASTEP code to compute the elastic constants (Cij) [22], [47], [48]. These calculated elastic constants (Cij) are presented in Table 2, alongside those of other 212 and 211 MAX phases. Due to the hexagonal crystal structure of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phase borides, five stiffness constants emerge: C11, C12, C13, C33, and C<sup>44</sup> [49]. The mechanical stability was evaluated using these stiffness constants in the preceding section. For instance, C<sup>11</sup> and C<sup>33</sup> determine stiffness when stress is exerted along the (100) and (001) directions, respectively, whereas C<sup>44</sup> evaluates resistance to shear deformation on the (100) and (001) planes. The stiffness constants C<sup>11</sup> and C<sup>12</sup> directly reflect the strength of atomic bonds along the a- and c-axes. When C<sup>11</sup> exceeds C<sup>33</sup> (or vice versa), it signifies stronger atomic bonding along the a-axis (or c-axis). In Table 2, for Mo2GaB2, Mo2GeB2, and Ta2GeB<sup>2</sup> compounds, C<sup>11</sup> surpasses C33, indicating superior atomic bonding along the a-axis compared to the c-axis. This robust bonding along the a-axis suggests heightened resistance to a-axial deformation. Conversely, in Ta2GeB2, where C<sup>11</sup> < C33, stronger atomic bonding along the c-axis translates to increased resistance against c-axial deformation. Analysis of Table-2 reveals that the values of C<sup>11</sup> and C<sup>22</sup> are notably higher for 212 MAX phase borides than 211 MAX phase borides. Consequently, it can be inferred that 212 MAX phases exhibit stronger resistance to axial deformation when juxtaposed with 211 MAX phases. C<sup>44</sup> is commonly utilized to gauge shear deformation tolerance among the elastic constants. Notably, Ta2GaB exhibits superior shear deformation resistance owing to its highest C<sup>44</sup> value among the studied compounds. Furthermore, the individual elastic constants C12, C13, and C<sup>44</sup> denote shear deformation response under external stress. The observation that C<sup>11</sup> and C<sup>33</sup> possess larger magnitudes than C<sup>44</sup> implies that shear deformation is more facile than axial strain. Another crucial parameter derived from the stiffness constants is the Cauchy pressure (CP), calculated as C<sup>12</sup> - C44, which provides vital insights relevant to the practical applications of solids [51]. Pettifor [51] emphasized the significance of Cauchy pressure (CP) in discerning the chemical bonding and ductile/brittle properties of solids. A negative CP value indicates covalently bonded brittle solids, whereas a positive value signifies isotropic ionic ductile solids. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> fall into covalently bonded brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit metallic ductile characteristics. Similarly, like Ga-containing 211 MAX phase borides, Ga-containing 212 MAX phases also demonstrate negative CP values and brittle behavior [52][53]. The elastic constants obtained are used to produce several bulk elastic parameters that are used to characterize polycrystalline materials, such as Young's modulus (Y), bulk modulus (B), and shear modulus (G). In Table 2, the bulk modulus (B) and shear modulus (G) calculated using Hill's approximation [54] are also presented. Hill's values represent the average of the upper limit (Voigt [55]) and lower limit (Reuss [56]) of B. The necessary equations for these calculations are provided in the supplementary document (S1). Young's modulus (Y) is a crucial indicator of material stiffness. A higher Y value indicates a stiffer material. It can be observed from the table that Mo2GaB<sup>2</sup> exhibits the highest Y value, signifying its greater stiffness compared to others. According to the sequence of Y values, stiffness can be ranked as follows: Mo2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Ta2GeB2. Therefore, Mo2GaB2, with its higher Young's modulus, is anticipated to demonstrate superior mechanical stability and deformation resistance compared to the other compounds under investigation. This quality is essential for aircraft parts or high-performance machinery applications where critical dimensional stability and structural integrity are required [57], [58]. Table 2 shows that 212 MAX phase materials exhibit larger Young's modulus (Y) values than 211 MAX phases. This suggests that 212 MAX phase borides are stiffer than their 211 MAX phase counterparts. Young's modulus (E) also correlates well with thermal shock resistance (R): R ∝ 1/E [59]. Lower Young's modulus values correspond to higher thermal shock resistance. Therefore, materials with higher thermal shock resistance (i.e., lower Young's modulus) are more suitable for use as Thermal Barrier Coating (TBC) materials. Given that Ta2GaB<sup>2</sup> possesses the lowest Y value among the materials studied, it should be considered a superior candidate for TBC material due to its higher thermal shock resistance. The material's ability to withstand shape distortion is elucidated by the shear modulus (G). On the other hand, the bulk modulus (B) indicates the strength of a material's chemical bonds and its ability to withstand uniform compression or volume change. We computed Young's modulus (Y), bulk modulus (B), and shear modulus for our analysis. Based on the bulk modulus values presented in Table 2, the sequence of a material that resists compression when pressure is applied can be outlined as follows: Mo2GeB<sup>2</sup> > Ta2GeB<sup>2</sup> > Mo2GaB<sup>2</sup> > Ta2GeB2. Mo2GeB2, boasting a higher bulk modulus, may exhibit reduced plastic deformation and superior resistance to stress-induced deformation compared to other compounds examined. The ductile and brittle characteristics of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds can also be assessed through Poisson's ratio (ν) and Pugh's ratio (G/B). A compound demonstrates ductile (brittle) behavior if the ν value surpasses (falls below) 0.26 [60]. Furthermore, if the G/B value exceeds (is less than) 0.571, then the compound exhibits brittle (ductile) behavior [19]. According to both criteria, Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are classified as brittle compounds, whereas Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> are categorized as ductile compounds. The ductile nature of Ge-based MAX phases has been reported previously [61]. Fracture toughness (KIC) is a vital property that gauges a material's ability to resist crack propagation. In the case of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, Equation (S2) was employed to determine KIC. The KIC values are documented in Table 2, and notably, these values surpass those of 211 MAX phases [16], [35]. Another parameter, the "f-index," characterizes the isotropic nature and strength of atom-atom bonds within a single hexagonal crystalline lattice along the a- and c-directions. If the f-index is less than 1, chemical bonds exhibit greater rigidity along the c-axis; conversely, if f exceeds 1, bonds are more rigid in the ab-plane. When the value of f is set to 1, atomic bonds exhibit similar strength and uniformity in all directions [15]. The fvalue is computed using Equation (S3) and displayed in Table 2. Table 2 shows that the f-values of Mo2GeB2, Ta2GaB2, and Ta2GeB2, which are close to one, indicate a slight anisotropic bonding strength. Nevertheless, substances with robust bonds in the horizontal plane (ab plane) (f > 1), such as Mo2GaB2, are deemed optimal candidates for the exfoliation process. In engineering applications, the hardness of a solid material serves as a valuable criterion for designing various devices. The elastic properties of polycrystalline materials can be utilized to calculate hardness values, as the ability to resist indentation is closely linked to a material's hardness. Both micro-hardness (Hmicro) and macro-hardness (Hmacro) were computed using Equation (S4) and are presented in Table 2. Based on the values of Hmicro and Hmacro, Mo2GaB<sup>2</sup> emerges as the toughest among the studied phases. The order of hardness is as follows: Mo2GaB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GeB2. Interestingly, Ga-containing MAX phase borides exhibit superior hardness compared to Ge-containing MAX phase borides. In Table 2, we compare our findings with previously reported MAX phases and observe that our data align perfectly with the earlier results. #### 3.4.2 Elastic anisotropy Additionally, anisotropy is linked to other crucial events like anisotropic plastic deformation and the development and spread of micro-cracks within mechanical stress. By providing directiondependent elastic constants, the understanding of anisotropy also offers a framework for improving the mechanical stability of materials in extreme circumstances. The following formulae are used for hexagonal structures to calculate the various anisotropic variables from elastic constants Cij [62]. $A\_{1} = \frac{\frac{1}{6}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{44}};$ $A\_{2} = \frac{2\mathcal{C}\_{44}}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ and $A\_{3} = A\_{1}.$ $A\_{2} = \frac{\frac{1}{3}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ Table 3 lists every anisotropy parameter. Since the value of A<sup>i</sup> should be 1 to be isotropic, the computed value of A<sup>i</sup> (i = 1-3) indicates that all of the compounds under research exhibit anisotropic behavior [63]. An additional method for estimating elastic anisotropy is to use the percentage anisotropy to compressibility and shear (A<sup>B</sup> & AG). This gives polycrystalline materials a helpful way to measure elastic anisotropy. They have been described as [45]; Zero values for AB, AG, and the universal anisotropy factor (AU) show elastic isotropy; the maximum amount of anisotropy is represented by a value of 1. Table 3 shows that, compared to other compounds, the values of AB, AG, and A U for Mo2GeB<sup>2</sup> and Ta2GaB<sup>2</sup> are incredibly close to zero, suggesting that these compounds have nearly isotropic characteristics. #### 3.4.3 Mulliken Populations The Mulliken charge assigned to an atomic species quantifies the effective valence by calculating the absolute difference between the formal ionic charge and the Mulliken charge. Equations (S5) and (S6) are employed to ascertain the Mulliken charge for each atom (α). Table-4 provides the Mulliken atomic population and effective valence charge. Transition metals Mo and Ta in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have pure valence states of 4d<sup>5</sup> and 5d<sup>3</sup> , respectively. The d- orbital electrons of transition metals have been found to influence their effective valence charge significantly. A non-zero positive value indicates a combination of covalent and ionic attributes within chemical bonds. As this value decreases towards zero, it signifies a rise in ionicity. A zero value suggests an ideal ionic character in the bond. Conversely, a progression from zero with a positive value indicates an elevation in the covalency level of the bonds. Based on their effective valence, M atoms move from the left to the right in the periodic table, increasing the covalency of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). Table-4 reveals that the Mulliken atomic charge ascribed to the B atoms is solely negative. Conversely, positive Mulliken atomic charges are associated with transition metals (M) and A. This suggests a charge transfer from M and A to B for each compound within M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge), thereby fostering ionic chemical bonds among these atoms. Bond population serves as another indicator of bond covalency within a crystal, as a high value of bond population essentially signifies a heightened degree of covalency within the chemical bond. The bonding and anti-bonding states influence the populations with positive and negative bond overlap. As demonstrated in Table-5, the B-B bond exhibits greater covalency compared to any other bond in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The presence of an antibonding state between two relevant atoms, which subsequently decreases their chemical bonding, is indicated by a hostile bond overlap population. In Mo2GeB<sup>2</sup> and Ta2GeB2, a hostile bond overlap population is observed in the Ge-Mo and Ge-Ta bonds, indicating the presence of an antibonding state. Therefore, the existence of ionic bonding is guaranteed by electronic charge transfer. On the other hand, the high positive value of the bond overlap population (BOP) denotes the presence of covalent bonding, a feature shared by materials in the MAX phase. ## 3.4.4 Theoretical Vickers Hardness Vickers hardness, derived from the atomic bonds found in solids, indicates how resistant a material is to deformation in extreme circumstances. Several variables influence this feature, such as the crystal flaws, solid structure, atomic arrangement, and bond strength. The Vickers hardness of the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) MAX phases is determined using the Mulliken bond population method, as described by Gou et al. [66] using the formula (S7-S10). This method is particularly suitable for partial metallic systems such as MAX phases. Table-5 lists the computed Vickers hardness values for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The calculated values are 3.35 GPa, 4.76 GPa, 8.21 GPa, and 8.74 GPa for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, respectively. We observed that Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> have much higher hardness values than Mo2GaB<sup>2</sup> and Mo2GeB2. These values are also higher than that of other 212 phases, like, Ti2InB<sup>2</sup> (4.05 GPa) [50], Hf2InB<sup>2</sup> (3.94 GPa), and Hf2SnB<sup>2</sup> (4.41 GPa) [26], Zr2InB<sup>2</sup> (2.92 GPa) and Zr2TlB<sup>2</sup> (2.19 GPa) [25], Zr2GaB<sup>2</sup> (2.53GPa), Zr2GeB<sup>2</sup> (3.31GPa), Hf2GaB<sup>2</sup> (4.73GPa) and Hf2GeB2(4.83GPa) [67]The Hv calculated by the geometrical average of the individual bonding, where the bonding strength mainly determined by the BOP values. In the case of Ta2GaB<sup>2</sup> and Ta2GeB2, the BOP of M-B bonding is much higher compared to Mo2GaB<sup>2</sup> and Mo2GeB2. Even though the BOP of M-B for Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> is higher than that of the other 212 phases mentioned earlier. Therefore, higher values of Hv are expected for Ta2GaB<sup>2</sup> and Ta2GeB2. Additionally, we looked at the Vickers hardness value between 211 and 212 compounds and discovered that the 212 MAX phase compounds had a higher H<sup>V</sup> value. This is because a 2D layer of B atoms is positioned between the M atoms. The B atoms share two center-two electrons to form an extremely strong B-B bond [3]. Table 5 Calculated data for Mulliken bond number (n μ ), bond length (d μ ), bond overlap populations BOP, (P μ ), metallic populations (P μˊ), Vickers hardness (HV) M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. # 3.5 Thermal Properties MAX phases are ideal for high-temperature applications due to their exceptional mechanical qualities at elevated temperatures. As a result, researching the fundamental parameters necessary for predicting their application is of great interest, and these can be obtained from the vibrations of atoms or phonons. The Debye temperature (ΘD) of a solid is directly connected to its bonding strength, melting temperature, thermal expansion, and conductivity. Using the sound velocity and Anderson's technique [68], the Θ<sup>D</sup> of the phases under study has been computed using the formula (S11). Equation (S12) can be used to get the average sound velocity (Vm) from the longitudinal and transverse sound velocities. Equations (S13–14) were used to determine v<sup>l</sup> and vt. The calculated values of Debye's temperature are shown in Table 6, where Mo2GaB<sup>2</sup> has the highest Θ<sup>D</sup> and Ta2GeB<sup>2</sup> has the lowest. If we rank them, it is as follows: Mo2GaB<sup>2</sup> < Mo2GeB<sup>2</sup> < Ta2GaB2 < Ta2GeB2. Hadi et al. recently reported a MAX phase (V2SnC) as a TBC material with a Θ<sup>D</sup> value of 472 K [69]. Thus, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) exhibit encouraging potential as TBC materials, as shown in Table 6. Table 6-Data for density (ρ), longitudinal, transverse, and average sound velocities (vl, vt, and vm), Debye temperature (ΘD), minimum thermal conductivity (Kmin), Grüneisen parameter (γ), thermal expansion coefficient (TEC) at 300K and melting temperature (Tm) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The constant thermal conductivity value at high temperatures is the minimum thermal conductivity (Kmin). As the name implies, this conductivity is minimal because, at high temperatures, phonon coupling breaks. The formula (S15) for the minimum thermal conductivity of solids was derived using the Clarke model [70]. Table 6 lists the calculated value of kmin, with Ta2GaB<sup>2</sup> having the lowest value and Mo2GaB<sup>2</sup> having the highest. When selecting suitable materials for TBC applications, a minimum thermal conductivity of 1.25 W/mK is used as a screening criterion [71]. Our compounds exhibit lower minimum thermal conductivity values, holding promising potential as TBC materials. Gd2Zr2O<sup>7</sup> and Y2SiO5, two recently developed thermal barrier coating (TBC) materials, have minimal thermal conductivities (Kmin) of 1.22 W/m.K. and 1.3 W/m.K. [72], respectively, as confirmed by experiment. These numbers roughly match the values we computed for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The Grüneisen parameter (γ) is a crucial thermal parameter that helps explain the anharmonic effects of lattice dynamics; solids utilized at high temperatures are expected to have lower anharmonic effects. The Grüneisen parameter (γ) can be determined with the help of Poisson's ratio using equation (S16) [73]. The computed γ values, as shown in Table 6, suggest that the compounds under investigation exhibit a weak anharmonic effect. Additionally, for solids with a Poisson's ratio between 0.05 and 0.46, the values similarly fall within the range of 0.85 and 3.53 [74]. The melting temperature (Tm) of the compounds under investigation has been calculated using the following formula (S17). The strength of atomic bonding is the primary factor determining the melting temperature of solids; the higher the Tm, the stronger the atomic bonding. The order of T<sup>m</sup> for the titled phases is found to mirror the Y-based (Young's modulus) order, indicating a close link between T<sup>m</sup> and Y [75]. As observed in Table 6, our compounds roughly follow the Ybased ranking with Mo2GeB<sup>2</sup> < Mo2GaB<sup>2</sup> < Ta2GeB<sup>2</sup> < Ta2GaB2. The T<sup>m</sup> value for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) is also comparable with the TBC material Y4Al2O<sup>9</sup> (2000 K)[75]. ![Fig. 6: Temperature dependence of the calculated thermodynamic parameters. (a) Specific heat at constant pressure, C<sup>p</sup> (b) Specific heat at constant volume, C<sup>v</sup> (c) Free energy, F (d) Entropy, S (e) Internal energy, E and (f) Thermal expansion coefficient, TEC of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge).](path) The temperature dependence of specific heats, Cv, Cp, for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds was obtained using the formulas (S19-S20), as shown in Fig.6(a,b). Assuming that the quasi-harmonic model is accurate and that phase transitions are not anticipated for the compounds under study, these properties are approximated across a temperature range of 0 to 1000 K. Because phonon thermal softening occurs at higher temperatures, the heat capacity rises with temperature. Heat capacities increase quickly and follow the Debye-T 3 power law at lower temperatures. At higher temperature regimes, where C<sup>v</sup> and C<sup>p</sup> do not greatly depend on temperature, they approach the Dulong-Petit (3) limit.[76]. Using the quasi-harmonic approximation, various temperature-dependent thermodynamic potential functions for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have been estimated at zero pressure and displayed in Fig.6 [77]. These functions include the Helmholtz free energy (F), internal energy (E), and entropy (S) within 0-1000 K temperature using equation (S21-S23). The free energy progressively decreases as the temperature rises, as shown in Fig.6(c). Free energy typically declines, and this trend becomes increasingly negative as a natural process proceeds. As demonstrated in Fig. 6(d), the internal energy (E) shows a rising trend with temperature, in contrast to the free energy. Since thermal agitation creates disorder, a system's entropy rises as temperature rises. This is illustrated in Fig.6(d). A material's thermal expansion coefficient (TEC) is derived from the anharmonicity in the lattice dynamics and can be found using equation (S18). The measure of a material's capacity to expand or contract with heat or cold is called the Thermal Expansion Coefficient, or TEC. As observed in Fig. 5(f), the Thermal Expansion Coefficient (TEC) increases rapidly up to 365 K. Then it approaches a constant value, which indicates lower saturations in the materials with temperature changes. The materials under study have a very low TEC value, a crucial characteristic of materials intended for application in high-temperature technology. To be effective as thermal barrier coating (TBC) materials, compounds must exhibit a low thermal conductivity (Kmin) to impede heat transfer, a high melting temperature to withstand extreme heat, and a low thermal expansion coefficient (TEC) to maintain dimensional stability under thermal stress. The compounds M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge) possess these properties, making them suitable candidates for use as TBC materials #### 3.6 Optical Properties Different materials exhibit unique behaviors when exposed to electromagnetic radiation. The optical constants determine the overall response of the sample to the incident radiation. The complex dielectric function, defined as ε(ω)=ε1(ω)+iε2(ω), is one of the main optical characteristics of solids. The following formula determines the imaginary part of the dielectric function ε2(ω) from the momentum matrix element between the occupied and unoccupied electronic states. In this formula, e stands for an electronic charge, ω for light angular frequency, u for the polarization vector of the incident electric field, and and for the conduction and valence band wave functions, respectively, at k. The Kramers-Kronig equation can estimate the real part of the dielectric function, ε1(ω). In contrast, ε2(ω) and ε1(ω) are utilized to evaluate all other optical parameters, such as the absorption coefficient, photoconductivity, reflectivity, and loss function [78]. In this part, several energy-dependent optical properties of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) 212 MAX phases are calculated and analyzed in detail for the photon energy range of 0 to 30 eV, for [100] plane enabling the first assessment of the compounds' practical applicability. Given the metallic conductivity of the MAX compounds' electronic structure, additional parameters were chosen to analyze the optical properties. These include a plasma frequency of 3 eV, damping of 0.05 eV, and Gaussian smearing of 0.5 eV [79]. Figure 7(a) shows the real component of the dielectric function, ε1, which exhibits metallic behavior. In metallic systems, ε<sup>1</sup> has a considerably high negative value in the low-energy range, with the real component reaching negative, which aligns with the band structure finding. Fig. 7(b) depicts the imaginary part of the dielectric function, ε2(ω), representing dielectric losses about frequency. Mo2GeB<sup>2</sup> demonstrates the highest peak in the low-energy region, with all compounds approaching zero from above at around 17 eV. This observation confirms Drude's behavior. ![Fig. 7- (a) Real part of dielectric constant, ε<sup>1</sup> (b) Imaginary part of dielectric constant, ε<sup>2</sup> (c) Reflectivity, R and (d) Loss function, (e) The coefficient of absorption, (α) (f) Refractive index, (n) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) for [100] electric field directions.](path) The reflectivity of a material measures the percentage of incident light energy reflected off it. Equation (S24) was employed to compute reflectivity using the dielectric function, as shown in Fig.7(c). The spectra's visible and infrared (IR) portions consistently exhibit values exceeding 55%. Consequently, the materials under examination are expected to appear metallic gray. In the infrared (IR) region, the maximum reflectivity for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> is 65% and 74%, respectively, occurring at 0.33 eV. Conversely, for Ta2GaB2, the maximum reflectivity is 59% at 14.02 eV, while for Ta2GeB2, it is 65% at 14.78 eV in the UV region. After 18 eV, the reflectance decreases significantly. According to reports, substances with an average reflectivity value above 44% in the visible light region can effectively reduce solar heating by reflecting a significant amount of the solar spectrum. Among all the compounds we examined, Ta2GaB<sup>2</sup> has a reflectivity in the visible range of more than 44% [78]. Therefore, Ta2GaB<sup>2</sup> can be utilized as a coating material and should be able to mitigate solar heating. plasma frequencies (ωp). The plasma frequency for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is observed at 23 eV, 25 eV, 21 eV, and 23 eV, respectively. At this specific frequency, the absorption coefficient rapidly decreases, ε<sup>1</sup> crosses zero from the negative side, and the reflectance R(ω) displays a falling tail. Above this distinctive frequency, the materials are transparent to the incident electromagnetic radiation. Fig. 7(e) illustrates the absorbance coefficient (α) of compounds M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) determined using equation (S26). As α begins to rise from zero photon energy, the metallic nature of the substances under study is again indicated. The visible light region experiences a sharp increase in absorption, peaking in the UV region at around 14 eV and then progressively declining. The IR region exhibits negligible absorption. However, the materials mentioned above appear to have a significant absorption band primarily located in the visible and ultraviolet spectrum. This suggests that the materials can be used in UV surface-disinfection devices, medical sterilization equipment, and other optoelectronic device designs. Its refractive index is crucial to a material's potential application in optical devices like waveguides and photonic crystals. Equations (S27) and (S28) were used to derive the refractive index (n) and extinction coefficient (k) for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), as depicted in Fig.7(c) and Fig.S-1(a). The variations of n and k in MAX phase carbides with incident photon energy closely resemble ε1(ω) and ε2(ω). The static refractive index n(0) for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is 9.4,13.07,7 and 9.2, respectively, and decreases gradually with the increase in photon energy. The extinction coefficient for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2 gradually increases in the IR region, reaching their maximum values of 2.56, 4.27, 2.42, and 3.28, respectively. Following this, they slowly decrease in the visible and UV regions. Figure S-1(b) illustrates the photoconductivity (σ) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) across various photon energies. The photoconductivity (σ) parameter quantifies the impact of photon irradiation on a material's electrical conductivity. Similar to the absorbance coefficient (α) spectrum, the σ spectrum aligns with the metallic nature of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). #### 4 Conclusions In summary, we employed DFT calculations to explore four 212 MAB phases, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), and investigated the structural, electronic, mechanical, lattice dynamical, and optical properties to predict their possible applications. The phonon dispersion curves, formation energy, and elastic constants collectively suggest that the M2AB<sup>2</sup> boride maintains dynamic, mechanical, chemical, and thermodynamic stability. The electronic band structure and density of states (DOS) offer evidence supporting the metallic nature of the compounds under investigation. Concurrently, the charge density mapping and atomic Mulliken population both confirm the presence of a strong B-B covalent bond. The stiffness constants, elastic moduli, findex, fracture toughness (KIC), Pugh's ratio (G/B), hardness parameters, and Cauchy Pressure (CP) of M2AB<sup>2</sup> were computed and compared with those of their 211 equivalents. We found that the values of the 212-phase borides are higher than those of the 211-phase carbides or borides. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are identified as brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit ductile characteristics, as indicated by Poisson's ratio (ν) and Pugh's ratio (B/G or G/B). The elastic characteristics display anisotropy due to the distinct atomic configurations along the aand c-directions. Vickers hardness calculations are considered reliable indicators of material hardness. The results suggest that Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> possess more pliable characteristics than Ta2GaB<sup>2</sup> and Ta2GeB2. The high hardness values of Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compared to the Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> due to higher BOP values of M-B bonding in Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> than in Mo2GaB<sup>2</sup> and Mo2GeB2. The value of minimum thermal conductivity (Kmin), thermal expansion coefficient, and melting temperature (Tm) collectively suggest the potential suitability of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) as a material for thermal barrier coating (TBC) applications in high-temperature devices. The optical conductivity and absorption coefficient corroborate the findings of the electronic band structure. Reflectivity is notably high in infrared (IR) regions and remains nearly constant in the visible and moderate ultraviolet (UV) regions, with an average value exceeding 44% for Ta2GaB2. This suggests that Ta2GaB<sup>2</sup> can be effectively utilized as a coating material to reduce solar heating. We expect that the comprehensive analysis of the diverse physical characteristics of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) presented in this study will establish a robust foundation for future theoretical and experimental explorations of these fascinating MAB phases. ## CRediT author statement A. K. M Naim Ishtiaq and Md Nasir Uddin: Data curation, Writing- Original draft preparation. Md. Rasel Rana, Shariful Islam and Noor Afsary: Reviewing and Editing. Md. 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Fig. 2 – Phonon DOS and Phonon dispersion curves of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB² compounds.	# Exploration of new 212 MAB phases: M2AB2 (M=Mo, Ta; A=Ga, Ge) via DFT calculations #### Abstract The recently developed MAB phases, an extension of the MAX phase, have sparked interest in research among scientists because of their better thermo-mechanical properties. In this paper, we have explored four new MAB phases M2AB2 (M=Mo, Ta and A=Ga, Ge) and studied the elastic, electronic, thermal, and optical properties to predict the possible applications. The stability of the new phases has been confirmed by calculating formation energy (Ef), formation enthalpy (∆H), phonon dispersion curve (PDC), and elastic constant (Cij). The study reveals that M2AB<sup>2</sup> (M=Mo, Ta and A=Ga, Ge) exhibit significantly higher elastic constants, elastic moduli, and Vickers hardness values than their counterpart 211 borides. Higher Vickers hardness values of Ta2AB2 (A=Ga, Ge) than Mo2AB2 (A=Ga, Ge) have been explained based on the values of the bond overlap population. The analysis of the density of states and electronic band structure revealed the metallic nature of the borides under examination. The thermodynamic characteristics of M2AB2 (M=Mo, Ta and A=Ga, Ge) under high temperatures (0–1000 K) are investigated using the quasi-harmonic Debye model. Critical thermal properties such as melting temperature (Tm), Grüneisen parameter (γ), minimum thermal conductivity (Kmin), Debye temperature (ΘD), and others are also computed. Compared with 211 MAX phases, the 212 phases exhibit higher values of (ΘD) and Tm, along with a lower value of Kmin. These findings suggest that the studied compounds exhibit superior thermal properties that are suitable for practical applications. The optical characteristics have been examined, and the reflectance spectrum indicates that the materials have the potential to mitigate solar heating across various energy regions. #First two authors contributed equally. ### 1. Introduction The MAX phase has garnered significant attention in the present era due to its outstanding mechanical and thermal characteristics at high temperatures, showcasing attributes shared by both metals and ceramics. The increased interest in MAX phase materials can be traced back to Barsoum's noteworthy contributions [1], [2]. The term MAX phase represents a family of multilayer solids where M is an earlier transition metal, A is an element from the IIIA or IVA group of the periodic table, and X is an atom of C/N/B [3]. MAX phase materials showcase metallic behavior due to alternate metallic A-layers and ceramic behavior attributed to the MX layers [4]. Like most metals and alloys, MAX phase materials have excellent thermal shock resistance, superior machinability, and enhanced thermal and electrical conductivities. On the other hand, they have high melting or decomposition temperatures and strong elastic stiffness, similar to many ceramics[5]. The unique combination of metallic and ceramic properties makes them versatile, with applications ranging from high-temperature coatings to nuclear accidenttolerant fuel (ATF), concentrated solar power (CSP), catalysis, and as precursors for MXenes [6], [7], [8], [9]. The diversity of MAX phases was confined to C and N as X elements for years (from 1960 to 2014). However, recent advancements have overcome this limitation by successfully synthesizing MAX phases that contain B [10]. The physical and chemical properties of B and Bcontaining compounds highlight the potential of MAX phase borides, which replace C/N with boron [11]. Due to the presence of B in the composition, they are also called the MAB phase. Furthermore, scientific communities are actively working to broaden the diversity of MAX phases by introducing structural changes, as seen in examples like Cr3AlB<sup>4</sup> (space group Immm), Cr4AlB<sup>6</sup> (space group Cmmm), Cr4AlB<sup>4</sup> [12], [13], [14]. Khazaei et al. reported on the first investigation of the theoretical MAX phase borides M2AlB (where M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta). They evaluated these compounds' electronic structure, mechanical characteristics, and dynamical stability in their investigation [15]. The diversity of MAX phases is a subject of research interest due to their structure variations and the number of atoms in the compounds, resulting in changes to their characteristics. In examples like the 211 MAX phase [16], 312 MAX phase [8][17], 212 MAX phase [3], and i-MAX phase [18], the recent additions to the MAX family include the 314 MAX phase and 212 MAX phase, where the element B has been incorporated as an X element [3] [19]. This introduces a novel aspect to the MAX phase family, contributing to its diversity. The unique structural features of MAB phases include both orthorhombic and hexagonal symmetry observed in crystals. These distinctive symmetries set MAB phases apart from the typical MAX phases. Yinqiao Liu's synthesis of the orthorhombic phase of M2AlB2, particularly the 212-MAB phases with M = Sc, Ti, Zr, Hf, V, Nb, Cr, Mo, W, Mn, Tc, Fe, Co, and Ni, has revealed unique structural stability and notable electrical and mechanical properties [20]. The MAB phase structure slightly differs from the standard MAX phases, which typically crystallize in the hexagonal system with a space group -6<sup>2</sup> (No. 187). Ali et al [3]. investigated the diverse physical properties of Zr2AB<sup>2</sup> (A = In, Tl). Martin Ade et al [13] synthesized ternary borides, namely Cr2AlB2, Cr3AlB4, and Cr4AlB6, and subsequently compared their mechanical properties. Qureshi et al. [19] investigated the 314 Zr3CdB<sup>4</sup> MAX phase boride, calculating its mechanical, thermodynamic, and optical properties. The 314 MAX phase Hf3PB<sup>4</sup> has been thoroughly studied using Density Functional Theory (DFT) and revealed that it was the hardest MAX phase compound discovered until that date [21]. The structure of 212 phases exhibits a slight deviation from 211 MAX phases. In the case of 212 phases, a 2D layer of B is situated between M layers, featuring an additional B atom at the X position, unlike in 211 phases [22] [23], [24]. The B-B bonding in 212 MAX phase borides has improved mechanical and thermal properties. To date, the physical properties of Zr2AB<sup>2</sup> (A = In, Tl) [25] and Hf2AB<sup>2</sup> (A = In, Sn) [26], M2AB (M= Ti, Zr, Hf; A=Al, Ga, In) [27] MAX phases, as well as Nb2AC (A = Ga, Ge, Tl, Zn, P, In, and Cd) MAX phases, have been investigated using density functional theory (DFT). In each instance, the mechanical properties of B-containing compounds show significant improvement compared to their traditional C/N containing 211 MAX phases. The Debye temperature and melting temperature are higher for boron-containing 212 phases than for 211 carbides/nitrides, while the minimum thermal conductivity is lower. The thermal expansion coefficient of borides remains well-suited for use as coating materials. Consequently, the superior thermomechanical properties of B-containing 212 MAX phases demonstrate their suitability for high-temperature technological applications, surpassing the commonly used 211 MAX phase carbides. It should be noted that the 212 MAB phase with a hexagonal structure has already been synthesized [24]. In addition, Ga and Ge-based MAX phases have also been synthesized previously [28]. Thus, the reports on the synthesis of 212 phase and Ga and Gebased MAX phases motivated us to select the 212 MAB phases: M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) for our present study, and we have performed an in-depth investigation of their physical properties through DFT method. Therefore, in this paper, the first-time prediction of the stability and mechanical, electronic, thermal, and optical properties of M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) phases has been presented. The results revealed that M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) compounds are stable and suitable for thermal barrier coating (TBC) and reflection coating applications. Additionally, to provide a comparison, the properties determined for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) are compared with those of other 212 and 211 compounds MAX phase borides. ### 2. Methods of calculations First principles density-functional theory (DFT) computations are conducted utilizing the Cambridge Serial Total Energy Package (CASTEP) module integrated within Materials Studio 2017 [29], [30]. The exchange-correlation function is estimated using the Generalized Gradient Approximation (GGA) method, originally suggested by Perdew, Burke, and Ernzerhof [31]. Pseudo-atomic simulations accounted for electronic orbitals corresponding to B (2s 2 2p 1 ), Ga (3d <sup>10</sup> 4s 2 4p 1 ), Ge (4s 2 4p 2 ), Mo (4s 2 4p 6 4d 5 5s 1 ), and Ta (5d 3 6s 2 ). The energy cutoff and k-point grids were established at 650 eV and 11 × 11 × 4, respectively. The structural relaxation was performed utilizing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique [32], while the electronic structure was computed employing density mixing. The parameters for relaxed structures incorporate the following tolerance thresholds: the self-consistent convergence of the total energy is set at 5 × 10-6 eV/atom, the maximum force exerted on the atom is limited to 0.01 eV/Å, the maximum ionic displacement is constrained to 5 × 10-4 Å, and a maximum stress threshold of 0.02 GPa is imposed. The finite strain method [33], grounded in density functional theory (DFT), is utilized to compute the elastic properties within this framework. All necessary equations for determining various properties are provided in the supplementary document. #### 3 Results and discussion #### 3.1 Structural properties The M2AB<sup>2</sup> compounds (where M=Mo or Ta; A=Ga or Ge) belong to the P6m<sup>2</sup> (No. 187)[23] space group and crystallize in the hexagonal system. Unlike conventional MAX phases, which typically belong to the P63/mmc (194) space group, the 212 MAX phases exhibit distinct characteristics. In Fig. 1, the unit cell structure of Mo2GaB<sup>2</sup> is depicted as a representative of M2AB<sup>2</sup> alongside Mo2GaB, facilitating a comparison to discern their differences easily. The atomic positions are as follows: M (Mo or Ta) at (0.3333, 0.6667, 0.6935), A at (0.6667, 0.3333, 0.0), and two B atoms positioned at (0.6667, 0.3333, 0.5) and (0.0, 0.0, 0.5). The B components are arranged at the corners of the unit cell in typical 211 MAX phases, but in 212 boride MAX phases, they form a 2D layer between the M layers. This structural arrangement results in B-B covalent bonds in the 2D layer, enhancing stability compared to conventional 211 MAX phases. ![Fig. 1 - The schematic unit cell of (a) Mo2GaB<sup>2</sup> and (b) Mo2GaB compound.](path) The calculated lattice parameters of M2AB<sup>2</sup> are presented in Table 1 alongside those of other 211 and 212 MAX phases for comparison, demonstrating consistency with prior results [16], [34], [35] and affirming the accuracy of the computational methodology used. The primary distinction between 212- and 211-unit cell structures arises from differences in the lattice parameter c, where the c value for 211 exceeds that of 212. Moreover, the volumes of Mo- and Ta-based 211 MAX phase borides surpass those of 212 MAX phase borides. #### 3.2 Stability Examining a compound's stability is significant for multiple purposes, as it yields valuable information regarding the compound's synthesis parameters and aids in assessing the material's resilience across diverse environments, including thermal, compressive, and mechanical pressures. In this section, we delve into a comprehensive theoretical analysis concerning the chemical, dynamic, and mechanical stability of M2AB<sup>2</sup> compounds. The compound's chemical stability is determined by computing its formation energy by the following equation [36]: 2<sup>2</sup> = 22−( + + ) ++ . In the context provided, 22 represents the total energy of the compound after optimizing the unit cell. , , and denote the energies of the individual elements M, A, and B, respectively. The variables x, y, and z correspond to the number of atoms in the unit cell for M, Ga, and B, respectively. For Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, the calculated formation energy (Ef) are -1.8079 eV/atom, -1.8859 eV/atom, -2.1252 eV/atom, and -2.1933 eV/atom, respectively. The negative values signify the chemical stability of all compounds. The order of chemical stability can be expressed as Ta2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Mo2GaB2, indicating that Ta2GeB<sup>2</sup> is the most stable. Additionally, it's observed that MAX phase borides containing Ta are more stable than those containing Mo. Negative formation energy alone may not fully explain the chemical stability of M2AB2 (M=Mo, Ta and A=Ga, Ge). We calculated its formation enthalpy by examining potential pathways to evaluate its thermodynamic stability. For this analysis, we used the experimentally identified stable phases of MoB[37] and TaB[38], Ga4Mo[39], and B2Mo[40]. The potential decomposition pathways for our compounds, as determined from the Open Quantum Materials Database (OQMD), are outlined below. ![We have calculated the reaction energy as follows [41]:](path) Where, M=Mo, Ta and A=Ga, Ge. The following formula can calculate the decomposition energy associated with the reaction energy. = . Where n is the number of participating atoms. The calculated decomposition energies for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> are -0.16, -0.17, -0.26, and -0.28 meV/atom, respectively. So, we can state that the M2AB2 (M=Mo, Ta and A=Ga, Ge) system exhibits thermodynamic stability. Phonon dispersion curves (PDCs) have been calculated at the ground state utilizing the density functional perturbation theory (DFPT) linear-response approach to evaluate the dynamic stability of the MAX phase borides under investigation [42]. The PDCs, depicting the phonon dispersion along the high symmetry directions of the crystal Brillouin zone (BZ), along with the total phonon density of states (PHDOS) of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, are illustrated in Fig. 2(a, b, c and d). Analysis of the PDCs reveals no negative phonon frequencies for any of the compounds, indicating their dynamic stability. The PHDOS of the M2AB<sup>2</sup> compounds are derived from the PDCs and are presented alongside the PDCs in Fig. 2(a, b, c, and d), facilitating band identification through comparison of corresponding peaks. From Fig. 2, it is observed that in Mo2GaB2, the flatness of the bands for the Transverse Optical (TO) modes results in a prominent peak in the PHDOS, whereas non-flat bands for the Longitudinal Optical (LO) modes lead to weaker peaks in the PHDOS. Similar trends are observed in Mo2GeB2, Ta2GaB2, and Ta2GeB2. Notably, a distinct discrepancy arises between the optical and acoustic branches, with the top of the LO and bottom of the TO modes situated at the G point, with separations of 7.49 THz, 6.39 THz, 9.39 THz, and 8.51 THz for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, respectively. ![Fig. 2 – Phonon DOS and Phonon dispersion curves of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) Materials are subjected to various forces and loads in practical applications, necessitating understanding their mechanical stability. The mechanical stability of a compound can be assessed using stiffness constants. For a hexagonal system, the conditions for mechanical stability are as follows [43]: C<sup>11</sup> > 0, C<sup>11</sup> > C12, C<sup>44</sup> > 0, and (C<sup>11</sup> + C12)C<sup>33</sup> - 2(C13)² > 0. As indicated in Table 2, the Cij values of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) satisfy these conditions, thus confirming the mechanical stability of herein predicted phases: M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge). #### 3.3 Electronic properties Analyzing the electronic band structure (EBS) is crucial for gaining insights into the electronic behavior of a compound. The EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) MAX phases are depicted in Fig. 3(a, b, c, and d), with the Fermi energy (EF) level set at 0 eV, represented by a horizontal line. Observing the EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge), it is evident that the conduction band overlaps with the valence band, indicating the absence of a band gap. This observation confirms that the M2AB<sup>2</sup> compounds exhibit metallic behavior, which aligns with conventional MAX phases. The red lines illustrate the overlapping band at the Fermi level. Fig. 3 (a, b) shows that for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> compounds, the maximum band overlap occurs along the A-H path. Conversely, in Fig. 3(c, d), the maximum band overlap is observed along the G-M paths. Utilizing the band structure, we can analyze the electrical anisotropy of M2AB<sup>2</sup> MAX phase compounds. The anisotropic nature can be understood by analyzing the energy dispersion in the basal plane and along the c-axis. The paths G-A, H-K, and M-L show energy dispersion along the c-direction, and A-H, K-G, G-M, and L-H show energy dispersion in the basal plane. In comparison to the paths A-H, K-G, G-M, and L-H (basal plane), there is less energy dispersion along the lines G-A, H–K, and M-L (c-direction), as shown by Fig. 2(a, b, c and d). Lower energy dispersion in the c-direction results from a higher effective mass [44], indicating the strong electronic anisotropy of the M2AB<sup>2</sup> MAX phase compound. Consequently, conductivity along the c-axis is expected to be lower than in the basal planes. These findings are consistent with prior studies [34], [45]. ![Fig. 3 – Band structure of (a) Mo2GaB2, (b) Mo2GeB2, (c) Ta2GaB2, and (d) Ta2GeB<sup>2</sup> compounds.](path) To investigate the bonding nature and electronic conductivity, the total density of states (TDOS) and partial density of states (PDOS) of M2AB<sup>2</sup> compounds were computed. Figure 4 (a, b, c, and d) illustrates the TDOS and PDOS of these compounds, with the Fermi energy (EF) set at zero energy level, indicated by a straight line. These profiles exhibit typical characteristics of MAX phase materials. The Mo or Ta-d electronic states predominantly contribute to the Fermi level, with a minor contribution from the B-p and Ga or Ge-p electronic states. To ascertain the hybridization characteristics of various electronic states within the valence band, the energy spectrum of the valence band has been partitioned into two distinct segments. The first segment encompasses the lower valence band region spanning from -7 eV to -3.5 eV, originating from the hybridization of Mo-p, Mo-d, and B-s orbitals in the case of the Mo2GaB<sup>2</sup> compound. Conversely, for the Mo2GeB2 compound, the lower valence band region arises from the hybridization of Mo-d, Ge-p, and B-s orbitals. Notably, for both the Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compounds, the dominance of the B-p state characterizes the lower valence band region. The second segment pertains to the upper valence band region from -3.5 eV to 0 eV. In the case of the Mo2GaB<sup>2</sup> compound, this region arises from the hybridization of Mo-p and Mo-d orbitals. However, for the Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, the upper valence band region results from the hybridization of Mo-d orbitals and (Ga/Ge)-p orbitals. Notably, the Fermi level of M2AB<sup>2</sup> resides near the pseudogap in the TDOS profile, indicating a high level of electronic stability. Similar trends are observed in other compounds like Zr2AlN, V2AlN, Sc2AlB, Sc2GaB, Ta2GaB, and Hf2GaB<sup>2</sup> [15], [35], [45]. ![Fig. 4 – TDOS and PDOS of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) The electron charge density mapping is helpful in understanding the distribution of electron densities linked to chemical bonds. It delineates areas of positive and negative charge densities, signifying the development and exhaustion of electrical charges, respectively. As depicted in the map, covalent bonds become apparent by accumulating charges between two atoms. Furthermore, the presence of ionic bonds can be inferred from a balance between negative and positive charges at specific atom positions[46]. The valence electronic CDM, denoted in units of eÅ-3 , for M2AB<sup>2</sup> (where M = Mo, Ta; A = Ga, Ge) is showcased in Fig. 5(a, b, c, and d) along the (110) crystallographic plane. The accompanying scale illustrates the intensity of electronic charge density, with red and blue colors indicating low and high electronic charge density, respectively. As depicted in Fig. 5(a, b, c, and d), it is evident that charges accumulate in the regions between the B sites. Consequently, it is anticipated that strong covalent B‒B bonding occurs through the formation of two center-two electron (2c‒2e) bonds in the M2AB<sup>2</sup> (where M=Mo, Ta; A=Ga, Ge) compound, similar to other 212 MAX phase borides like Ti2PB2, Zr2PbB2, Nb2SB2, Zr2GaB<sup>2</sup> and Hf2GaB<sup>2</sup> [34], [45]. Mulliken analysis has corroborated the charge transfer from Mo/Ta atoms to B atoms. The charge received from Mo/Ta atoms is distributed among the B atoms positioned at the transitions and those located at the edges, facilitating the formation of a two center-two electron (2c‒2e) bond between B atoms within the 2D layer of B, as illustrated in Fig. 1(a). The hardness of each bond value presented in Table 5 also aligns with the results obtained from charge density mapping (CDM) and our analysis using moduli and elastic stiffness constants. ![Fig. 5 – The charge density mapping of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) ## 3.4.1 Mechanical properties The mechanical properties of materials play a pivotal role in determining their potential applications, serving as crucial indicators of their behavior and suitability in materials engineering endeavors. These properties are equally applicable to MAX phase materials. Initially, to assess the mechanical properties of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phases, we employed the stress-strain method within the CASTEP code to compute the elastic constants (Cij) [22], [47], [48]. These calculated elastic constants (Cij) are presented in Table 2, alongside those of other 212 and 211 MAX phases. Due to the hexagonal crystal structure of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phase borides, five stiffness constants emerge: C11, C12, C13, C33, and C<sup>44</sup> [49]. The mechanical stability was evaluated using these stiffness constants in the preceding section. For instance, C<sup>11</sup> and C<sup>33</sup> determine stiffness when stress is exerted along the (100) and (001) directions, respectively, whereas C<sup>44</sup> evaluates resistance to shear deformation on the (100) and (001) planes. The stiffness constants C<sup>11</sup> and C<sup>12</sup> directly reflect the strength of atomic bonds along the a- and c-axes. When C<sup>11</sup> exceeds C<sup>33</sup> (or vice versa), it signifies stronger atomic bonding along the a-axis (or c-axis). In Table 2, for Mo2GaB2, Mo2GeB2, and Ta2GeB<sup>2</sup> compounds, C<sup>11</sup> surpasses C33, indicating superior atomic bonding along the a-axis compared to the c-axis. This robust bonding along the a-axis suggests heightened resistance to a-axial deformation. Conversely, in Ta2GeB2, where C<sup>11</sup> < C33, stronger atomic bonding along the c-axis translates to increased resistance against c-axial deformation. Analysis of Table-2 reveals that the values of C<sup>11</sup> and C<sup>22</sup> are notably higher for 212 MAX phase borides than 211 MAX phase borides. Consequently, it can be inferred that 212 MAX phases exhibit stronger resistance to axial deformation when juxtaposed with 211 MAX phases. C<sup>44</sup> is commonly utilized to gauge shear deformation tolerance among the elastic constants. Notably, Ta2GaB exhibits superior shear deformation resistance owing to its highest C<sup>44</sup> value among the studied compounds. Furthermore, the individual elastic constants C12, C13, and C<sup>44</sup> denote shear deformation response under external stress. The observation that C<sup>11</sup> and C<sup>33</sup> possess larger magnitudes than C<sup>44</sup> implies that shear deformation is more facile than axial strain. Another crucial parameter derived from the stiffness constants is the Cauchy pressure (CP), calculated as C<sup>12</sup> - C44, which provides vital insights relevant to the practical applications of solids [51]. Pettifor [51] emphasized the significance of Cauchy pressure (CP) in discerning the chemical bonding and ductile/brittle properties of solids. A negative CP value indicates covalently bonded brittle solids, whereas a positive value signifies isotropic ionic ductile solids. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> fall into covalently bonded brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit metallic ductile characteristics. Similarly, like Ga-containing 211 MAX phase borides, Ga-containing 212 MAX phases also demonstrate negative CP values and brittle behavior [52][53]. The elastic constants obtained are used to produce several bulk elastic parameters that are used to characterize polycrystalline materials, such as Young's modulus (Y), bulk modulus (B), and shear modulus (G). In Table 2, the bulk modulus (B) and shear modulus (G) calculated using Hill's approximation [54] are also presented. Hill's values represent the average of the upper limit (Voigt [55]) and lower limit (Reuss [56]) of B. The necessary equations for these calculations are provided in the supplementary document (S1). Young's modulus (Y) is a crucial indicator of material stiffness. A higher Y value indicates a stiffer material. It can be observed from the table that Mo2GaB<sup>2</sup> exhibits the highest Y value, signifying its greater stiffness compared to others. According to the sequence of Y values, stiffness can be ranked as follows: Mo2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Ta2GeB2. Therefore, Mo2GaB2, with its higher Young's modulus, is anticipated to demonstrate superior mechanical stability and deformation resistance compared to the other compounds under investigation. This quality is essential for aircraft parts or high-performance machinery applications where critical dimensional stability and structural integrity are required [57], [58]. Table 2 shows that 212 MAX phase materials exhibit larger Young's modulus (Y) values than 211 MAX phases. This suggests that 212 MAX phase borides are stiffer than their 211 MAX phase counterparts. Young's modulus (E) also correlates well with thermal shock resistance (R): R ∝ 1/E [59]. Lower Young's modulus values correspond to higher thermal shock resistance. Therefore, materials with higher thermal shock resistance (i.e., lower Young's modulus) are more suitable for use as Thermal Barrier Coating (TBC) materials. Given that Ta2GaB<sup>2</sup> possesses the lowest Y value among the materials studied, it should be considered a superior candidate for TBC material due to its higher thermal shock resistance. The material's ability to withstand shape distortion is elucidated by the shear modulus (G). On the other hand, the bulk modulus (B) indicates the strength of a material's chemical bonds and its ability to withstand uniform compression or volume change. We computed Young's modulus (Y), bulk modulus (B), and shear modulus for our analysis. Based on the bulk modulus values presented in Table 2, the sequence of a material that resists compression when pressure is applied can be outlined as follows: Mo2GeB<sup>2</sup> > Ta2GeB<sup>2</sup> > Mo2GaB<sup>2</sup> > Ta2GeB2. Mo2GeB2, boasting a higher bulk modulus, may exhibit reduced plastic deformation and superior resistance to stress-induced deformation compared to other compounds examined. The ductile and brittle characteristics of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds can also be assessed through Poisson's ratio (ν) and Pugh's ratio (G/B). A compound demonstrates ductile (brittle) behavior if the ν value surpasses (falls below) 0.26 [60]. Furthermore, if the G/B value exceeds (is less than) 0.571, then the compound exhibits brittle (ductile) behavior [19]. According to both criteria, Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are classified as brittle compounds, whereas Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> are categorized as ductile compounds. The ductile nature of Ge-based MAX phases has been reported previously [61]. Fracture toughness (KIC) is a vital property that gauges a material's ability to resist crack propagation. In the case of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, Equation (S2) was employed to determine KIC. The KIC values are documented in Table 2, and notably, these values surpass those of 211 MAX phases [16], [35]. Another parameter, the "f-index," characterizes the isotropic nature and strength of atom-atom bonds within a single hexagonal crystalline lattice along the a- and c-directions. If the f-index is less than 1, chemical bonds exhibit greater rigidity along the c-axis; conversely, if f exceeds 1, bonds are more rigid in the ab-plane. When the value of f is set to 1, atomic bonds exhibit similar strength and uniformity in all directions [15]. The fvalue is computed using Equation (S3) and displayed in Table 2. Table 2 shows that the f-values of Mo2GeB2, Ta2GaB2, and Ta2GeB2, which are close to one, indicate a slight anisotropic bonding strength. Nevertheless, substances with robust bonds in the horizontal plane (ab plane) (f > 1), such as Mo2GaB2, are deemed optimal candidates for the exfoliation process. In engineering applications, the hardness of a solid material serves as a valuable criterion for designing various devices. The elastic properties of polycrystalline materials can be utilized to calculate hardness values, as the ability to resist indentation is closely linked to a material's hardness. Both micro-hardness (Hmicro) and macro-hardness (Hmacro) were computed using Equation (S4) and are presented in Table 2. Based on the values of Hmicro and Hmacro, Mo2GaB<sup>2</sup> emerges as the toughest among the studied phases. The order of hardness is as follows: Mo2GaB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GeB2. Interestingly, Ga-containing MAX phase borides exhibit superior hardness compared to Ge-containing MAX phase borides. In Table 2, we compare our findings with previously reported MAX phases and observe that our data align perfectly with the earlier results. #### 3.4.2 Elastic anisotropy Additionally, anisotropy is linked to other crucial events like anisotropic plastic deformation and the development and spread of micro-cracks within mechanical stress. By providing directiondependent elastic constants, the understanding of anisotropy also offers a framework for improving the mechanical stability of materials in extreme circumstances. The following formulae are used for hexagonal structures to calculate the various anisotropic variables from elastic constants Cij [62]. $A\_{1} = \frac{\frac{1}{6}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{44}};$ $A\_{2} = \frac{2\mathcal{C}\_{44}}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ and $A\_{3} = A\_{1}.$ $A\_{2} = \frac{\frac{1}{3}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ Table 3 lists every anisotropy parameter. Since the value of A<sup>i</sup> should be 1 to be isotropic, the computed value of A<sup>i</sup> (i = 1-3) indicates that all of the compounds under research exhibit anisotropic behavior [63]. An additional method for estimating elastic anisotropy is to use the percentage anisotropy to compressibility and shear (A<sup>B</sup> & AG). This gives polycrystalline materials a helpful way to measure elastic anisotropy. They have been described as [45]; Zero values for AB, AG, and the universal anisotropy factor (AU) show elastic isotropy; the maximum amount of anisotropy is represented by a value of 1. Table 3 shows that, compared to other compounds, the values of AB, AG, and A U for Mo2GeB<sup>2</sup> and Ta2GaB<sup>2</sup> are incredibly close to zero, suggesting that these compounds have nearly isotropic characteristics. #### 3.4.3 Mulliken Populations The Mulliken charge assigned to an atomic species quantifies the effective valence by calculating the absolute difference between the formal ionic charge and the Mulliken charge. Equations (S5) and (S6) are employed to ascertain the Mulliken charge for each atom (α). Table-4 provides the Mulliken atomic population and effective valence charge. Transition metals Mo and Ta in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have pure valence states of 4d<sup>5</sup> and 5d<sup>3</sup> , respectively. The d- orbital electrons of transition metals have been found to influence their effective valence charge significantly. A non-zero positive value indicates a combination of covalent and ionic attributes within chemical bonds. As this value decreases towards zero, it signifies a rise in ionicity. A zero value suggests an ideal ionic character in the bond. Conversely, a progression from zero with a positive value indicates an elevation in the covalency level of the bonds. Based on their effective valence, M atoms move from the left to the right in the periodic table, increasing the covalency of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). Table-4 reveals that the Mulliken atomic charge ascribed to the B atoms is solely negative. Conversely, positive Mulliken atomic charges are associated with transition metals (M) and A. This suggests a charge transfer from M and A to B for each compound within M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge), thereby fostering ionic chemical bonds among these atoms. Bond population serves as another indicator of bond covalency within a crystal, as a high value of bond population essentially signifies a heightened degree of covalency within the chemical bond. The bonding and anti-bonding states influence the populations with positive and negative bond overlap. As demonstrated in Table-5, the B-B bond exhibits greater covalency compared to any other bond in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The presence of an antibonding state between two relevant atoms, which subsequently decreases their chemical bonding, is indicated by a hostile bond overlap population. In Mo2GeB<sup>2</sup> and Ta2GeB2, a hostile bond overlap population is observed in the Ge-Mo and Ge-Ta bonds, indicating the presence of an antibonding state. Therefore, the existence of ionic bonding is guaranteed by electronic charge transfer. On the other hand, the high positive value of the bond overlap population (BOP) denotes the presence of covalent bonding, a feature shared by materials in the MAX phase. ## 3.4.4 Theoretical Vickers Hardness Vickers hardness, derived from the atomic bonds found in solids, indicates how resistant a material is to deformation in extreme circumstances. Several variables influence this feature, such as the crystal flaws, solid structure, atomic arrangement, and bond strength. The Vickers hardness of the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) MAX phases is determined using the Mulliken bond population method, as described by Gou et al. [66] using the formula (S7-S10). This method is particularly suitable for partial metallic systems such as MAX phases. Table-5 lists the computed Vickers hardness values for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The calculated values are 3.35 GPa, 4.76 GPa, 8.21 GPa, and 8.74 GPa for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, respectively. We observed that Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> have much higher hardness values than Mo2GaB<sup>2</sup> and Mo2GeB2. These values are also higher than that of other 212 phases, like, Ti2InB<sup>2</sup> (4.05 GPa) [50], Hf2InB<sup>2</sup> (3.94 GPa), and Hf2SnB<sup>2</sup> (4.41 GPa) [26], Zr2InB<sup>2</sup> (2.92 GPa) and Zr2TlB<sup>2</sup> (2.19 GPa) [25], Zr2GaB<sup>2</sup> (2.53GPa), Zr2GeB<sup>2</sup> (3.31GPa), Hf2GaB<sup>2</sup> (4.73GPa) and Hf2GeB2(4.83GPa) [67]The Hv calculated by the geometrical average of the individual bonding, where the bonding strength mainly determined by the BOP values. In the case of Ta2GaB<sup>2</sup> and Ta2GeB2, the BOP of M-B bonding is much higher compared to Mo2GaB<sup>2</sup> and Mo2GeB2. Even though the BOP of M-B for Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> is higher than that of the other 212 phases mentioned earlier. Therefore, higher values of Hv are expected for Ta2GaB<sup>2</sup> and Ta2GeB2. Additionally, we looked at the Vickers hardness value between 211 and 212 compounds and discovered that the 212 MAX phase compounds had a higher H<sup>V</sup> value. This is because a 2D layer of B atoms is positioned between the M atoms. The B atoms share two center-two electrons to form an extremely strong B-B bond [3]. Table 5 Calculated data for Mulliken bond number (n μ ), bond length (d μ ), bond overlap populations BOP, (P μ ), metallic populations (P μˊ), Vickers hardness (HV) M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. # 3.5 Thermal Properties MAX phases are ideal for high-temperature applications due to their exceptional mechanical qualities at elevated temperatures. As a result, researching the fundamental parameters necessary for predicting their application is of great interest, and these can be obtained from the vibrations of atoms or phonons. The Debye temperature (ΘD) of a solid is directly connected to its bonding strength, melting temperature, thermal expansion, and conductivity. Using the sound velocity and Anderson's technique [68], the Θ<sup>D</sup> of the phases under study has been computed using the formula (S11). Equation (S12) can be used to get the average sound velocity (Vm) from the longitudinal and transverse sound velocities. Equations (S13–14) were used to determine v<sup>l</sup> and vt. The calculated values of Debye's temperature are shown in Table 6, where Mo2GaB<sup>2</sup> has the highest Θ<sup>D</sup> and Ta2GeB<sup>2</sup> has the lowest. If we rank them, it is as follows: Mo2GaB<sup>2</sup> < Mo2GeB<sup>2</sup> < Ta2GaB2 < Ta2GeB2. Hadi et al. recently reported a MAX phase (V2SnC) as a TBC material with a Θ<sup>D</sup> value of 472 K [69]. Thus, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) exhibit encouraging potential as TBC materials, as shown in Table 6. Table 6-Data for density (ρ), longitudinal, transverse, and average sound velocities (vl, vt, and vm), Debye temperature (ΘD), minimum thermal conductivity (Kmin), Grüneisen parameter (γ), thermal expansion coefficient (TEC) at 300K and melting temperature (Tm) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The constant thermal conductivity value at high temperatures is the minimum thermal conductivity (Kmin). As the name implies, this conductivity is minimal because, at high temperatures, phonon coupling breaks. The formula (S15) for the minimum thermal conductivity of solids was derived using the Clarke model [70]. Table 6 lists the calculated value of kmin, with Ta2GaB<sup>2</sup> having the lowest value and Mo2GaB<sup>2</sup> having the highest. When selecting suitable materials for TBC applications, a minimum thermal conductivity of 1.25 W/mK is used as a screening criterion [71]. Our compounds exhibit lower minimum thermal conductivity values, holding promising potential as TBC materials. Gd2Zr2O<sup>7</sup> and Y2SiO5, two recently developed thermal barrier coating (TBC) materials, have minimal thermal conductivities (Kmin) of 1.22 W/m.K. and 1.3 W/m.K. [72], respectively, as confirmed by experiment. These numbers roughly match the values we computed for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The Grüneisen parameter (γ) is a crucial thermal parameter that helps explain the anharmonic effects of lattice dynamics; solids utilized at high temperatures are expected to have lower anharmonic effects. The Grüneisen parameter (γ) can be determined with the help of Poisson's ratio using equation (S16) [73]. The computed γ values, as shown in Table 6, suggest that the compounds under investigation exhibit a weak anharmonic effect. Additionally, for solids with a Poisson's ratio between 0.05 and 0.46, the values similarly fall within the range of 0.85 and 3.53 [74]. The melting temperature (Tm) of the compounds under investigation has been calculated using the following formula (S17). The strength of atomic bonding is the primary factor determining the melting temperature of solids; the higher the Tm, the stronger the atomic bonding. The order of T<sup>m</sup> for the titled phases is found to mirror the Y-based (Young's modulus) order, indicating a close link between T<sup>m</sup> and Y [75]. As observed in Table 6, our compounds roughly follow the Ybased ranking with Mo2GeB<sup>2</sup> < Mo2GaB<sup>2</sup> < Ta2GeB<sup>2</sup> < Ta2GaB2. The T<sup>m</sup> value for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) is also comparable with the TBC material Y4Al2O<sup>9</sup> (2000 K)[75]. ![Fig. 6: Temperature dependence of the calculated thermodynamic parameters. (a) Specific heat at constant pressure, C<sup>p</sup> (b) Specific heat at constant volume, C<sup>v</sup> (c) Free energy, F (d) Entropy, S (e) Internal energy, E and (f) Thermal expansion coefficient, TEC of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge).](path) The temperature dependence of specific heats, Cv, Cp, for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds was obtained using the formulas (S19-S20), as shown in Fig.6(a,b). Assuming that the quasi-harmonic model is accurate and that phase transitions are not anticipated for the compounds under study, these properties are approximated across a temperature range of 0 to 1000 K. Because phonon thermal softening occurs at higher temperatures, the heat capacity rises with temperature. Heat capacities increase quickly and follow the Debye-T 3 power law at lower temperatures. At higher temperature regimes, where C<sup>v</sup> and C<sup>p</sup> do not greatly depend on temperature, they approach the Dulong-Petit (3) limit.[76]. Using the quasi-harmonic approximation, various temperature-dependent thermodynamic potential functions for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have been estimated at zero pressure and displayed in Fig.6 [77]. These functions include the Helmholtz free energy (F), internal energy (E), and entropy (S) within 0-1000 K temperature using equation (S21-S23). The free energy progressively decreases as the temperature rises, as shown in Fig.6(c). Free energy typically declines, and this trend becomes increasingly negative as a natural process proceeds. As demonstrated in Fig. 6(d), the internal energy (E) shows a rising trend with temperature, in contrast to the free energy. Since thermal agitation creates disorder, a system's entropy rises as temperature rises. This is illustrated in Fig.6(d). A material's thermal expansion coefficient (TEC) is derived from the anharmonicity in the lattice dynamics and can be found using equation (S18). The measure of a material's capacity to expand or contract with heat or cold is called the Thermal Expansion Coefficient, or TEC. As observed in Fig. 5(f), the Thermal Expansion Coefficient (TEC) increases rapidly up to 365 K. Then it approaches a constant value, which indicates lower saturations in the materials with temperature changes. The materials under study have a very low TEC value, a crucial characteristic of materials intended for application in high-temperature technology. To be effective as thermal barrier coating (TBC) materials, compounds must exhibit a low thermal conductivity (Kmin) to impede heat transfer, a high melting temperature to withstand extreme heat, and a low thermal expansion coefficient (TEC) to maintain dimensional stability under thermal stress. The compounds M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge) possess these properties, making them suitable candidates for use as TBC materials #### 3.6 Optical Properties Different materials exhibit unique behaviors when exposed to electromagnetic radiation. The optical constants determine the overall response of the sample to the incident radiation. The complex dielectric function, defined as ε(ω)=ε1(ω)+iε2(ω), is one of the main optical characteristics of solids. The following formula determines the imaginary part of the dielectric function ε2(ω) from the momentum matrix element between the occupied and unoccupied electronic states. In this formula, e stands for an electronic charge, ω for light angular frequency, u for the polarization vector of the incident electric field, and and for the conduction and valence band wave functions, respectively, at k. The Kramers-Kronig equation can estimate the real part of the dielectric function, ε1(ω). In contrast, ε2(ω) and ε1(ω) are utilized to evaluate all other optical parameters, such as the absorption coefficient, photoconductivity, reflectivity, and loss function [78]. In this part, several energy-dependent optical properties of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) 212 MAX phases are calculated and analyzed in detail for the photon energy range of 0 to 30 eV, for [100] plane enabling the first assessment of the compounds' practical applicability. Given the metallic conductivity of the MAX compounds' electronic structure, additional parameters were chosen to analyze the optical properties. These include a plasma frequency of 3 eV, damping of 0.05 eV, and Gaussian smearing of 0.5 eV [79]. Figure 7(a) shows the real component of the dielectric function, ε1, which exhibits metallic behavior. In metallic systems, ε<sup>1</sup> has a considerably high negative value in the low-energy range, with the real component reaching negative, which aligns with the band structure finding. Fig. 7(b) depicts the imaginary part of the dielectric function, ε2(ω), representing dielectric losses about frequency. Mo2GeB<sup>2</sup> demonstrates the highest peak in the low-energy region, with all compounds approaching zero from above at around 17 eV. This observation confirms Drude's behavior. ![Fig. 7- (a) Real part of dielectric constant, ε<sup>1</sup> (b) Imaginary part of dielectric constant, ε<sup>2</sup> (c) Reflectivity, R and (d) Loss function, (e) The coefficient of absorption, (α) (f) Refractive index, (n) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) for [100] electric field directions.](path) The reflectivity of a material measures the percentage of incident light energy reflected off it. Equation (S24) was employed to compute reflectivity using the dielectric function, as shown in Fig.7(c). The spectra's visible and infrared (IR) portions consistently exhibit values exceeding 55%. Consequently, the materials under examination are expected to appear metallic gray. In the infrared (IR) region, the maximum reflectivity for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> is 65% and 74%, respectively, occurring at 0.33 eV. Conversely, for Ta2GaB2, the maximum reflectivity is 59% at 14.02 eV, while for Ta2GeB2, it is 65% at 14.78 eV in the UV region. After 18 eV, the reflectance decreases significantly. According to reports, substances with an average reflectivity value above 44% in the visible light region can effectively reduce solar heating by reflecting a significant amount of the solar spectrum. Among all the compounds we examined, Ta2GaB<sup>2</sup> has a reflectivity in the visible range of more than 44% [78]. Therefore, Ta2GaB<sup>2</sup> can be utilized as a coating material and should be able to mitigate solar heating. plasma frequencies (ωp). The plasma frequency for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is observed at 23 eV, 25 eV, 21 eV, and 23 eV, respectively. At this specific frequency, the absorption coefficient rapidly decreases, ε<sup>1</sup> crosses zero from the negative side, and the reflectance R(ω) displays a falling tail. Above this distinctive frequency, the materials are transparent to the incident electromagnetic radiation. Fig. 7(e) illustrates the absorbance coefficient (α) of compounds M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) determined using equation (S26). As α begins to rise from zero photon energy, the metallic nature of the substances under study is again indicated. The visible light region experiences a sharp increase in absorption, peaking in the UV region at around 14 eV and then progressively declining. The IR region exhibits negligible absorption. However, the materials mentioned above appear to have a significant absorption band primarily located in the visible and ultraviolet spectrum. This suggests that the materials can be used in UV surface-disinfection devices, medical sterilization equipment, and other optoelectronic device designs. Its refractive index is crucial to a material's potential application in optical devices like waveguides and photonic crystals. Equations (S27) and (S28) were used to derive the refractive index (n) and extinction coefficient (k) for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), as depicted in Fig.7(c) and Fig.S-1(a). The variations of n and k in MAX phase carbides with incident photon energy closely resemble ε1(ω) and ε2(ω). The static refractive index n(0) for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is 9.4,13.07,7 and 9.2, respectively, and decreases gradually with the increase in photon energy. The extinction coefficient for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2 gradually increases in the IR region, reaching their maximum values of 2.56, 4.27, 2.42, and 3.28, respectively. Following this, they slowly decrease in the visible and UV regions. Figure S-1(b) illustrates the photoconductivity (σ) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) across various photon energies. The photoconductivity (σ) parameter quantifies the impact of photon irradiation on a material's electrical conductivity. Similar to the absorbance coefficient (α) spectrum, the σ spectrum aligns with the metallic nature of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). #### 4 Conclusions In summary, we employed DFT calculations to explore four 212 MAB phases, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), and investigated the structural, electronic, mechanical, lattice dynamical, and optical properties to predict their possible applications. The phonon dispersion curves, formation energy, and elastic constants collectively suggest that the M2AB<sup>2</sup> boride maintains dynamic, mechanical, chemical, and thermodynamic stability. The electronic band structure and density of states (DOS) offer evidence supporting the metallic nature of the compounds under investigation. Concurrently, the charge density mapping and atomic Mulliken population both confirm the presence of a strong B-B covalent bond. The stiffness constants, elastic moduli, findex, fracture toughness (KIC), Pugh's ratio (G/B), hardness parameters, and Cauchy Pressure (CP) of M2AB<sup>2</sup> were computed and compared with those of their 211 equivalents. We found that the values of the 212-phase borides are higher than those of the 211-phase carbides or borides. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are identified as brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit ductile characteristics, as indicated by Poisson's ratio (ν) and Pugh's ratio (B/G or G/B). The elastic characteristics display anisotropy due to the distinct atomic configurations along the aand c-directions. Vickers hardness calculations are considered reliable indicators of material hardness. The results suggest that Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> possess more pliable characteristics than Ta2GaB<sup>2</sup> and Ta2GeB2. The high hardness values of Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compared to the Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> due to higher BOP values of M-B bonding in Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> than in Mo2GaB<sup>2</sup> and Mo2GeB2. The value of minimum thermal conductivity (Kmin), thermal expansion coefficient, and melting temperature (Tm) collectively suggest the potential suitability of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) as a material for thermal barrier coating (TBC) applications in high-temperature devices. The optical conductivity and absorption coefficient corroborate the findings of the electronic band structure. Reflectivity is notably high in infrared (IR) regions and remains nearly constant in the visible and moderate ultraviolet (UV) regions, with an average value exceeding 44% for Ta2GaB2. This suggests that Ta2GaB<sup>2</sup> can be effectively utilized as a coating material to reduce solar heating. We expect that the comprehensive analysis of the diverse physical characteristics of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) presented in this study will establish a robust foundation for future theoretical and experimental explorations of these fascinating MAB phases. ## CRediT author statement A. K. M Naim Ishtiaq and Md Nasir Uddin: Data curation, Writing- Original draft preparation. Md. Rasel Rana, Shariful Islam and Noor Afsary: Reviewing and Editing. Md. Ashraf Ali: Methodology, Reviewing and Editing, conceptualization, supervision; and Karimul Hoque: conceptualization, supervision, editing, and reviewing. # Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. # Acknowledgments The authors acknowledge Physics Discipline, Khulna University, Khulna for the logistic support and Advanced Computational Materials Research Laboratory (ACMRL), Department of Physics at Chittagong University of Engineering & Technology (CUET), Chattogram-4349, Bangladesh for laboratory facilities. - [1] M. W. Barsoum, G. Yaroschuk, and S. Tyagi, "Fabrication and characterization of M2SnC (M = Ti, Zr, Hf and Nb)," Scr. Mater., vol. 37, no. 10, pp. 1583–1591, 1997, doi: 10.1016/S1359-6462(97)00288-1. - [2] M. W. Barsoum and T. 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Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening.	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. Schnyder, Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery, Phys. Rev. Mat. 5, 124202 (2021). - 2. X. Zhou, Ch.-H. Hsu, H. Aramberri, M. Iraola, Ch.-Yi Huang, J. L. Mañes, M. G. Vergniory, H. Lin, and N. Kioussis, Novel family of topological semimetals with butterflylike nodal lines, Phys. Rev. B 104, 125135 (2021). - 3. J. Wang, H. Yuan, Z.-M. Yu, Z. Zhang, and X. Wang, Coexistence of symmetryenforced phononic Dirac nodal-line net and three-nodal surfaces phonons in solid-state materials: Theory and materials realization, Phys. Rev. Mat. 5, 124203 (2021). - 4. J. Bannies, E. Razzoli, M. Michiardi, H.-H. Kung, I. S. Elfimov, M. Yao, A. Fedorov, J. Fink , C. Jozwiak, A. Bostwick, E. Rotenberg, A. Damascelli , and C. Felser Extremely large magnetoresistance from electron-hole compensation in the nodal-loop semimetal ZrP2, Phys. Rev. B 103, 155144 (2021). - 5. A. S. Wadge, K. Zberecki, B. J. Kowalski, D. Jastrzebski, P. K. Tanwar, P. Iwanowski, R. Diduszko, A. Moosarikandy, M. Rosmus, N. Olszowska, and A. Wisniewski, Emergent impervious band crossing in the bulk in topological nodal line semimetal ZrAs<sup>2</sup> [https://arxiv.org/abs/2403.04434.](https://arxiv.org/abs/2403.04434) - 6. S. Nandi, B. B. Maity, V. Sharma, R. Verma, V. Saini, B. Singh, D. Aoki, and A. Thamizhavel, Magnetotransport and Fermi surface studies of a purported nodal line semimetal ZrAs2, Phys. Rev. B 109, 075155 (2024). - 8. G. B. Osterhoudt, Y. Wang, C. A. C. Garcia, V. M. Plisson, J. Gooth, C. Felser, P. Narang, and K. S. Burch, Evidence for Dominant Phonon-Electron Scattering in Weyl Semimetal WP2, Phys. Rev. X 11, 011017 (2021). - 9. K. Zhang, T. Wang, X. Pang, F. Han, S. Shang, N.T. Hung , Zi-Kui Liu, M. Li, R. Saito, and S. Huang, Anisotropic Fano resonance in the Weyl semimetal candidate LaAlSi Phys. Rev. B 102, 235162 (2020). - 10. K. Tanaka, T. Nishihara, A. Takakura,\|Y. Segawa, K. Matsuda, and Y. Miyauchi, Polarized Raman spectroscopy on topological semimetal Co3Sn2S<sup>2</sup> J. Raman Spectroscopy 54(1), 93 (2023). - 21. K. Zhang, X. Pang, T. Wang, F. Han, Shun-Li Shang, N. T. Hung, A.R.T. Nugraha, Zi-Kui Liu, M. Li, R, Saito, and S. Huang, Anomalous phonon-mode dependence in polarized Raman spectroscopy of the topological Weyl semimetal TaP, Phys. Rev. B 101, 014308 (2020). - 22. H. P. C. Miranda, S. Reichardt, G. Froehlicher, A. Molina-Sanchez, S. Berciaud, and L. Wirtz, Quantum Interference Effects in Resonant Raman Spectroscopy of Single- and Triple-Layer MoTe<sup>2</sup> from First-Principles, Nano Lett. 17, 2381 (2017). - 23. M. Osiekowicz, D. Staszczuk, K. Olkowska‑Pucko , Ł. Kipczak , M. Grzeszczyk, M. Zinkiewicz, K. Nogajewski, Z. R. Kudrynskyi, Z. D. Kovalyuk, A. Patané, A. Babiński, and M. R. Molas, Resonance and antiresonance in Raman scattering in GaSe and InSe crystals, Sci. Rep. 11, 924 (2021). - 24. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. 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Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![![](_page_14_Figure_1.jpeg)](path) ![Fig.2. Experiment geometry and crystal orientation. The crystal frame of reference (XYZ) and the lab frame of reference (X'Y'Z'). Light polarization (incident and scattered) is in the (X'Y) plane. The red arrow shows the light propagation vector, and the blue denotes the light polarization vector.](path) ![Fig.3. Room temperature Raman spectra for three different excitations (as measured). The continuous background is present with the dashed lines as reference levels for every plot. The green line is the fitted model of quasi-elastic scattering background using expression (3) to the 785 nm plot. The inset shows the temperature dependence of the integrated background intensity for the 785 nm excitation.](path) ![Fig.4. Comparison of the parallel (YY) and the (X'Y) crossed polarization spectra recorded with the 785 nm excitation at room temperature.](path) ![Fig. 5. Room temperature Raman spectra in the (YY) (a) and (X'X') (b) configurations for 473 nm, 633 nm and 785 nm excitation wavelengths.](path) ![Fig. 6. Band structure of ZrAs<sup>2</sup> with a few allowed electronic transitions to/from intermediate states in the Raman process, which are shown with the vertical arrows: black - 785 nm excitation, red 633 nm, and blue 473 nm (a); Density of states (b); Details of the electronic bands close to the Fermi level (c); shaded areas indicate regions of the Brillouin zone where phonon decay via e-h pairs excitation is energetically possible; Brillouin zone of ZrAs<sup>2</sup> (d).](path) ![Fig. 7. Fano profile fitted to the 171 cm-1 A<sup>g</sup> phonon peak measured with the 785 nm excitation at room temperature for two parallel light polarizations (YY) and (X'X').](path) ![Fig. 8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).](path) ![![](_page_20_Figure_1.jpeg)](path) Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening. # SUPPORTING INFORMATION # Temperature and excitation energy dependence of Raman scattering in nodal line Dirac semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland #### S1. Angular dependence of Raman scattering intensity According to the factor group analysis, 18 Raman active modes in ZrAs<sup>2</sup> are distributed as follows: 6A 3B 6B 3B g 1g 2g 3g + + + . In the identification process of modes symmetry, we measured angular dependence of the modes intensity as a function of the angle (Fig.2 in the main text) in parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) polarization configuration. We expect the following expressions for the intensity of the modes: Fitting the angular dependence of intensity with real elements of Raman tensor fails for A<sup>g</sup> modes. To properly describe this dependence, we have to assume complex Raman elements [S1,S2]: a i a a e = , b i b b e = c i c c e = . It leads to a complicated expression that cannot be deconvoluted to find modules or phase factors. Operationally, it comes down to expressions: For B modes phase factors do not play any role and the real values of Raman tensors are good enough to fit the angular dependence for parallel and perpendicular configurations. Figure S1 shows the angular dependence for six representative modes of different symmetry. The fitted value of the angle offset was 4. ![Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.](path) ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
Fig. 8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. 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Miyauchi, Polarized Raman spectroscopy on topological semimetal Co3Sn2S<sup>2</sup> J. Raman Spectroscopy 54(1), 93 (2023). - 21. K. Zhang, X. Pang, T. Wang, F. Han, Shun-Li Shang, N. T. Hung, A.R.T. Nugraha, Zi-Kui Liu, M. Li, R, Saito, and S. Huang, Anomalous phonon-mode dependence in polarized Raman spectroscopy of the topological Weyl semimetal TaP, Phys. Rev. B 101, 014308 (2020). - 22. H. P. C. Miranda, S. Reichardt, G. Froehlicher, A. Molina-Sanchez, S. Berciaud, and L. Wirtz, Quantum Interference Effects in Resonant Raman Spectroscopy of Single- and Triple-Layer MoTe<sup>2</sup> from First-Principles, Nano Lett. 17, 2381 (2017). - 23. M. Osiekowicz, D. Staszczuk, K. Olkowska‑Pucko , Ł. Kipczak , M. Grzeszczyk, M. Zinkiewicz, K. Nogajewski, Z. R. Kudrynskyi, Z. D. Kovalyuk, A. Patané, A. Babiński, and M. R. Molas, Resonance and antiresonance in Raman scattering in GaSe and InSe crystals, Sci. Rep. 11, 924 (2021). - 24. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. 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Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![![](_page_20_Figure_1.jpeg)](path) Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening. # SUPPORTING INFORMATION # Temperature and excitation energy dependence of Raman scattering in nodal line Dirac semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland #### S1. Angular dependence of Raman scattering intensity According to the factor group analysis, 18 Raman active modes in ZrAs<sup>2</sup> are distributed as follows: 6A 3B 6B 3B g 1g 2g 3g + + + . In the identification process of modes symmetry, we measured angular dependence of the modes intensity as a function of the angle (Fig.2 in the main text) in parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) polarization configuration. We expect the following expressions for the intensity of the modes: Fitting the angular dependence of intensity with real elements of Raman tensor fails for A<sup>g</sup> modes. To properly describe this dependence, we have to assume complex Raman elements [S1,S2]: a i a a e = , b i b b e = c i c c e = . It leads to a complicated expression that cannot be deconvoluted to find modules or phase factors. Operationally, it comes down to expressions: For B modes phase factors do not play any role and the real values of Raman tensors are good enough to fit the angular dependence for parallel and perpendicular configurations. Figure S1 shows the angular dependence for six representative modes of different symmetry. The fitted value of the angle offset was 4. ![Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.](path) ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
Fig.3. Room temperature Raman spectra for three different excitations (as measured). The continuous background is present with the dashed lines as reference levels for every plot. The green line is the fitted model of quasi-elastic scattering background using expression (3) to the 785 nm plot. The inset shows the temperature dependence of the integrated background intensity for the 785 nm excitation.	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. 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Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![Fig.4. Comparison of the parallel (YY) and the (X'Y) crossed polarization spectra recorded with the 785 nm excitation at room temperature.](path) ![Fig. 5. Room temperature Raman spectra in the (YY) (a) and (X'X') (b) configurations for 473 nm, 633 nm and 785 nm excitation wavelengths.](path) ![Fig. 6. Band structure of ZrAs<sup>2</sup> with a few allowed electronic transitions to/from intermediate states in the Raman process, which are shown with the vertical arrows: black - 785 nm excitation, red 633 nm, and blue 473 nm (a); Density of states (b); Details of the electronic bands close to the Fermi level (c); shaded areas indicate regions of the Brillouin zone where phonon decay via e-h pairs excitation is energetically possible; Brillouin zone of ZrAs<sup>2</sup> (d).](path) ![Fig. 7. Fano profile fitted to the 171 cm-1 A<sup>g</sup> phonon peak measured with the 785 nm excitation at room temperature for two parallel light polarizations (YY) and (X'X').](path) ![Fig. 8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).](path) ![![](_page_20_Figure_1.jpeg)](path) Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening. # SUPPORTING INFORMATION # Temperature and excitation energy dependence of Raman scattering in nodal line Dirac semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland #### S1. Angular dependence of Raman scattering intensity According to the factor group analysis, 18 Raman active modes in ZrAs<sup>2</sup> are distributed as follows: 6A 3B 6B 3B g 1g 2g 3g + + + . In the identification process of modes symmetry, we measured angular dependence of the modes intensity as a function of the angle (Fig.2 in the main text) in parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) polarization configuration. We expect the following expressions for the intensity of the modes: Fitting the angular dependence of intensity with real elements of Raman tensor fails for A<sup>g</sup> modes. To properly describe this dependence, we have to assume complex Raman elements [S1,S2]: a i a a e = , b i b b e = c i c c e = . It leads to a complicated expression that cannot be deconvoluted to find modules or phase factors. Operationally, it comes down to expressions: For B modes phase factors do not play any role and the real values of Raman tensors are good enough to fit the angular dependence for parallel and perpendicular configurations. Figure S1 shows the angular dependence for six representative modes of different symmetry. The fitted value of the angle offset was 4. ![Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.](path) ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
Fig. 1. Orthorhombic unit cell of ZrAs2.	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. Schnyder, Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery, Phys. Rev. Mat. 5, 124202 (2021). - 2. X. Zhou, Ch.-H. Hsu, H. Aramberri, M. Iraola, Ch.-Yi Huang, J. L. Mañes, M. G. Vergniory, H. Lin, and N. Kioussis, Novel family of topological semimetals with butterflylike nodal lines, Phys. Rev. B 104, 125135 (2021). - 3. J. Wang, H. Yuan, Z.-M. Yu, Z. Zhang, and X. Wang, Coexistence of symmetryenforced phononic Dirac nodal-line net and three-nodal surfaces phonons in solid-state materials: Theory and materials realization, Phys. Rev. Mat. 5, 124203 (2021). - 4. J. Bannies, E. Razzoli, M. Michiardi, H.-H. Kung, I. S. Elfimov, M. Yao, A. Fedorov, J. Fink , C. Jozwiak, A. Bostwick, E. Rotenberg, A. Damascelli , and C. Felser Extremely large magnetoresistance from electron-hole compensation in the nodal-loop semimetal ZrP2, Phys. Rev. B 103, 155144 (2021). - 5. A. S. Wadge, K. Zberecki, B. J. Kowalski, D. Jastrzebski, P. K. Tanwar, P. Iwanowski, R. Diduszko, A. Moosarikandy, M. Rosmus, N. Olszowska, and A. Wisniewski, Emergent impervious band crossing in the bulk in topological nodal line semimetal ZrAs<sup>2</sup> [https://arxiv.org/abs/2403.04434.](https://arxiv.org/abs/2403.04434) - 6. S. Nandi, B. B. Maity, V. Sharma, R. Verma, V. Saini, B. Singh, D. Aoki, and A. Thamizhavel, Magnetotransport and Fermi surface studies of a purported nodal line semimetal ZrAs2, Phys. Rev. B 109, 075155 (2024). - 8. G. B. Osterhoudt, Y. Wang, C. A. C. Garcia, V. M. Plisson, J. Gooth, C. Felser, P. Narang, and K. S. Burch, Evidence for Dominant Phonon-Electron Scattering in Weyl Semimetal WP2, Phys. Rev. X 11, 011017 (2021). - 9. K. Zhang, T. Wang, X. Pang, F. Han, S. Shang, N.T. Hung , Zi-Kui Liu, M. Li, R. Saito, and S. Huang, Anisotropic Fano resonance in the Weyl semimetal candidate LaAlSi Phys. Rev. B 102, 235162 (2020). - 10. K. Tanaka, T. Nishihara, A. Takakura,\|Y. Segawa, K. Matsuda, and Y. Miyauchi, Polarized Raman spectroscopy on topological semimetal Co3Sn2S<sup>2</sup> J. Raman Spectroscopy 54(1), 93 (2023). - 21. K. Zhang, X. Pang, T. Wang, F. Han, Shun-Li Shang, N. T. Hung, A.R.T. Nugraha, Zi-Kui Liu, M. Li, R, Saito, and S. Huang, Anomalous phonon-mode dependence in polarized Raman spectroscopy of the topological Weyl semimetal TaP, Phys. Rev. B 101, 014308 (2020). - 22. H. P. C. Miranda, S. Reichardt, G. Froehlicher, A. Molina-Sanchez, S. Berciaud, and L. Wirtz, Quantum Interference Effects in Resonant Raman Spectroscopy of Single- and Triple-Layer MoTe<sup>2</sup> from First-Principles, Nano Lett. 17, 2381 (2017). - 23. M. Osiekowicz, D. Staszczuk, K. Olkowska‑Pucko , Ł. Kipczak , M. Grzeszczyk, M. Zinkiewicz, K. Nogajewski, Z. R. Kudrynskyi, Z. D. Kovalyuk, A. Patané, A. Babiński, and M. R. Molas, Resonance and antiresonance in Raman scattering in GaSe and InSe crystals, Sci. Rep. 11, 924 (2021). - 24. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron– phonon interactions in the Weyl semimetal NbIrTe4, Sci. Rep. 11, 8155 (2021). - 25. S. Nath Gupta, D. V. S. Muthu, C. Shekhar, R. Sankar, C. Felser, and A. K. Sood Pressure-induced electronic and structural phase transitions in Dirac semimetal Cd3As2: Raman study, EPL, 120, 57003 (2017). - 26. D. Wulferding, P. Lemmens, F. Büscher, D. Schmeltzer, C. Felser, and Ch. Shekhar, Effect of topology on quasiparticle interactions in the Weyl semimetal WP<sup>2</sup> Phys. Rev. B 102, 075116 (2020). - 32. J. Coulter, G. B. Osterhoudt, C A. C. Garcia, Y. Wang, V. M. Plisson, B. Shen, N. Ni, K. S. Burch, and P. Narang, Uncovering electron-phonon scattering and phonon dynamics in type-I Weyl semimetals, Phys. Rev. B 100, 220301(R) (2019). - 33. Xu, Y.M. Da, L.X. Zha, K. Wang, R. Yang, W. Zhang, J.Y. Liu, H. Xiao, G.F. Chen, S.A. Trugman, J.-X. Zhu, A.J. Taylor, D.A. Yarotski, R.P. Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![![](_page_14_Figure_1.jpeg)](path) ![Fig.2. Experiment geometry and crystal orientation. The crystal frame of reference (XYZ) and the lab frame of reference (X'Y'Z'). Light polarization (incident and scattered) is in the (X'Y) plane. The red arrow shows the light propagation vector, and the blue denotes the light polarization vector.](path) ![Fig.3. Room temperature Raman spectra for three different excitations (as measured). The continuous background is present with the dashed lines as reference levels for every plot. The green line is the fitted model of quasi-elastic scattering background using expression (3) to the 785 nm plot. The inset shows the temperature dependence of the integrated background intensity for the 785 nm excitation.](path) ![Fig.4. Comparison of the parallel (YY) and the (X'Y) crossed polarization spectra recorded with the 785 nm excitation at room temperature.](path) ![Fig. 5. Room temperature Raman spectra in the (YY) (a) and (X'X') (b) configurations for 473 nm, 633 nm and 785 nm excitation wavelengths.](path) ![Fig. 6. Band structure of ZrAs<sup>2</sup> with a few allowed electronic transitions to/from intermediate states in the Raman process, which are shown with the vertical arrows: black - 785 nm excitation, red 633 nm, and blue 473 nm (a); Density of states (b); Details of the electronic bands close to the Fermi level (c); shaded areas indicate regions of the Brillouin zone where phonon decay via e-h pairs excitation is energetically possible; Brillouin zone of ZrAs<sup>2</sup> (d).](path) ![Fig. 7. Fano profile fitted to the 171 cm-1 A<sup>g</sup> phonon peak measured with the 785 nm excitation at room temperature for two parallel light polarizations (YY) and (X'X').](path) ![Fig. 8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).](path) ![![](_page_20_Figure_1.jpeg)](path) Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening. # SUPPORTING INFORMATION # Temperature and excitation energy dependence of Raman scattering in nodal line Dirac semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland #### S1. Angular dependence of Raman scattering intensity According to the factor group analysis, 18 Raman active modes in ZrAs<sup>2</sup> are distributed as follows: 6A 3B 6B 3B g 1g 2g 3g + + + . In the identification process of modes symmetry, we measured angular dependence of the modes intensity as a function of the angle (Fig.2 in the main text) in parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) polarization configuration. We expect the following expressions for the intensity of the modes: Fitting the angular dependence of intensity with real elements of Raman tensor fails for A<sup>g</sup> modes. To properly describe this dependence, we have to assume complex Raman elements [S1,S2]: a i a a e = , b i b b e = c i c c e = . It leads to a complicated expression that cannot be deconvoluted to find modules or phase factors. Operationally, it comes down to expressions: For B modes phase factors do not play any role and the real values of Raman tensors are good enough to fit the angular dependence for parallel and perpendicular configurations. Figure S1 shows the angular dependence for six representative modes of different symmetry. The fitted value of the angle offset was 4. ![Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.](path) ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
Fig. 6. Band structure of ZrAs<sup>2</sup> with a few allowed electronic transitions to/from intermediate states in the Raman process, which are shown with the vertical arrows: black - 785 nm excitation, red 633 nm, and blue 473 nm (a); Density of states (b); Details of the electronic bands close to the Fermi level (c); shaded areas indicate regions of the Brillouin zone where phonon decay via e-h pairs excitation is energetically possible; Brillouin zone of ZrAs<sup>2</sup> (d).	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. 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Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron– phonon interactions in the Weyl semimetal NbIrTe4, Sci. Rep. 11, 8155 (2021). - 25. S. Nath Gupta, D. V. S. Muthu, C. Shekhar, R. Sankar, C. Felser, and A. K. Sood Pressure-induced electronic and structural phase transitions in Dirac semimetal Cd3As2: Raman study, EPL, 120, 57003 (2017). - 26. D. Wulferding, P. Lemmens, F. Büscher, D. Schmeltzer, C. Felser, and Ch. Shekhar, Effect of topology on quasiparticle interactions in the Weyl semimetal WP<sup>2</sup> Phys. Rev. B 102, 075116 (2020). - 32. J. Coulter, G. B. Osterhoudt, C A. C. Garcia, Y. Wang, V. M. Plisson, B. Shen, N. Ni, K. S. Burch, and P. Narang, Uncovering electron-phonon scattering and phonon dynamics in type-I Weyl semimetals, Phys. Rev. B 100, 220301(R) (2019). - 33. Xu, Y.M. Da, L.X. Zha, K. Wang, R. Yang, W. Zhang, J.Y. Liu, H. Xiao, G.F. Chen, S.A. Trugman, J.-X. Zhu, A.J. Taylor, D.A. Yarotski, R.P. Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![![](_page_14_Figure_1.jpeg)](path) ![Fig.2. Experiment geometry and crystal orientation. The crystal frame of reference (XYZ) and the lab frame of reference (X'Y'Z'). Light polarization (incident and scattered) is in the (X'Y) plane. The red arrow shows the light propagation vector, and the blue denotes the light polarization vector.](path) ![Fig.3. Room temperature Raman spectra for three different excitations (as measured). The continuous background is present with the dashed lines as reference levels for every plot. The green line is the fitted model of quasi-elastic scattering background using expression (3) to the 785 nm plot. The inset shows the temperature dependence of the integrated background intensity for the 785 nm excitation.](path) ![Fig.4. Comparison of the parallel (YY) and the (X'Y) crossed polarization spectra recorded with the 785 nm excitation at room temperature.](path) ![Fig. 5. Room temperature Raman spectra in the (YY) (a) and (X'X') (b) configurations for 473 nm, 633 nm and 785 nm excitation wavelengths.](path) ![Fig. 6. Band structure of ZrAs<sup>2</sup> with a few allowed electronic transitions to/from intermediate states in the Raman process, which are shown with the vertical arrows: black - 785 nm excitation, red 633 nm, and blue 473 nm (a); Density of states (b); Details of the electronic bands close to the Fermi level (c); shaded areas indicate regions of the Brillouin zone where phonon decay via e-h pairs excitation is energetically possible; Brillouin zone of ZrAs<sup>2</sup> (d).](path) ![Fig. 7. Fano profile fitted to the 171 cm-1 A<sup>g</sup> phonon peak measured with the 785 nm excitation at room temperature for two parallel light polarizations (YY) and (X'X').](path) ![Fig. 8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).](path) ![![](_page_20_Figure_1.jpeg)](path) Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening. # SUPPORTING INFORMATION # Temperature and excitation energy dependence of Raman scattering in nodal line Dirac semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland #### S1. Angular dependence of Raman scattering intensity According to the factor group analysis, 18 Raman active modes in ZrAs<sup>2</sup> are distributed as follows: 6A 3B 6B 3B g 1g 2g 3g + + + . In the identification process of modes symmetry, we measured angular dependence of the modes intensity as a function of the angle (Fig.2 in the main text) in parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) polarization configuration. We expect the following expressions for the intensity of the modes: Fitting the angular dependence of intensity with real elements of Raman tensor fails for A<sup>g</sup> modes. To properly describe this dependence, we have to assume complex Raman elements [S1,S2]: a i a a e = , b i b b e = c i c c e = . It leads to a complicated expression that cannot be deconvoluted to find modules or phase factors. Operationally, it comes down to expressions: For B modes phase factors do not play any role and the real values of Raman tensors are good enough to fit the angular dependence for parallel and perpendicular configurations. Figure S1 shows the angular dependence for six representative modes of different symmetry. The fitted value of the angle offset was 4. ![Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.](path) ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. 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Miyauchi, Polarized Raman spectroscopy on topological semimetal Co3Sn2S<sup>2</sup> J. Raman Spectroscopy 54(1), 93 (2023). - 21. K. Zhang, X. Pang, T. Wang, F. Han, Shun-Li Shang, N. T. Hung, A.R.T. Nugraha, Zi-Kui Liu, M. Li, R, Saito, and S. Huang, Anomalous phonon-mode dependence in polarized Raman spectroscopy of the topological Weyl semimetal TaP, Phys. Rev. B 101, 014308 (2020). - 22. H. P. C. Miranda, S. Reichardt, G. Froehlicher, A. Molina-Sanchez, S. Berciaud, and L. Wirtz, Quantum Interference Effects in Resonant Raman Spectroscopy of Single- and Triple-Layer MoTe<sup>2</sup> from First-Principles, Nano Lett. 17, 2381 (2017). - 23. M. Osiekowicz, D. Staszczuk, K. Olkowska‑Pucko , Ł. Kipczak , M. Grzeszczyk, M. Zinkiewicz, K. Nogajewski, Z. R. Kudrynskyi, Z. D. Kovalyuk, A. Patané, A. Babiński, and M. R. Molas, Resonance and antiresonance in Raman scattering in GaSe and InSe crystals, Sci. Rep. 11, 924 (2021). - 24. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron– phonon interactions in the Weyl semimetal NbIrTe4, Sci. Rep. 11, 8155 (2021). - 25. S. Nath Gupta, D. V. S. Muthu, C. Shekhar, R. Sankar, C. Felser, and A. K. Sood Pressure-induced electronic and structural phase transitions in Dirac semimetal Cd3As2: Raman study, EPL, 120, 57003 (2017). - 26. D. Wulferding, P. Lemmens, F. Büscher, D. Schmeltzer, C. Felser, and Ch. Shekhar, Effect of topology on quasiparticle interactions in the Weyl semimetal WP<sup>2</sup> Phys. Rev. B 102, 075116 (2020). - 32. J. Coulter, G. B. Osterhoudt, C A. C. Garcia, Y. Wang, V. M. Plisson, B. Shen, N. Ni, K. S. Burch, and P. Narang, Uncovering electron-phonon scattering and phonon dynamics in type-I Weyl semimetals, Phys. Rev. B 100, 220301(R) (2019). - 33. Xu, Y.M. Da, L.X. Zha, K. Wang, R. Yang, W. Zhang, J.Y. Liu, H. Xiao, G.F. Chen, S.A. Trugman, J.-X. Zhu, A.J. Taylor, D.A. Yarotski, R.P. Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
Fig.8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. Schnyder, Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery, Phys. Rev. Mat. 5, 124202 (2021). - 2. X. Zhou, Ch.-H. Hsu, H. Aramberri, M. Iraola, Ch.-Yi Huang, J. L. Mañes, M. G. Vergniory, H. Lin, and N. Kioussis, Novel family of topological semimetals with butterflylike nodal lines, Phys. Rev. B 104, 125135 (2021). - 3. J. Wang, H. Yuan, Z.-M. Yu, Z. Zhang, and X. Wang, Coexistence of symmetryenforced phononic Dirac nodal-line net and three-nodal surfaces phonons in solid-state materials: Theory and materials realization, Phys. Rev. Mat. 5, 124203 (2021). - 4. J. Bannies, E. Razzoli, M. Michiardi, H.-H. Kung, I. S. Elfimov, M. Yao, A. Fedorov, J. Fink , C. Jozwiak, A. Bostwick, E. Rotenberg, A. Damascelli , and C. Felser Extremely large magnetoresistance from electron-hole compensation in the nodal-loop semimetal ZrP2, Phys. Rev. B 103, 155144 (2021). - 5. A. S. Wadge, K. Zberecki, B. J. Kowalski, D. Jastrzebski, P. K. 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Miyauchi, Polarized Raman spectroscopy on topological semimetal Co3Sn2S<sup>2</sup> J. Raman Spectroscopy 54(1), 93 (2023). - 21. K. Zhang, X. Pang, T. Wang, F. Han, Shun-Li Shang, N. T. Hung, A.R.T. Nugraha, Zi-Kui Liu, M. Li, R, Saito, and S. Huang, Anomalous phonon-mode dependence in polarized Raman spectroscopy of the topological Weyl semimetal TaP, Phys. Rev. B 101, 014308 (2020). - 22. H. P. C. Miranda, S. Reichardt, G. Froehlicher, A. Molina-Sanchez, S. Berciaud, and L. Wirtz, Quantum Interference Effects in Resonant Raman Spectroscopy of Single- and Triple-Layer MoTe<sup>2</sup> from First-Principles, Nano Lett. 17, 2381 (2017). - 23. M. Osiekowicz, D. Staszczuk, K. Olkowska‑Pucko , Ł. Kipczak , M. Grzeszczyk, M. Zinkiewicz, K. Nogajewski, Z. R. Kudrynskyi, Z. D. Kovalyuk, A. Patané, A. Babiński, and M. R. Molas, Resonance and antiresonance in Raman scattering in GaSe and InSe crystals, Sci. Rep. 11, 924 (2021). - 24. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron– phonon interactions in the Weyl semimetal NbIrTe4, Sci. Rep. 11, 8155 (2021). - 25. S. Nath Gupta, D. V. S. Muthu, C. Shekhar, R. Sankar, C. Felser, and A. K. Sood Pressure-induced electronic and structural phase transitions in Dirac semimetal Cd3As2: Raman study, EPL, 120, 57003 (2017). - 26. D. Wulferding, P. Lemmens, F. Büscher, D. Schmeltzer, C. Felser, and Ch. Shekhar, Effect of topology on quasiparticle interactions in the Weyl semimetal WP<sup>2</sup> Phys. Rev. B 102, 075116 (2020). - 32. J. Coulter, G. B. Osterhoudt, C A. C. Garcia, Y. Wang, V. M. Plisson, B. Shen, N. Ni, K. S. Burch, and P. Narang, Uncovering electron-phonon scattering and phonon dynamics in type-I Weyl semimetals, Phys. Rev. B 100, 220301(R) (2019). - 33. Xu, Y.M. Da, L.X. Zha, K. Wang, R. Yang, W. Zhang, J.Y. Liu, H. Xiao, G.F. Chen, S.A. Trugman, J.-X. Zhu, A.J. Taylor, D.A. Yarotski, R.P. Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![![](_page_14_Figure_1.jpeg)](path) ![Fig.2. Experiment geometry and crystal orientation. The crystal frame of reference (XYZ) and the lab frame of reference (X'Y'Z'). Light polarization (incident and scattered) is in the (X'Y) plane. The red arrow shows the light propagation vector, and the blue denotes the light polarization vector.](path) ![Fig.3. Room temperature Raman spectra for three different excitations (as measured). The continuous background is present with the dashed lines as reference levels for every plot. The green line is the fitted model of quasi-elastic scattering background using expression (3) to the 785 nm plot. The inset shows the temperature dependence of the integrated background intensity for the 785 nm excitation.](path) ![Fig.4. Comparison of the parallel (YY) and the (X'Y) crossed polarization spectra recorded with the 785 nm excitation at room temperature.](path) ![Fig. 5. Room temperature Raman spectra in the (YY) (a) and (X'X') (b) configurations for 473 nm, 633 nm and 785 nm excitation wavelengths.](path) ![Fig. 6. Band structure of ZrAs<sup>2</sup> with a few allowed electronic transitions to/from intermediate states in the Raman process, which are shown with the vertical arrows: black - 785 nm excitation, red 633 nm, and blue 473 nm (a); Density of states (b); Details of the electronic bands close to the Fermi level (c); shaded areas indicate regions of the Brillouin zone where phonon decay via e-h pairs excitation is energetically possible; Brillouin zone of ZrAs<sup>2</sup> (d).](path) ![Fig. 7. Fano profile fitted to the 171 cm-1 A<sup>g</sup> phonon peak measured with the 785 nm excitation at room temperature for two parallel light polarizations (YY) and (X'X').](path) ![Fig. 8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).](path) ![![](_page_20_Figure_1.jpeg)](path) Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening. # SUPPORTING INFORMATION # Temperature and excitation energy dependence of Raman scattering in nodal line Dirac semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland #### S1. Angular dependence of Raman scattering intensity According to the factor group analysis, 18 Raman active modes in ZrAs<sup>2</sup> are distributed as follows: 6A 3B 6B 3B g 1g 2g 3g + + + . In the identification process of modes symmetry, we measured angular dependence of the modes intensity as a function of the angle (Fig.2 in the main text) in parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) polarization configuration. We expect the following expressions for the intensity of the modes: Fitting the angular dependence of intensity with real elements of Raman tensor fails for A<sup>g</sup> modes. To properly describe this dependence, we have to assume complex Raman elements [S1,S2]: a i a a e = , b i b b e = c i c c e = . It leads to a complicated expression that cannot be deconvoluted to find modules or phase factors. Operationally, it comes down to expressions: For B modes phase factors do not play any role and the real values of Raman tensors are good enough to fit the angular dependence for parallel and perpendicular configurations. Figure S1 shows the angular dependence for six representative modes of different symmetry. The fitted value of the angle offset was 4. ![Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.](path) ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 5. Positron diffraction analysis with the Al-embedded silicene model. (a,b) Total-reflection high-energy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with an 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions. (c) Schematic of the model structure. A 3×3 unit cell is depicted.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. Figure 5(a,b) is a set of TRHEPD rocking curves of the Al(111)3×3-Si surface, with comparisons to simulation curves (z1−<sup>8</sup> = 2.40 ˚A) of the flat Al-embedded silicene model that replaces one of the Si atoms with an Al atom [Fig. 5 (c)]. The results show good agreements that confirm the Al-Si mixed model. A R-factor value of the Al-embedded model was R = 2.12%, which was slightly smaller than R = 2.38% for the flat silicene model (Fig. 2). The R-factor values derived from Figs.4 and 5 were different because the former TRHEPD results were obtained with only a specular beam, while the latter were obtained with many diffracted beams. Previous reports proposed a "kagome-like silicene" model as the surface structure[\[21\]](#page-20-10). However, the TRHEPD analysis here was in disagreement, as shown in Fig. 12 in the Appendix. The first-principles calculation with OpenMX has also resulted in the flat structure for the Al-embedded silicene layer on Al(111) after the optimization. A standard derivation of the height (SDH) of the Al-embedded silicene layer on the Al(111) was only 0.06 ˚A. The result is consistent to the flat surface structure, as determined by the TRHEPD analyses. ## B. Core-level spectra Positron diffraction experiments revealed that the Al(111)3×3-Si surface structure could be described by a flat honeycomb lattice model. However, substitution of Al atoms in the Si overlayer was uncertain because of indistinguishable positron scattering. The Si atoms could be bonded with neighboring Si and/or Al atoms. This creates different chemical environments for these Si sites that should result in corresponding chemical shifts in Si core-level photoemission spectra[\[1\]](#page-19-0). Figure 6(a) shows Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface. There were at least five peaks, as reported previously[\[21\]](#page-20-10). Because a single chemical site for Si generates a doublet state, Si 2p3/<sup>2</sup> and 2p1/2, from spin-orbit splitting in the binding energy range, the spectrum indicated the existence of multiple chemical environments for Si atoms at the surface. To reveal the components, spectral curve-fittings were conducted using Doniach-Sunjic functions for various emission angles, θ's. The spin-orbit splitting and the branching ratio were fixed at 0.4 eV and 0.5, respectively. The Doniach-Sunjic width was fixed at 0.0425 eV to be consistent to the 0.085 eV reference Lorentzian width reported previously[\[44\]](#page-22-4). The curve-fit spectra are shown in Fig. 6(b-d), with parameters given in Table I. The results indicated four Si components, or four chemical sites, on the Al(111)3×3-Si surface. To analyze chemical environments at the Al(111)3×3-Si surface, the first-principles spectral simulations were calculated using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0). Figure 7(a) shows four prominent peaks in a simulated spectrum for the silicene layer on Al(111). Because of spin-orbit splitting, each Si 2p core-level appeared as a doublet state, and the simulations indicated two ![FIG. 6. Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface, taken at emission angles of (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with 150 eV photons. In (b-d), experimental spectra are dotted lines, while curve-fit spectra of the Si components are given by green and red curves, respectively. The curve-fit parameters are summarized in Table I.](path) TABLE I. Fitting parameters of the Si 2p core-level spectra for the Al(111)3×3-Si surface, shown in Fig. 6. The spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.6 eV, 0.0425 eV, and 0.5, respectively. Si components for the model structure. As depicted in the inset, one was attributed to Si atoms at atop Al sites and the other to bridge Al sites. While the calculations successfully linked the surface structure and the core-level spectra, the simulated results for the silicene layer were completely different from the experimental data (Fig. 6). Thus, it was necessity to consider a surface structure beyond a simple silicene layer for the Al(111)3×3-Si surface. ![FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦ , (c) θ = 30◦ , and (d) θ = 60◦ . In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b-d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.](path) Si sites in the Al-embedded silicene layer model (Fig. 8). As noted earlier, Si atoms in the surface layer can bond with different numbers of intralayer Si and/or Al atoms. For example, in Fig. 8(a), one Si atom used for the calculations was bonded with two Si atoms and one Al nearest neighbor (NN). Table II lists binding energies of the Si 2p3/<sup>2</sup> level for various structure models of silicene and Al-embedded silicene layers on Al(111). The core-level binding energies varied with respect to the number of NN Al atoms in the Si-Al bonding. Smaller NN values had higher binding energies, which was consistent with the silicene model results given in Table II. Moreover, there was a quantitative relationship between the binding energy and the NN value: 98.4±0.1eV (NN=3), 98.6±0.1eV (NN=2), 98.7±0.1eV (NN=1), and 98.8±0.1eV (NN=0). Because the model calculations in Fig. 8 considered only a single (atop) site on the Al substrate, differences in the substitution sites should be considered in TABLE III. Simulation parameters for the Si 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 7. The spectral components were given by Voigt functions with 0.085 eV Lorentzian widths and Gaussian widths given in the table. Spin-orbit splitting and a branching ratio were set at 0.6 eV and 0.5, respectively. The simulation was performed using OpenMX[\[34,](#page-21-8) [38\]](#page-22-0). On the basis of the calculated binding energies, the experimental spectra of the Al(111)3×3- Si surface were fairly well reproduced by five components, S1, S2, S2', S3 and S4, with the parameters in Table III. From the binding energies, S3 and S4 could be assigned to Si atoms that bonded only with NN Si atoms, while S1, S2, and S2' were also bonded to NN Al ![FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.](path) atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 6. Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface, taken at emission angles of (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with 150 eV photons. In (b-d), experimental spectra are dotted lines, while curve-fit spectra of the Si components are given by green and red curves, respectively. The curve-fit parameters are summarized in Table I.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. TABLE I. Fitting parameters of the Si 2p core-level spectra for the Al(111)3×3-Si surface, shown in Fig. 6. The spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.6 eV, 0.0425 eV, and 0.5, respectively. Si components for the model structure. As depicted in the inset, one was attributed to Si atoms at atop Al sites and the other to bridge Al sites. While the calculations successfully linked the surface structure and the core-level spectra, the simulated results for the silicene layer were completely different from the experimental data (Fig. 6). Thus, it was necessity to consider a surface structure beyond a simple silicene layer for the Al(111)3×3-Si surface. ![FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦ , (c) θ = 30◦ , and (d) θ = 60◦ . In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b-d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.](path) Si sites in the Al-embedded silicene layer model (Fig. 8). As noted earlier, Si atoms in the surface layer can bond with different numbers of intralayer Si and/or Al atoms. For example, in Fig. 8(a), one Si atom used for the calculations was bonded with two Si atoms and one Al nearest neighbor (NN). Table II lists binding energies of the Si 2p3/<sup>2</sup> level for various structure models of silicene and Al-embedded silicene layers on Al(111). The core-level binding energies varied with respect to the number of NN Al atoms in the Si-Al bonding. Smaller NN values had higher binding energies, which was consistent with the silicene model results given in Table II. Moreover, there was a quantitative relationship between the binding energy and the NN value: 98.4±0.1eV (NN=3), 98.6±0.1eV (NN=2), 98.7±0.1eV (NN=1), and 98.8±0.1eV (NN=0). Because the model calculations in Fig. 8 considered only a single (atop) site on the Al substrate, differences in the substitution sites should be considered in TABLE III. Simulation parameters for the Si 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 7. The spectral components were given by Voigt functions with 0.085 eV Lorentzian widths and Gaussian widths given in the table. Spin-orbit splitting and a branching ratio were set at 0.6 eV and 0.5, respectively. The simulation was performed using OpenMX[\[34,](#page-21-8) [38\]](#page-22-0). On the basis of the calculated binding energies, the experimental spectra of the Al(111)3×3- Si surface were fairly well reproduced by five components, S1, S2, S2', S3 and S4, with the parameters in Table III. From the binding energies, S3 and S4 could be assigned to Si atoms that bonded only with NN Si atoms, while S1, S2, and S2' were also bonded to NN Al ![FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.](path) atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 4. (a) A contour plot of the R-factor with respect to z positions of one (z<sup>7</sup> in Fig. 2) and the other Si atoms (z1,z2,z3,z4,z5,z6,z<sup>8</sup> in Fig. 2) in the Si honeycomb lattice layer on Al(111). (b) Total-reflection high-energy positron diffraction rocking curves for the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam under the one-beam condition.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. Figure 4(a) shows the 2D R-factor results for TRHEPD rocking curves under the onebeam condition [Fig. 4(b)] for the silicene model on Al(111), calculated via massively parallel data-analysis using 2DMAT[\[32\]](#page-21-5). In the model, there were eight atoms in the 3×3 unit cell and the height was z<sup>i</sup> for the i th atom (i = 1 ∼ 8). To examine deformation of the flat structure, the z<sup>i</sup> parameters were classified into two groups, one atom (z7) and the other containing atoms with the same height (z1=z2=z3=z4=z5=z6=z8). R-factor values were then plotted with respect to these heights [Fig. 4(a)]. In the 2D diagram, the minimum R-factor of 1.26 % was found at z<sup>7</sup> = 2.40 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.35 ˚A. The heights, z<sup>i</sup> 's, were essentially equal at each Si site. Further searches for optimum values around the minimum of R-factors were performed using Nelder-Mead methods with 2DMAT[\[32\]](#page-21-5). The optimum silicene atomic positions had nearly the same heights (z<sup>7</sup> = 2.38 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.37 ˚A). As shown in Fig. 4(b), the TRHEPD rocking curve calculated from the optimum heights was in good agreement with the experimental curve. Consequently, the analysis confirmed that the silicene layer was flat with no buckling structure. ![FIG. 5. Positron diffraction analysis with the Al-embedded silicene model. (a,b) Total-reflection high-energy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with an 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions. (c) Schematic of the model structure. A 3×3 unit cell is depicted.](path) Figure 5(a,b) is a set of TRHEPD rocking curves of the Al(111)3×3-Si surface, with comparisons to simulation curves (z1−<sup>8</sup> = 2.40 ˚A) of the flat Al-embedded silicene model that replaces one of the Si atoms with an Al atom [Fig. 5 (c)]. The results show good agreements that confirm the Al-Si mixed model. A R-factor value of the Al-embedded model was R = 2.12%, which was slightly smaller than R = 2.38% for the flat silicene model (Fig. 2). The R-factor values derived from Figs.4 and 5 were different because the former TRHEPD results were obtained with only a specular beam, while the latter were obtained with many diffracted beams. Previous reports proposed a "kagome-like silicene" model as the surface structure[\[21\]](#page-20-10). However, the TRHEPD analysis here was in disagreement, as shown in Fig. 12 in the Appendix. The first-principles calculation with OpenMX has also resulted in the flat structure for the Al-embedded silicene layer on Al(111) after the optimization. A standard derivation of the height (SDH) of the Al-embedded silicene layer on the Al(111) was only 0.06 ˚A. The result is consistent to the flat surface structure, as determined by the TRHEPD analyses. ## B. Core-level spectra Positron diffraction experiments revealed that the Al(111)3×3-Si surface structure could be described by a flat honeycomb lattice model. However, substitution of Al atoms in the Si overlayer was uncertain because of indistinguishable positron scattering. The Si atoms could be bonded with neighboring Si and/or Al atoms. This creates different chemical environments for these Si sites that should result in corresponding chemical shifts in Si core-level photoemission spectra[\[1\]](#page-19-0). Figure 6(a) shows Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface. There were at least five peaks, as reported previously[\[21\]](#page-20-10). Because a single chemical site for Si generates a doublet state, Si 2p3/<sup>2</sup> and 2p1/2, from spin-orbit splitting in the binding energy range, the spectrum indicated the existence of multiple chemical environments for Si atoms at the surface. To reveal the components, spectral curve-fittings were conducted using Doniach-Sunjic functions for various emission angles, θ's. The spin-orbit splitting and the branching ratio were fixed at 0.4 eV and 0.5, respectively. The Doniach-Sunjic width was fixed at 0.0425 eV to be consistent to the 0.085 eV reference Lorentzian width reported previously[\[44\]](#page-22-4). The curve-fit spectra are shown in Fig. 6(b-d), with parameters given in Table I. The results indicated four Si components, or four chemical sites, on the Al(111)3×3-Si surface. To analyze chemical environments at the Al(111)3×3-Si surface, the first-principles spectral simulations were calculated using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0). Figure 7(a) shows four prominent peaks in a simulated spectrum for the silicene layer on Al(111). Because of spin-orbit splitting, each Si 2p core-level appeared as a doublet state, and the simulations indicated two ![FIG. 6. Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface, taken at emission angles of (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with 150 eV photons. In (b-d), experimental spectra are dotted lines, while curve-fit spectra of the Si components are given by green and red curves, respectively. The curve-fit parameters are summarized in Table I.](path) TABLE I. Fitting parameters of the Si 2p core-level spectra for the Al(111)3×3-Si surface, shown in Fig. 6. The spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.6 eV, 0.0425 eV, and 0.5, respectively. Si components for the model structure. As depicted in the inset, one was attributed to Si atoms at atop Al sites and the other to bridge Al sites. While the calculations successfully linked the surface structure and the core-level spectra, the simulated results for the silicene layer were completely different from the experimental data (Fig. 6). Thus, it was necessity to consider a surface structure beyond a simple silicene layer for the Al(111)3×3-Si surface. ![FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦ , (c) θ = 30◦ , and (d) θ = 60◦ . In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b-d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.](path) Si sites in the Al-embedded silicene layer model (Fig. 8). As noted earlier, Si atoms in the surface layer can bond with different numbers of intralayer Si and/or Al atoms. For example, in Fig. 8(a), one Si atom used for the calculations was bonded with two Si atoms and one Al nearest neighbor (NN). Table II lists binding energies of the Si 2p3/<sup>2</sup> level for various structure models of silicene and Al-embedded silicene layers on Al(111). The core-level binding energies varied with respect to the number of NN Al atoms in the Si-Al bonding. Smaller NN values had higher binding energies, which was consistent with the silicene model results given in Table II. Moreover, there was a quantitative relationship between the binding energy and the NN value: 98.4±0.1eV (NN=3), 98.6±0.1eV (NN=2), 98.7±0.1eV (NN=1), and 98.8±0.1eV (NN=0). Because the model calculations in Fig. 8 considered only a single (atop) site on the Al substrate, differences in the substitution sites should be considered in TABLE III. Simulation parameters for the Si 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 7. The spectral components were given by Voigt functions with 0.085 eV Lorentzian widths and Gaussian widths given in the table. Spin-orbit splitting and a branching ratio were set at 0.6 eV and 0.5, respectively. The simulation was performed using OpenMX[\[34,](#page-21-8) [38\]](#page-22-0). On the basis of the calculated binding energies, the experimental spectra of the Al(111)3×3- Si surface were fairly well reproduced by five components, S1, S2, S2', S3 and S4, with the parameters in Table III. From the binding energies, S3 and S4 could be assigned to Si atoms that bonded only with NN Si atoms, while S1, S2, and S2' were also bonded to NN Al ![FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.](path) atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 1. Low-energy electron diffraction patterns acquired with 88.2 eV electrons. (a) A pristine Al(111) surface, and (b) the Al(111)3×3-Si surface[22], .	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. ![FIG. 1. Low-energy electron diffraction patterns acquired with 88.2 eV electrons. (a) A pristine Al(111) surface, and (b) the Al(111)3×3-Si surface[\[22\]](#page-20-9), .](path) # II. EXPERIMENTS AND SIMULATIONS ## A. Sample preparation The 3×3 phase was prepared via Si deposition on the Al(111) substrate in an ultrahigh vacuum chamber that had a base pressure of 2 × 10<sup>−</sup><sup>10</sup> mbar. A clean Al(111) surface was prepared via cycles of Ar<sup>+</sup> sputtering and annealing at 670 K. A clean and ordered surface was confirmed by elemental analyses via X-ray photoelectron spectroscopy and electron (positron) diffraction patterns, respectively. Here, various diffraction methods were used to characterize the sample surface: low-energy electron diffraction (LEED), reflection highenergy electron diffraction, and total-reflection high-energy positron diffraction (TRHEPD). Figure 1(a) shows a LEED pattern of a pristine Al(111) surface. Si was then deposited on this Al(111) substrate at 350 K. The Si source was based on sublimation of a Si wafer via Joule heating. Formation of the Al(111)3×3-Si phase was checked with a electron diffraction pattern, as shown in Fig. 1(b). The sample surface was then transferred in situ to perform positron diffraction and core-level spectroscopy. #### B. Positron diffraction The Al(111)3×3-Si structure was examined with TRHEPD [\[1,](#page-19-0) [11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1) performed at the Slow Positron Facility at KEK[\[28\]](#page-21-2). Details of the experimental setup were reported previously[\[29\]](#page-21-3). The energy of the incident positron beam was 10 keV, and rocking curves and spot intensities as a function of glancing angle of the beam were measured by rotating the sample holder up to 6◦ with step sizes of 0.1 ◦ . The surface structure was determined by searching for a model that had good agreement between experimental and simulated rocking curves. It was also quantitatively judged by a R-factor with the minimum value for the most appropriate structure model[\[11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1). The R-factor (R) was defined by: The data-driven analyses used TRHEPD results taken under the one-beam condition, and simulation results obtained with the 2DMAT software package[\[32\]](#page-21-5). Bayesian inference was performed via the massively parallel Monte Carlo method on the "Fugaku" supercomputer. #### C. Core-level photoemission spectroscopy Core-level photoemission spectroscopy measurements were performed at a vacuum ultraviolet photoemission beamline (Elettra, Trieste). Details were described previously[\[1,](#page-19-0) [12,](#page-20-4) [13\]](#page-20-11). The beamline covered a photon energy range of 20 eV to 750 eV, and the base pressure was 1 × 10<sup>−</sup><sup>11</sup> mbar. Simulations of the spectroscopy were performed with OpenMX[\[33–](#page-21-6)[38\]](#page-22-0). In the calculations, 3,3,2 optimized radial functions were allocated for Al s, p, and d orbitals with a cutoff radius of 7 Bohr. For Si atoms, 2, 2, 1 optimized radial functions were allocated for s, p, and d orbitals with a cutoff radius of 7 Bohr. A Perdew-Burke-Ernzerhof generalized gradient approximation was used as the exchange correlation functional[\[39\]](#page-22-1). In the spectral simulations, both atomic orbitals and pseudo-potentials included contributions from the 2p orbital. The energy cutoff and the energy convergence criterion were 220 Rydberg and 1.0× 10<sup>−</sup><sup>8</sup> Hartree, respectively. For the wavevector space, a 6×6×1 grid was used. In the calculations, an Al(111) slab of seven layers was used, where one side was covered with the Si(Al) surface layer. The slabs were repeatedly separated by a vacuum at a distance of 17 ˚A and facing the bare and overlayer surfaces. The binding energy of each core-level was obtained by calculating the total energy difference before and after creation of the core-hole state[\[38\]](#page-22-0). #### D. Electronic structure calculations Calculations of structural optimizations and band dispersion curves were performed using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0) with the same conditions used for the spectral simulations described above for optimized radial functions, a cutoff radius, and an exchange correlation functional. The energy cutoff was 300 Rydberg, while the energy convergence criterion was 1.0× 10<sup>−</sup><sup>8</sup> Hartree. A grid of 6×6×1 was used for the wavevector space. In the calculations, an Al(111) slab of five layers was adopted, where one side was covered with the Si(Al) surface layer. The slabs were repeatedly separated by vacuum at a distance of 23 ˚A. Structural optimization was performed by minimizing the force upon each atom. At the bottom of the Al(111) slab, the coordinates of the atoms were fixed, while those of the other atoms were relaxed. The force convergence criterion was 1.0 × 10<sup>−</sup><sup>4</sup> Hartree/Bohr. After the convergence of energy and force was achieved, the total energy of the system was calculated to assess the system stability. To simulate the 2D slab, an effective screening medium method was applied in which a semi-infinite vacuum was inserted at the bottom and top of the system[\[40](#page-22-2)[–43\]](#page-22-3). ![FIG. 2. Schematic of the flat silicene model. Si atoms are labeled with numbers 1 to 8. A 3×3 unit cell is depicted.](path) #### III. RESULTS AND DISCUSSION #### A. Positron diffraction TRHEPD was conducted to investigate the structure of the Al(111)3×3-Si surface. TRHEPD rocking curves were measured at 13◦ off from the [11¯2] direction (one-beam condition), and along the [11¯2] and [1¯10] directions. Under the one-beam condition, the specular (00) spot intensities depended mainly on the surface-normal components (z) of atomic positions because simultaneous reflections in the surface-parallel plane were sufficiently suppressed[\[31\]](#page-21-7). In the TRHEPD analyses, a surface structure was determined by searching for a model that had agreements between simulated and experimental curves[\[11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1). A honeycomb lattice of Si atoms, shown in Fig. 2, was adopted as an initial model because it had previously described the Si layer (silicene) on Ag(111) and the Ge layer (germanene) on Al(111)[\[11,](#page-20-3) [23\]](#page-21-0). ![FIG. 3. Real parts of scattering potentials, Re[U00], averaged in the surface-parallel plane for the Si honeycomb lattice layer on Al(111) (red) without or (blue) with Al substitution. Bulk and surface layers are indicated by labels.](path) When performing the surface analysis via positron diffraction, it is important to note that Al and Si are located next to each other in the periodic table and thus have similar scattering potentials for a positron beam. This was demonstrated by the calculations shown in Fig. 3. The scattering event can be evaluated by the positron scattering potential. The scattering potential at the Si surface layer appeared nearly identical without or with Al substitution. This indicated that the surface structure model created uncertainty regarding embedded Al atoms. However, the structure parameters of the Si(Al) surface layer can be elaborated by considering only Si atoms. ![FIG. 4. (a) A contour plot of the R-factor with respect to z positions of one (z<sup>7</sup> in Fig. 2) and the other Si atoms (z1,z2,z3,z4,z5,z6,z<sup>8</sup> in Fig. 2) in the Si honeycomb lattice layer on Al(111). (b) Total-reflection high-energy positron diffraction rocking curves for the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam under the one-beam condition.](path) Figure 4(a) shows the 2D R-factor results for TRHEPD rocking curves under the onebeam condition [Fig. 4(b)] for the silicene model on Al(111), calculated via massively parallel data-analysis using 2DMAT[\[32\]](#page-21-5). In the model, there were eight atoms in the 3×3 unit cell and the height was z<sup>i</sup> for the i th atom (i = 1 ∼ 8). To examine deformation of the flat structure, the z<sup>i</sup> parameters were classified into two groups, one atom (z7) and the other containing atoms with the same height (z1=z2=z3=z4=z5=z6=z8). R-factor values were then plotted with respect to these heights [Fig. 4(a)]. In the 2D diagram, the minimum R-factor of 1.26 % was found at z<sup>7</sup> = 2.40 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.35 ˚A. The heights, z<sup>i</sup> 's, were essentially equal at each Si site. Further searches for optimum values around the minimum of R-factors were performed using Nelder-Mead methods with 2DMAT[\[32\]](#page-21-5). The optimum silicene atomic positions had nearly the same heights (z<sup>7</sup> = 2.38 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.37 ˚A). As shown in Fig. 4(b), the TRHEPD rocking curve calculated from the optimum heights was in good agreement with the experimental curve. Consequently, the analysis confirmed that the silicene layer was flat with no buckling structure. ![FIG. 5. Positron diffraction analysis with the Al-embedded silicene model. (a,b) Total-reflection high-energy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with an 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions. (c) Schematic of the model structure. A 3×3 unit cell is depicted.](path) Figure 5(a,b) is a set of TRHEPD rocking curves of the Al(111)3×3-Si surface, with comparisons to simulation curves (z1−<sup>8</sup> = 2.40 ˚A) of the flat Al-embedded silicene model that replaces one of the Si atoms with an Al atom [Fig. 5 (c)]. The results show good agreements that confirm the Al-Si mixed model. A R-factor value of the Al-embedded model was R = 2.12%, which was slightly smaller than R = 2.38% for the flat silicene model (Fig. 2). The R-factor values derived from Figs.4 and 5 were different because the former TRHEPD results were obtained with only a specular beam, while the latter were obtained with many diffracted beams. Previous reports proposed a "kagome-like silicene" model as the surface structure[\[21\]](#page-20-10). However, the TRHEPD analysis here was in disagreement, as shown in Fig. 12 in the Appendix. The first-principles calculation with OpenMX has also resulted in the flat structure for the Al-embedded silicene layer on Al(111) after the optimization. A standard derivation of the height (SDH) of the Al-embedded silicene layer on the Al(111) was only 0.06 ˚A. The result is consistent to the flat surface structure, as determined by the TRHEPD analyses. ## B. Core-level spectra Positron diffraction experiments revealed that the Al(111)3×3-Si surface structure could be described by a flat honeycomb lattice model. However, substitution of Al atoms in the Si overlayer was uncertain because of indistinguishable positron scattering. The Si atoms could be bonded with neighboring Si and/or Al atoms. This creates different chemical environments for these Si sites that should result in corresponding chemical shifts in Si core-level photoemission spectra[\[1\]](#page-19-0). Figure 6(a) shows Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface. There were at least five peaks, as reported previously[\[21\]](#page-20-10). Because a single chemical site for Si generates a doublet state, Si 2p3/<sup>2</sup> and 2p1/2, from spin-orbit splitting in the binding energy range, the spectrum indicated the existence of multiple chemical environments for Si atoms at the surface. To reveal the components, spectral curve-fittings were conducted using Doniach-Sunjic functions for various emission angles, θ's. The spin-orbit splitting and the branching ratio were fixed at 0.4 eV and 0.5, respectively. The Doniach-Sunjic width was fixed at 0.0425 eV to be consistent to the 0.085 eV reference Lorentzian width reported previously[\[44\]](#page-22-4). The curve-fit spectra are shown in Fig. 6(b-d), with parameters given in Table I. The results indicated four Si components, or four chemical sites, on the Al(111)3×3-Si surface. To analyze chemical environments at the Al(111)3×3-Si surface, the first-principles spectral simulations were calculated using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0). Figure 7(a) shows four prominent peaks in a simulated spectrum for the silicene layer on Al(111). Because of spin-orbit splitting, each Si 2p core-level appeared as a doublet state, and the simulations indicated two ![FIG. 6. Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface, taken at emission angles of (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with 150 eV photons. In (b-d), experimental spectra are dotted lines, while curve-fit spectra of the Si components are given by green and red curves, respectively. The curve-fit parameters are summarized in Table I.](path) TABLE I. Fitting parameters of the Si 2p core-level spectra for the Al(111)3×3-Si surface, shown in Fig. 6. The spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.6 eV, 0.0425 eV, and 0.5, respectively. Si components for the model structure. As depicted in the inset, one was attributed to Si atoms at atop Al sites and the other to bridge Al sites. While the calculations successfully linked the surface structure and the core-level spectra, the simulated results for the silicene layer were completely different from the experimental data (Fig. 6). Thus, it was necessity to consider a surface structure beyond a simple silicene layer for the Al(111)3×3-Si surface. ![FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦ , (c) θ = 30◦ , and (d) θ = 60◦ . In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b-d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.](path) Si sites in the Al-embedded silicene layer model (Fig. 8). As noted earlier, Si atoms in the surface layer can bond with different numbers of intralayer Si and/or Al atoms. For example, in Fig. 8(a), one Si atom used for the calculations was bonded with two Si atoms and one Al nearest neighbor (NN). Table II lists binding energies of the Si 2p3/<sup>2</sup> level for various structure models of silicene and Al-embedded silicene layers on Al(111). The core-level binding energies varied with respect to the number of NN Al atoms in the Si-Al bonding. Smaller NN values had higher binding energies, which was consistent with the silicene model results given in Table II. Moreover, there was a quantitative relationship between the binding energy and the NN value: 98.4±0.1eV (NN=3), 98.6±0.1eV (NN=2), 98.7±0.1eV (NN=1), and 98.8±0.1eV (NN=0). Because the model calculations in Fig. 8 considered only a single (atop) site on the Al substrate, differences in the substitution sites should be considered in TABLE III. Simulation parameters for the Si 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 7. The spectral components were given by Voigt functions with 0.085 eV Lorentzian widths and Gaussian widths given in the table. Spin-orbit splitting and a branching ratio were set at 0.6 eV and 0.5, respectively. The simulation was performed using OpenMX[\[34,](#page-21-8) [38\]](#page-22-0). On the basis of the calculated binding energies, the experimental spectra of the Al(111)3×3- Si surface were fairly well reproduced by five components, S1, S2, S2', S3 and S4, with the parameters in Table III. From the binding energies, S3 and S4 could be assigned to Si atoms that bonded only with NN Si atoms, while S1, S2, and S2' were also bonded to NN Al ![FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.](path) atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[46].	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. ![FIG. 1. Low-energy electron diffraction patterns acquired with 88.2 eV electrons. (a) A pristine Al(111) surface, and (b) the Al(111)3×3-Si surface[\[22\]](#page-20-9), .](path) # II. EXPERIMENTS AND SIMULATIONS ## A. Sample preparation The 3×3 phase was prepared via Si deposition on the Al(111) substrate in an ultrahigh vacuum chamber that had a base pressure of 2 × 10<sup>−</sup><sup>10</sup> mbar. A clean Al(111) surface was prepared via cycles of Ar<sup>+</sup> sputtering and annealing at 670 K. A clean and ordered surface was confirmed by elemental analyses via X-ray photoelectron spectroscopy and electron (positron) diffraction patterns, respectively. Here, various diffraction methods were used to characterize the sample surface: low-energy electron diffraction (LEED), reflection highenergy electron diffraction, and total-reflection high-energy positron diffraction (TRHEPD). Figure 1(a) shows a LEED pattern of a pristine Al(111) surface. Si was then deposited on this Al(111) substrate at 350 K. The Si source was based on sublimation of a Si wafer via Joule heating. Formation of the Al(111)3×3-Si phase was checked with a electron diffraction pattern, as shown in Fig. 1(b). The sample surface was then transferred in situ to perform positron diffraction and core-level spectroscopy. #### B. Positron diffraction The Al(111)3×3-Si structure was examined with TRHEPD [\[1,](#page-19-0) [11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1) performed at the Slow Positron Facility at KEK[\[28\]](#page-21-2). Details of the experimental setup were reported previously[\[29\]](#page-21-3). The energy of the incident positron beam was 10 keV, and rocking curves and spot intensities as a function of glancing angle of the beam were measured by rotating the sample holder up to 6◦ with step sizes of 0.1 ◦ . The surface structure was determined by searching for a model that had good agreement between experimental and simulated rocking curves. It was also quantitatively judged by a R-factor with the minimum value for the most appropriate structure model[\[11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1). The R-factor (R) was defined by: The data-driven analyses used TRHEPD results taken under the one-beam condition, and simulation results obtained with the 2DMAT software package[\[32\]](#page-21-5). Bayesian inference was performed via the massively parallel Monte Carlo method on the "Fugaku" supercomputer. #### C. Core-level photoemission spectroscopy Core-level photoemission spectroscopy measurements were performed at a vacuum ultraviolet photoemission beamline (Elettra, Trieste). Details were described previously[\[1,](#page-19-0) [12,](#page-20-4) [13\]](#page-20-11). The beamline covered a photon energy range of 20 eV to 750 eV, and the base pressure was 1 × 10<sup>−</sup><sup>11</sup> mbar. Simulations of the spectroscopy were performed with OpenMX[\[33–](#page-21-6)[38\]](#page-22-0). In the calculations, 3,3,2 optimized radial functions were allocated for Al s, p, and d orbitals with a cutoff radius of 7 Bohr. For Si atoms, 2, 2, 1 optimized radial functions were allocated for s, p, and d orbitals with a cutoff radius of 7 Bohr. A Perdew-Burke-Ernzerhof generalized gradient approximation was used as the exchange correlation functional[\[39\]](#page-22-1). In the spectral simulations, both atomic orbitals and pseudo-potentials included contributions from the 2p orbital. The energy cutoff and the energy convergence criterion were 220 Rydberg and 1.0× 10<sup>−</sup><sup>8</sup> Hartree, respectively. For the wavevector space, a 6×6×1 grid was used. In the calculations, an Al(111) slab of seven layers was used, where one side was covered with the Si(Al) surface layer. The slabs were repeatedly separated by a vacuum at a distance of 17 ˚A and facing the bare and overlayer surfaces. The binding energy of each core-level was obtained by calculating the total energy difference before and after creation of the core-hole state[\[38\]](#page-22-0). #### D. Electronic structure calculations Calculations of structural optimizations and band dispersion curves were performed using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0) with the same conditions used for the spectral simulations described above for optimized radial functions, a cutoff radius, and an exchange correlation functional. The energy cutoff was 300 Rydberg, while the energy convergence criterion was 1.0× 10<sup>−</sup><sup>8</sup> Hartree. A grid of 6×6×1 was used for the wavevector space. In the calculations, an Al(111) slab of five layers was adopted, where one side was covered with the Si(Al) surface layer. The slabs were repeatedly separated by vacuum at a distance of 23 ˚A. Structural optimization was performed by minimizing the force upon each atom. At the bottom of the Al(111) slab, the coordinates of the atoms were fixed, while those of the other atoms were relaxed. The force convergence criterion was 1.0 × 10<sup>−</sup><sup>4</sup> Hartree/Bohr. After the convergence of energy and force was achieved, the total energy of the system was calculated to assess the system stability. To simulate the 2D slab, an effective screening medium method was applied in which a semi-infinite vacuum was inserted at the bottom and top of the system[\[40](#page-22-2)[–43\]](#page-22-3). ![FIG. 2. Schematic of the flat silicene model. Si atoms are labeled with numbers 1 to 8. A 3×3 unit cell is depicted.](path) #### III. RESULTS AND DISCUSSION #### A. Positron diffraction TRHEPD was conducted to investigate the structure of the Al(111)3×3-Si surface. TRHEPD rocking curves were measured at 13◦ off from the [11¯2] direction (one-beam condition), and along the [11¯2] and [1¯10] directions. Under the one-beam condition, the specular (00) spot intensities depended mainly on the surface-normal components (z) of atomic positions because simultaneous reflections in the surface-parallel plane were sufficiently suppressed[\[31\]](#page-21-7). In the TRHEPD analyses, a surface structure was determined by searching for a model that had agreements between simulated and experimental curves[\[11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1). A honeycomb lattice of Si atoms, shown in Fig. 2, was adopted as an initial model because it had previously described the Si layer (silicene) on Ag(111) and the Ge layer (germanene) on Al(111)[\[11,](#page-20-3) [23\]](#page-21-0). ![FIG. 3. Real parts of scattering potentials, Re[U00], averaged in the surface-parallel plane for the Si honeycomb lattice layer on Al(111) (red) without or (blue) with Al substitution. Bulk and surface layers are indicated by labels.](path) When performing the surface analysis via positron diffraction, it is important to note that Al and Si are located next to each other in the periodic table and thus have similar scattering potentials for a positron beam. This was demonstrated by the calculations shown in Fig. 3. The scattering event can be evaluated by the positron scattering potential. The scattering potential at the Si surface layer appeared nearly identical without or with Al substitution. This indicated that the surface structure model created uncertainty regarding embedded Al atoms. However, the structure parameters of the Si(Al) surface layer can be elaborated by considering only Si atoms. ![FIG. 4. (a) A contour plot of the R-factor with respect to z positions of one (z<sup>7</sup> in Fig. 2) and the other Si atoms (z1,z2,z3,z4,z5,z6,z<sup>8</sup> in Fig. 2) in the Si honeycomb lattice layer on Al(111). (b) Total-reflection high-energy positron diffraction rocking curves for the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam under the one-beam condition.](path) Figure 4(a) shows the 2D R-factor results for TRHEPD rocking curves under the onebeam condition [Fig. 4(b)] for the silicene model on Al(111), calculated via massively parallel data-analysis using 2DMAT[\[32\]](#page-21-5). In the model, there were eight atoms in the 3×3 unit cell and the height was z<sup>i</sup> for the i th atom (i = 1 ∼ 8). To examine deformation of the flat structure, the z<sup>i</sup> parameters were classified into two groups, one atom (z7) and the other containing atoms with the same height (z1=z2=z3=z4=z5=z6=z8). R-factor values were then plotted with respect to these heights [Fig. 4(a)]. In the 2D diagram, the minimum R-factor of 1.26 % was found at z<sup>7</sup> = 2.40 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.35 ˚A. The heights, z<sup>i</sup> 's, were essentially equal at each Si site. Further searches for optimum values around the minimum of R-factors were performed using Nelder-Mead methods with 2DMAT[\[32\]](#page-21-5). The optimum silicene atomic positions had nearly the same heights (z<sup>7</sup> = 2.38 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.37 ˚A). As shown in Fig. 4(b), the TRHEPD rocking curve calculated from the optimum heights was in good agreement with the experimental curve. Consequently, the analysis confirmed that the silicene layer was flat with no buckling structure. ![FIG. 5. Positron diffraction analysis with the Al-embedded silicene model. (a,b) Total-reflection high-energy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with an 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions. (c) Schematic of the model structure. A 3×3 unit cell is depicted.](path) Figure 5(a,b) is a set of TRHEPD rocking curves of the Al(111)3×3-Si surface, with comparisons to simulation curves (z1−<sup>8</sup> = 2.40 ˚A) of the flat Al-embedded silicene model that replaces one of the Si atoms with an Al atom [Fig. 5 (c)]. The results show good agreements that confirm the Al-Si mixed model. A R-factor value of the Al-embedded model was R = 2.12%, which was slightly smaller than R = 2.38% for the flat silicene model (Fig. 2). The R-factor values derived from Figs.4 and 5 were different because the former TRHEPD results were obtained with only a specular beam, while the latter were obtained with many diffracted beams. Previous reports proposed a "kagome-like silicene" model as the surface structure[\[21\]](#page-20-10). However, the TRHEPD analysis here was in disagreement, as shown in Fig. 12 in the Appendix. The first-principles calculation with OpenMX has also resulted in the flat structure for the Al-embedded silicene layer on Al(111) after the optimization. A standard derivation of the height (SDH) of the Al-embedded silicene layer on the Al(111) was only 0.06 ˚A. The result is consistent to the flat surface structure, as determined by the TRHEPD analyses. ## B. Core-level spectra Positron diffraction experiments revealed that the Al(111)3×3-Si surface structure could be described by a flat honeycomb lattice model. However, substitution of Al atoms in the Si overlayer was uncertain because of indistinguishable positron scattering. The Si atoms could be bonded with neighboring Si and/or Al atoms. This creates different chemical environments for these Si sites that should result in corresponding chemical shifts in Si core-level photoemission spectra[\[1\]](#page-19-0). Figure 6(a) shows Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface. There were at least five peaks, as reported previously[\[21\]](#page-20-10). Because a single chemical site for Si generates a doublet state, Si 2p3/<sup>2</sup> and 2p1/2, from spin-orbit splitting in the binding energy range, the spectrum indicated the existence of multiple chemical environments for Si atoms at the surface. To reveal the components, spectral curve-fittings were conducted using Doniach-Sunjic functions for various emission angles, θ's. The spin-orbit splitting and the branching ratio were fixed at 0.4 eV and 0.5, respectively. The Doniach-Sunjic width was fixed at 0.0425 eV to be consistent to the 0.085 eV reference Lorentzian width reported previously[\[44\]](#page-22-4). The curve-fit spectra are shown in Fig. 6(b-d), with parameters given in Table I. The results indicated four Si components, or four chemical sites, on the Al(111)3×3-Si surface. To analyze chemical environments at the Al(111)3×3-Si surface, the first-principles spectral simulations were calculated using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0). Figure 7(a) shows four prominent peaks in a simulated spectrum for the silicene layer on Al(111). Because of spin-orbit splitting, each Si 2p core-level appeared as a doublet state, and the simulations indicated two ![FIG. 6. Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface, taken at emission angles of (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with 150 eV photons. In (b-d), experimental spectra are dotted lines, while curve-fit spectra of the Si components are given by green and red curves, respectively. The curve-fit parameters are summarized in Table I.](path) TABLE I. Fitting parameters of the Si 2p core-level spectra for the Al(111)3×3-Si surface, shown in Fig. 6. The spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.6 eV, 0.0425 eV, and 0.5, respectively. Si components for the model structure. As depicted in the inset, one was attributed to Si atoms at atop Al sites and the other to bridge Al sites. While the calculations successfully linked the surface structure and the core-level spectra, the simulated results for the silicene layer were completely different from the experimental data (Fig. 6). Thus, it was necessity to consider a surface structure beyond a simple silicene layer for the Al(111)3×3-Si surface. ![FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦ , (c) θ = 30◦ , and (d) θ = 60◦ . In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b-d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.](path) Si sites in the Al-embedded silicene layer model (Fig. 8). As noted earlier, Si atoms in the surface layer can bond with different numbers of intralayer Si and/or Al atoms. For example, in Fig. 8(a), one Si atom used for the calculations was bonded with two Si atoms and one Al nearest neighbor (NN). Table II lists binding energies of the Si 2p3/<sup>2</sup> level for various structure models of silicene and Al-embedded silicene layers on Al(111). The core-level binding energies varied with respect to the number of NN Al atoms in the Si-Al bonding. Smaller NN values had higher binding energies, which was consistent with the silicene model results given in Table II. Moreover, there was a quantitative relationship between the binding energy and the NN value: 98.4±0.1eV (NN=3), 98.6±0.1eV (NN=2), 98.7±0.1eV (NN=1), and 98.8±0.1eV (NN=0). Because the model calculations in Fig. 8 considered only a single (atop) site on the Al substrate, differences in the substitution sites should be considered in TABLE III. Simulation parameters for the Si 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 7. The spectral components were given by Voigt functions with 0.085 eV Lorentzian widths and Gaussian widths given in the table. Spin-orbit splitting and a branching ratio were set at 0.6 eV and 0.5, respectively. The simulation was performed using OpenMX[\[34,](#page-21-8) [38\]](#page-22-0). On the basis of the calculated binding energies, the experimental spectra of the Al(111)3×3- Si surface were fairly well reproduced by five components, S1, S2, S2', S3 and S4, with the parameters in Table III. From the binding energies, S3 and S4 could be assigned to Si atoms that bonded only with NN Si atoms, while S1, S2, and S2' were also bonded to NN Al ![FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.](path) atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦, (c) θ = 30◦, and (d) θ = 60◦. In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b–d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. ![FIG. 1. Low-energy electron diffraction patterns acquired with 88.2 eV electrons. (a) A pristine Al(111) surface, and (b) the Al(111)3×3-Si surface[\[22\]](#page-20-9), .](path) # II. EXPERIMENTS AND SIMULATIONS ## A. Sample preparation The 3×3 phase was prepared via Si deposition on the Al(111) substrate in an ultrahigh vacuum chamber that had a base pressure of 2 × 10<sup>−</sup><sup>10</sup> mbar. A clean Al(111) surface was prepared via cycles of Ar<sup>+</sup> sputtering and annealing at 670 K. A clean and ordered surface was confirmed by elemental analyses via X-ray photoelectron spectroscopy and electron (positron) diffraction patterns, respectively. Here, various diffraction methods were used to characterize the sample surface: low-energy electron diffraction (LEED), reflection highenergy electron diffraction, and total-reflection high-energy positron diffraction (TRHEPD). Figure 1(a) shows a LEED pattern of a pristine Al(111) surface. Si was then deposited on this Al(111) substrate at 350 K. The Si source was based on sublimation of a Si wafer via Joule heating. Formation of the Al(111)3×3-Si phase was checked with a electron diffraction pattern, as shown in Fig. 1(b). The sample surface was then transferred in situ to perform positron diffraction and core-level spectroscopy. #### B. Positron diffraction The Al(111)3×3-Si structure was examined with TRHEPD [\[1,](#page-19-0) [11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1) performed at the Slow Positron Facility at KEK[\[28\]](#page-21-2). Details of the experimental setup were reported previously[\[29\]](#page-21-3). The energy of the incident positron beam was 10 keV, and rocking curves and spot intensities as a function of glancing angle of the beam were measured by rotating the sample holder up to 6◦ with step sizes of 0.1 ◦ . The surface structure was determined by searching for a model that had good agreement between experimental and simulated rocking curves. It was also quantitatively judged by a R-factor with the minimum value for the most appropriate structure model[\[11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1). The R-factor (R) was defined by: The data-driven analyses used TRHEPD results taken under the one-beam condition, and simulation results obtained with the 2DMAT software package[\[32\]](#page-21-5). Bayesian inference was performed via the massively parallel Monte Carlo method on the "Fugaku" supercomputer. #### C. Core-level photoemission spectroscopy Core-level photoemission spectroscopy measurements were performed at a vacuum ultraviolet photoemission beamline (Elettra, Trieste). Details were described previously[\[1,](#page-19-0) [12,](#page-20-4) [13\]](#page-20-11). The beamline covered a photon energy range of 20 eV to 750 eV, and the base pressure was 1 × 10<sup>−</sup><sup>11</sup> mbar. Simulations of the spectroscopy were performed with OpenMX[\[33–](#page-21-6)[38\]](#page-22-0). In the calculations, 3,3,2 optimized radial functions were allocated for Al s, p, and d orbitals with a cutoff radius of 7 Bohr. For Si atoms, 2, 2, 1 optimized radial functions were allocated for s, p, and d orbitals with a cutoff radius of 7 Bohr. A Perdew-Burke-Ernzerhof generalized gradient approximation was used as the exchange correlation functional[\[39\]](#page-22-1). In the spectral simulations, both atomic orbitals and pseudo-potentials included contributions from the 2p orbital. The energy cutoff and the energy convergence criterion were 220 Rydberg and 1.0× 10<sup>−</sup><sup>8</sup> Hartree, respectively. For the wavevector space, a 6×6×1 grid was used. In the calculations, an Al(111) slab of seven layers was used, where one side was covered with the Si(Al) surface layer. The slabs were repeatedly separated by a vacuum at a distance of 17 ˚A and facing the bare and overlayer surfaces. The binding energy of each core-level was obtained by calculating the total energy difference before and after creation of the core-hole state[\[38\]](#page-22-0). #### D. Electronic structure calculations Calculations of structural optimizations and band dispersion curves were performed using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0) with the same conditions used for the spectral simulations described above for optimized radial functions, a cutoff radius, and an exchange correlation functional. The energy cutoff was 300 Rydberg, while the energy convergence criterion was 1.0× 10<sup>−</sup><sup>8</sup> Hartree. A grid of 6×6×1 was used for the wavevector space. In the calculations, an Al(111) slab of five layers was adopted, where one side was covered with the Si(Al) surface layer. The slabs were repeatedly separated by vacuum at a distance of 23 ˚A. Structural optimization was performed by minimizing the force upon each atom. At the bottom of the Al(111) slab, the coordinates of the atoms were fixed, while those of the other atoms were relaxed. The force convergence criterion was 1.0 × 10<sup>−</sup><sup>4</sup> Hartree/Bohr. After the convergence of energy and force was achieved, the total energy of the system was calculated to assess the system stability. To simulate the 2D slab, an effective screening medium method was applied in which a semi-infinite vacuum was inserted at the bottom and top of the system[\[40](#page-22-2)[–43\]](#page-22-3). ![FIG. 2. Schematic of the flat silicene model. Si atoms are labeled with numbers 1 to 8. A 3×3 unit cell is depicted.](path) #### III. RESULTS AND DISCUSSION #### A. Positron diffraction TRHEPD was conducted to investigate the structure of the Al(111)3×3-Si surface. TRHEPD rocking curves were measured at 13◦ off from the [11¯2] direction (one-beam condition), and along the [11¯2] and [1¯10] directions. Under the one-beam condition, the specular (00) spot intensities depended mainly on the surface-normal components (z) of atomic positions because simultaneous reflections in the surface-parallel plane were sufficiently suppressed[\[31\]](#page-21-7). In the TRHEPD analyses, a surface structure was determined by searching for a model that had agreements between simulated and experimental curves[\[11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1). A honeycomb lattice of Si atoms, shown in Fig. 2, was adopted as an initial model because it had previously described the Si layer (silicene) on Ag(111) and the Ge layer (germanene) on Al(111)[\[11,](#page-20-3) [23\]](#page-21-0). ![FIG. 3. Real parts of scattering potentials, Re[U00], averaged in the surface-parallel plane for the Si honeycomb lattice layer on Al(111) (red) without or (blue) with Al substitution. Bulk and surface layers are indicated by labels.](path) When performing the surface analysis via positron diffraction, it is important to note that Al and Si are located next to each other in the periodic table and thus have similar scattering potentials for a positron beam. This was demonstrated by the calculations shown in Fig. 3. The scattering event can be evaluated by the positron scattering potential. The scattering potential at the Si surface layer appeared nearly identical without or with Al substitution. This indicated that the surface structure model created uncertainty regarding embedded Al atoms. However, the structure parameters of the Si(Al) surface layer can be elaborated by considering only Si atoms. ![FIG. 4. (a) A contour plot of the R-factor with respect to z positions of one (z<sup>7</sup> in Fig. 2) and the other Si atoms (z1,z2,z3,z4,z5,z6,z<sup>8</sup> in Fig. 2) in the Si honeycomb lattice layer on Al(111). (b) Total-reflection high-energy positron diffraction rocking curves for the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam under the one-beam condition.](path) Figure 4(a) shows the 2D R-factor results for TRHEPD rocking curves under the onebeam condition [Fig. 4(b)] for the silicene model on Al(111), calculated via massively parallel data-analysis using 2DMAT[\[32\]](#page-21-5). In the model, there were eight atoms in the 3×3 unit cell and the height was z<sup>i</sup> for the i th atom (i = 1 ∼ 8). To examine deformation of the flat structure, the z<sup>i</sup> parameters were classified into two groups, one atom (z7) and the other containing atoms with the same height (z1=z2=z3=z4=z5=z6=z8). R-factor values were then plotted with respect to these heights [Fig. 4(a)]. In the 2D diagram, the minimum R-factor of 1.26 % was found at z<sup>7</sup> = 2.40 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.35 ˚A. The heights, z<sup>i</sup> 's, were essentially equal at each Si site. Further searches for optimum values around the minimum of R-factors were performed using Nelder-Mead methods with 2DMAT[\[32\]](#page-21-5). The optimum silicene atomic positions had nearly the same heights (z<sup>7</sup> = 2.38 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.37 ˚A). As shown in Fig. 4(b), the TRHEPD rocking curve calculated from the optimum heights was in good agreement with the experimental curve. Consequently, the analysis confirmed that the silicene layer was flat with no buckling structure. ![FIG. 5. Positron diffraction analysis with the Al-embedded silicene model. (a,b) Total-reflection high-energy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with an 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions. (c) Schematic of the model structure. A 3×3 unit cell is depicted.](path) Figure 5(a,b) is a set of TRHEPD rocking curves of the Al(111)3×3-Si surface, with comparisons to simulation curves (z1−<sup>8</sup> = 2.40 ˚A) of the flat Al-embedded silicene model that replaces one of the Si atoms with an Al atom [Fig. 5 (c)]. The results show good agreements that confirm the Al-Si mixed model. A R-factor value of the Al-embedded model was R = 2.12%, which was slightly smaller than R = 2.38% for the flat silicene model (Fig. 2). The R-factor values derived from Figs.4 and 5 were different because the former TRHEPD results were obtained with only a specular beam, while the latter were obtained with many diffracted beams. Previous reports proposed a "kagome-like silicene" model as the surface structure[\[21\]](#page-20-10). However, the TRHEPD analysis here was in disagreement, as shown in Fig. 12 in the Appendix. The first-principles calculation with OpenMX has also resulted in the flat structure for the Al-embedded silicene layer on Al(111) after the optimization. A standard derivation of the height (SDH) of the Al-embedded silicene layer on the Al(111) was only 0.06 ˚A. The result is consistent to the flat surface structure, as determined by the TRHEPD analyses. ## B. Core-level spectra Positron diffraction experiments revealed that the Al(111)3×3-Si surface structure could be described by a flat honeycomb lattice model. However, substitution of Al atoms in the Si overlayer was uncertain because of indistinguishable positron scattering. The Si atoms could be bonded with neighboring Si and/or Al atoms. This creates different chemical environments for these Si sites that should result in corresponding chemical shifts in Si core-level photoemission spectra[\[1\]](#page-19-0). Figure 6(a) shows Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface. There were at least five peaks, as reported previously[\[21\]](#page-20-10). Because a single chemical site for Si generates a doublet state, Si 2p3/<sup>2</sup> and 2p1/2, from spin-orbit splitting in the binding energy range, the spectrum indicated the existence of multiple chemical environments for Si atoms at the surface. To reveal the components, spectral curve-fittings were conducted using Doniach-Sunjic functions for various emission angles, θ's. The spin-orbit splitting and the branching ratio were fixed at 0.4 eV and 0.5, respectively. The Doniach-Sunjic width was fixed at 0.0425 eV to be consistent to the 0.085 eV reference Lorentzian width reported previously[\[44\]](#page-22-4). The curve-fit spectra are shown in Fig. 6(b-d), with parameters given in Table I. The results indicated four Si components, or four chemical sites, on the Al(111)3×3-Si surface. To analyze chemical environments at the Al(111)3×3-Si surface, the first-principles spectral simulations were calculated using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0). Figure 7(a) shows four prominent peaks in a simulated spectrum for the silicene layer on Al(111). Because of spin-orbit splitting, each Si 2p core-level appeared as a doublet state, and the simulations indicated two ![FIG. 6. Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface, taken at emission angles of (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with 150 eV photons. In (b-d), experimental spectra are dotted lines, while curve-fit spectra of the Si components are given by green and red curves, respectively. The curve-fit parameters are summarized in Table I.](path) TABLE I. Fitting parameters of the Si 2p core-level spectra for the Al(111)3×3-Si surface, shown in Fig. 6. The spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.6 eV, 0.0425 eV, and 0.5, respectively. Si components for the model structure. As depicted in the inset, one was attributed to Si atoms at atop Al sites and the other to bridge Al sites. While the calculations successfully linked the surface structure and the core-level spectra, the simulated results for the silicene layer were completely different from the experimental data (Fig. 6). Thus, it was necessity to consider a surface structure beyond a simple silicene layer for the Al(111)3×3-Si surface. ![FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦ , (c) θ = 30◦ , and (d) θ = 60◦ . In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b-d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.](path) Si sites in the Al-embedded silicene layer model (Fig. 8). As noted earlier, Si atoms in the surface layer can bond with different numbers of intralayer Si and/or Al atoms. For example, in Fig. 8(a), one Si atom used for the calculations was bonded with two Si atoms and one Al nearest neighbor (NN). Table II lists binding energies of the Si 2p3/<sup>2</sup> level for various structure models of silicene and Al-embedded silicene layers on Al(111). The core-level binding energies varied with respect to the number of NN Al atoms in the Si-Al bonding. Smaller NN values had higher binding energies, which was consistent with the silicene model results given in Table II. Moreover, there was a quantitative relationship between the binding energy and the NN value: 98.4±0.1eV (NN=3), 98.6±0.1eV (NN=2), 98.7±0.1eV (NN=1), and 98.8±0.1eV (NN=0). Because the model calculations in Fig. 8 considered only a single (atop) site on the Al substrate, differences in the substitution sites should be considered in TABLE III. Simulation parameters for the Si 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 7. The spectral components were given by Voigt functions with 0.085 eV Lorentzian widths and Gaussian widths given in the table. Spin-orbit splitting and a branching ratio were set at 0.6 eV and 0.5, respectively. The simulation was performed using OpenMX[\[34,](#page-21-8) [38\]](#page-22-0). On the basis of the calculated binding energies, the experimental spectra of the Al(111)3×3- Si surface were fairly well reproduced by five components, S1, S2, S2', S3 and S4, with the parameters in Table III. From the binding energies, S3 and S4 could be assigned to Si atoms that bonded only with NN Si atoms, while S1, S2, and S2' were also bonded to NN Al ![FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.](path) atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
Figure 1.5 Flexoelectric voltage generated in the cantilever around its resonance frequency.	# Measuring direct flexoelectricity at the nanoscale #### 1.1 Abstract Flexoelectricity is a property of all dielectric materials, where inhomogeneous strain induces electrical polarization. is effect becomes particularly prominent at the nanoscale where larger strain gradients can be obtained. While flexoelectric charges have been measured in mm-scale systems, direct measurements in nanoscale-thickness materials have not yet been achieved. Given that one of the most prominent applications of flexoelectricity is in nanoelectro-mechanical systems (NEMS), confirming the presence and magnitude of the effect at these scales is essential. is study presents the first-ever measurements of flexoelectricgenerated charges (direct effect) in nanoscale-thickness materials, using cantilevers with a 50 nm hafnium oxide layer. We confirm that the estimated flexoelectric coefficient from said measurements aligns with the values obtained from complementary experiments using the flexoelectric inverse effect. Additionally, by changing the cantilever geometry (modifying the width of the cantilevers), we demonstrate a 40% increase in the effective flexoelectric coefficient, explained by the interplay of different flexoelectric tensor components. These findings not only validate the presence of flexoelectric effects at the nanoscale but also open the possibility for full flexoelectric transduction of the motion in NEMS/MEMS devices. ### 1.2 Introduction Flexoelectricity is a property of dielectric materials where an inhomogeneous mechanical deformation, such as bending, induces electrical polarization (direct effect) [1]. Conversely, an applied electric field induces mechanical deformation (inverse effect) [2]. Unlike piezoelectricity, which requires non-centrosymmetric materials, flexoelectricity can occur in all dielectric materials [3]. is effect is particularly significant at the nanoscale, where large strain gradients are easier to achieve [4]. e simplest expression of this relationship is captured in Equation [1.1,](#page-1-0) where the flexoelectric coefficient is represented by , the material strain gradient is ⁄ and the generated polarization is [5]. A conceptual visualization of flexoelectricity can be illustrated using a simple ionic lattice [\(Figure 1.1a](#page-1-1)). When the material bends, the strain gradient causes a relative displacement between positively and negatively charged ions, leading to a local redistribution of charges and inducing a net polarization [\(Figure 1.1c](#page-1-1)). In materials where ionic movement is minimal, flexoelectricity can also be understood as a redistribution of electron density under strain gradients [6], enabling the effect in any type of dielectrics. ![<span id="page-1-1"></span>![](_page_1_Figure_3.jpeg)](path) Direct flexoelectricity in a mechanical structure [\(Figure 1.1b](#page-1-1)) generates surface charges across the beam when it bends along its length. To facilitate the reading of small charges, studies in the literature tend to use samples of large lateral dimensions (millimeters to centimeters). Consequently, to avoid breaking the sample when applying the load, the samples have thicknesses that range from 0.5 to 3 mm. Typically these measurements employ a three- or four-point bending tester, connected to an amplifier [7], [8], [9], [10], [11]. Other authors produce the strain in the material with a loudspeaker and measure charges with a lock-in amplifier. In these cases, the displacement of the loudspeaker is measured using a Displacement Voltage Ratio Transformer (DVRT) [1], [12], [13], [14] or an optical sensor [15], [16]. Additionally, some approaches utilize a piezoelectric shaker to flex the sample, with the displacement controlled by a Laser Doppler Vibrometer (LDV). In these cases, charge measurements are conducted using an amplifier [17], [18]. To date, no method reported in the literature has successfully measured direct flexoelectric effect in materials with nanoscale thicknesses. is is primarily due to the challenges in achieving sufficient sensitivity and accurately controlling strain gradients at such small scales [2]. Given that one of the most promising applications of flexoelectricity lies in Nano-Electro-Mechanical Systems (NEMS), where device dimensions are on the nanometer scale, confirming the presence and magnitude of the flexoelectric effect at these scales is crucial. is could enable the development of novel sensors, actuators, and energy-harvesting devices that use nanoscale flexoelectricity [19]. # 1.3 Methodology Our approach for measuring flexoelectric charges at the nanoscale starts by fabricating cantilevers with a thin (50 nm) layer of hafnium oxide, sandwiched between two platinum electrodes (20 nm). Hafnium oxide is selected due to its high dielectric constant (within microelectronic compatible materials), well-characterized coefficient [20], [21], and its wide availability in cleanroom facilities. e cantilevers measure 20 µm in length and vary in width from 5 µm to 50 µm [$Figure 1.2$](#page-2-0). Even though we favor narrower devices, we include a range of widths in this study to facilitate charge detection. ![<span id="page-2-0"></span>Figure 1.2 SEM images of fabricated cantilevers. a) Cantilever with a length of 20 µm and a width of 5 µm. b) Cantilever with a length of 20 µm and a width of 50 µm.](path) To maximize the flexoelectric response, we excite the cantilevers at their resonance frequency using a piezoelectric shaker under the cantilevers chip. Operating at resonance amplifies the mechanical displacement, improving the signal-to-noise ratio. Measurements are conducted under vacuum (3 · 10−4 ) to reduce air damping, further enhancing displacement amplitude. e generated flexoelectric charges are collected by platinum electrodes and amplified using a low-noise voltage amplifier (Sierra Amps [22]) placed in close proximity to minimize the load capacitance and signal loss. e experimental setup [$Figure 1.3$](#page-3-0) includes a Laser Doppler Vibrometer to monitor cantilever displacement. All signals are processed through a lock-in amplifier (Zurich Instruments UHFLI), which controls the shaker, reads the electrical charges, and monitors the cantilever's displacement. Importantly, the electrical charges and the vibrometer signal are not monitored simultaneously, but subsequently. ![<span id="page-3-0"></span>Figure 1.3 Schematic of the experimental setup to measure flexoelectric charges in nanoscale-thickness materials. e cantilever is actuated by a piezoelectric shaker at its resonance frequency under vacuum. A low-noise amplifier amplifies the generated flexoelectric charges () and parasitic charges ( ). A Laser Doppler Vibrometer monitors the displacement of the cantilever. All signals are processed in a lockin amplifier.](path) e first measurement quantifies the cantilever's displacement for a certain shaker actuation. By sweeping the excitation frequency around the cantilever's resonance frequency, we measure the displacement at the laser's measurement point [$Figure 1.4$](#page-4-0). However, our primary interest is the displacement at the cantilever's tip, which is essential for calculating the strain gradient (beam curvature). To extrapolate the tip displacement from the measured point, we analyze the thermomechanical noise and use a standard cantilever beam model. is method allows us to translate the measurements at any point along the cantilever to the tip [23], [24]. ![<span id="page-4-0"></span>Figure 1.4 Laser Doppler Vibrometer measurements of cantilever displacement around the resonance frequency. e displacement amplitude is linearly proportional to the actuation voltage applied to the shaker. ermomechanical noise measurements, conducted without actuation, are used to verify the laser position on the cantilever and translate the amplitude measurements to cantilever's tip displacement.](path) ![<span id="page-4-1"></span>Figure 1.5 Flexoelectric voltage generated in the cantilever around its resonance frequency. a) Normalized output voltage for a cantilever with a width of 40 µm and a length of 20 µm. We observe the flexoelectric charges superimposed on the background signal. b) Lorentzian signal corresponding to the cantilever's motion, obtained by subtracting the background signal from the output signal.](path) e second measurement evaluates the electrical charges generated by the flexoelectric effect. We perform the measurements without the laser, as its illumination can induce photoelectric effects and interfere with the flexoelectric charges [25]. [Figure 1.5](#page-4-1) shows the electrical measurements obtained from a cantilever with dimensions of 40 µm in width and 20 µm in length. In [Figure 1.5a](#page-4-1), the normalized transfer function is plotted as a function of frequency, revealing the cantilever's resonance peak superimposed on a background signal. is background arises from parasitic capacitance pathways between the shaker's actuation and the cantilever. To isolate the flexoelectric signal, we perform a linear baseline subtraction for both components in phase and in quadrature with the drive, to remove the background contribution, resulting in the cleaned data presented in [Figure 1.5b](#page-4-1). e extracted signal corresponds to the flexoelectric charges generated due to the cantilever's motion, which are linearly dependent on the shaker's actuation. By combining the measured electrical amplitudes () with the cantilever displacements at the tip () at various shaker actuations, we calculate the effective flexoelectric coefficient () using Equation [1.2.](#page-5-0) is calculation also involves the cantilever's width (), the capacitance of the measurement path leading to the low-noise amplifier ( ) and the derivative of the normalized n-mode shape (′ ()). For the cantilever's first mode, 1 ′ () = − 1.3765⁄, where L represents the length of the cantilever. Detailed derivation is provided in the supplementary material. #### 1.4 Results and discussion [Figure 1.6](#page-6-0) presents the effective flexoelectric coefficient for cantilevers of varying widths. We observe that increases with increasing width, reaching a saturation value for widths larger than 40 µm. is trend is attributed to a regime shift, as the cantilevers transition from the < condition to > . is change affects how the flexoelectric tensor components 11 and 12 [4] interact to determine the generated charges. [Table 1.1](#page-6-1) describes the theoretical asymptotic limits of the effective flexoelectric coefficient for both narrow and wide cantilevers (compared to the length). Detailed derivations are provided in the supplementary material. By making the ratio between the limits we obtain ,⁄, = 1⁄(1 − ), which depends solely on the material's Poisson's ratio (), and not on the flexoelectric tensor terms. For hafnium oxide, with = 0.3, the ratio evaluates to approximately 1.42. is theoretical prediction aligns well with the data i[n Figure 1.6,](#page-6-0) where increasing the width results in a 42% increase in . <span id="page-6-1"></span>Table 1.1 Asymptotical effective flexoelectric coefficients. Asymptotical limits of the effective flexoelectric coefficient for narrow and wide cantilevers. e ratio between these limits is independent of the flexoelectric tensor components, depending only on Poisson's ratio of the material. (t = Cantilever's thickness). ![<span id="page-6-0"></span>Figure 1.6 Effective flexoelectric coefficient for cantilevers with different widths. e growth in the effective flexoelectric coefficient is theoretically expected, as the interplay between the tensorial components depend on the width of the cantilever. e figure shows the asymptotical limits for the extreme cases of the width.](path) e direct flexoelectric measurements presented in this work [$Figure 1.6$](#page-6-0) align with the published coefficients we previously obtained from inverse flexoelectric measurements [20]. In that study, we report a value of = 105 ± 10 / for narrow cantilevers, which aligns closely with the direct flexoelectric measurements. # 1.5 Conclusion is study presents the first flexoelectric charge measurements in nanometrically thin materials. We employ microcantilevers with a 50 nm layer of hafnium oxide between platinum electrodes. Electrical signals are captured using a low-noise amplifier and they are compared with the cantilever displacement which is measured with a Laser Doppler Vibrometer. e results align with hafnium oxide coefficient values obtained through inverse methods. Additionally, we demonstrate that the effective flexoelectric coefficient increases with cantilever width (for a fixed length), showing the dependence on cantilever geometry. ese findings could help in the fabrication of next-generation NEMS devices. ## References - [8] P. Zubko, G. Catalan, A. Buckley, P. R. L. Welche, and J. F. Scott, "Strain-Gradient-Induced Polarization in SrTiO 3 Single Crystals," Phys. Rev. Lett., vol. 99, no. 16, p. 167601, Oct. 2007, doi: 10.1103/PhysRevLett.99.167601. - [10] J. Narvaez, S. Saremi, J. Hong, M. Stengel, and G. Catalan, "Large Flexoelectric Anisotropy in Paraelectric Barium Titanate," Phys. Rev. Lett., vol. 115, no. 3, p. 037601, Jul. 2015, doi: 10.1103/PhysRevLett.115.037601. - [12] W. Ma and L. E. Cross, "Flexoelectric polarization of barium strontium titanate in the paraelectric state," Applied Physics Letters, vol. 81, no. 18, pp. 3440–3442, Oct. 2002, doi: 10.1063/1.1518559. - [16] Y. Li, L. Shu, W. Huang, X. Jiang, and H. Wang, "Giant flexoelectricity in Ba0.6Sr0.4TiO3/Ni0.8Zn0.2Fe2O4 composite," Applied Physics Letters, vol. 105, no. 16, p. 162906, Oct. 2014, doi: 10.1063/1.4899060. - [17] W. Huang, K. Kim, S. Zhang, F.-G. Yuan, and X. Jiang, "Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers: Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers," Phys. Status Solidi RRL, vol. 5, no. 9, pp. 350–352, Sep. 2011, doi: 10.1002/pssr.201105326. - [18] S. Huang, T. Kim, D. Hou, D. Cann, J. L. Jones, and X. Jiang, "Flexoelectric characterization of BaTiO3-0.08Bi(Zn1/2Ti1/2)O3," Applied Physics Letters, vol. 110, no. 22, p. 222904, May 2017, doi: 10.1063/1.4984212. - [19] Q. Deng, M. Kammoun, A. Erturk, and P. Sharma, "Nanoscale flexoelectric energy harvesting," International Journal of Solids and Structures, vol. 51, no. 18, pp. 3218– 3225, Sep. 2014, doi: 10.1016/j.ijsolstr.2014.05.018. - [23] L. G. Villanueva, R. B. Karabalin, M. H. Matheny, D. Chi, J. E. Sader, and M. L. Roukes, "Nonlinearity in nanomechanical cantilevers," Phys. Rev. B, vol. 87, no. 2, p. 024304, Jan. 2013, doi: 10.1103/PhysRevB.87.024304. # 1.6 Supplementary Material #### 1.6.1 Flexoelectric formula derivation Derivation of the formula to convert the read voltage (after background subtraction) and cantilever tip displacement into effective flexoelectric coefficient. ![Figure 1.7 Equivalent circuit of the electronic setup. e cantilever is represented as a current source ( ), which has an equivalent parallel resistance () and capacitance ( ). e capacitance of the connector is represented by ( ), the input lownoise amplifier resistance ( ) and capacitance ( ).](path) $$\phi\_n(\mathbf{x}) = \frac{1}{2} \left[ \cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} (\sin(\beta\_n \mathbf{x}) - \sinh(\beta\_n \mathbf{x})) \right] \tag{1.12}$$ $$\phi\_n'(\mathbf{x}) = \frac{1}{2} \left[ -\beta\_n \cdot \sin(\beta\_n \mathbf{x}) - \beta\_n \cdot \sinh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} \cdot \beta\_n \cdot (\cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x})) \right] \tag{1.13}$$ #### 1.6.2 Measurement uncertainty evaluation Extended formula for the direct measurement of flexoelectricity, including all dependencies. $$\mu\_{direct} = \frac{\left(\mathcal{C}\_{device} + \mathcal{C}\_{connect} + \mathcal{C}\_{amplitude}\right) \cdot L}{1.3765 \cdot W} \cdot \frac{V\_{initial}}{Amp\_{sweep} \cdot \sqrt{\frac{4 \cdot Q \cdot K\_B T}{1 \cdot L \cdot W \cdot \Sigma \rho\_l t\_l \cdot (2\pi f\_R)^3 \cdot P \text{SD}/2}}} \tag{1.15}$$ ## 1.6.3 Flexoelectric tensor contributions to the effective flexoelectric coefficient ![∂z](path) ## 1.6.4 Equivalent flexoelectric charges for the measured voltages Here, just for curiosity, we calculate what are the equivalent flexoelectric charges for the measured flexoelectric voltages . Our methodology does not measure flexoelectric charges directly, we measure currents, which is an integration of the charges at a relatively high frequency (200 kHz). We found these calculations interesting to have an estimated electron charge produced by the flexoelectric effect for hafnium oxide.
Figure 1.6 Effective flexoelectric coefficient for cantilevers with different widths. e growth in the effective flexoelectric coefficient is theoretically expected, as the interplay between the tensorial components depend on the width of the cantilever. e figure shows the asymptotical limits for the extreme cases of the width.	# Measuring direct flexoelectricity at the nanoscale #### 1.1 Abstract Flexoelectricity is a property of all dielectric materials, where inhomogeneous strain induces electrical polarization. is effect becomes particularly prominent at the nanoscale where larger strain gradients can be obtained. While flexoelectric charges have been measured in mm-scale systems, direct measurements in nanoscale-thickness materials have not yet been achieved. Given that one of the most prominent applications of flexoelectricity is in nanoelectro-mechanical systems (NEMS), confirming the presence and magnitude of the effect at these scales is essential. is study presents the first-ever measurements of flexoelectricgenerated charges (direct effect) in nanoscale-thickness materials, using cantilevers with a 50 nm hafnium oxide layer. We confirm that the estimated flexoelectric coefficient from said measurements aligns with the values obtained from complementary experiments using the flexoelectric inverse effect. Additionally, by changing the cantilever geometry (modifying the width of the cantilevers), we demonstrate a 40% increase in the effective flexoelectric coefficient, explained by the interplay of different flexoelectric tensor components. These findings not only validate the presence of flexoelectric effects at the nanoscale but also open the possibility for full flexoelectric transduction of the motion in NEMS/MEMS devices. ### 1.2 Introduction Flexoelectricity is a property of dielectric materials where an inhomogeneous mechanical deformation, such as bending, induces electrical polarization (direct effect) [1]. Conversely, an applied electric field induces mechanical deformation (inverse effect) [2]. Unlike piezoelectricity, which requires non-centrosymmetric materials, flexoelectricity can occur in all dielectric materials [3]. is effect is particularly significant at the nanoscale, where large strain gradients are easier to achieve [4]. e simplest expression of this relationship is captured in Equation [1.1,](#page-1-0) where the flexoelectric coefficient is represented by , the material strain gradient is ⁄ and the generated polarization is [5]. A conceptual visualization of flexoelectricity can be illustrated using a simple ionic lattice [\(Figure 1.1a](#page-1-1)). When the material bends, the strain gradient causes a relative displacement between positively and negatively charged ions, leading to a local redistribution of charges and inducing a net polarization [\(Figure 1.1c](#page-1-1)). In materials where ionic movement is minimal, flexoelectricity can also be understood as a redistribution of electron density under strain gradients [6], enabling the effect in any type of dielectrics. ![<span id="page-1-1"></span>![](_page_1_Figure_3.jpeg)](path) Direct flexoelectricity in a mechanical structure [\(Figure 1.1b](#page-1-1)) generates surface charges across the beam when it bends along its length. To facilitate the reading of small charges, studies in the literature tend to use samples of large lateral dimensions (millimeters to centimeters). Consequently, to avoid breaking the sample when applying the load, the samples have thicknesses that range from 0.5 to 3 mm. Typically these measurements employ a three- or four-point bending tester, connected to an amplifier [7], [8], [9], [10], [11]. Other authors produce the strain in the material with a loudspeaker and measure charges with a lock-in amplifier. In these cases, the displacement of the loudspeaker is measured using a Displacement Voltage Ratio Transformer (DVRT) [1], [12], [13], [14] or an optical sensor [15], [16]. Additionally, some approaches utilize a piezoelectric shaker to flex the sample, with the displacement controlled by a Laser Doppler Vibrometer (LDV). In these cases, charge measurements are conducted using an amplifier [17], [18]. To date, no method reported in the literature has successfully measured direct flexoelectric effect in materials with nanoscale thicknesses. is is primarily due to the challenges in achieving sufficient sensitivity and accurately controlling strain gradients at such small scales [2]. Given that one of the most promising applications of flexoelectricity lies in Nano-Electro-Mechanical Systems (NEMS), where device dimensions are on the nanometer scale, confirming the presence and magnitude of the flexoelectric effect at these scales is crucial. is could enable the development of novel sensors, actuators, and energy-harvesting devices that use nanoscale flexoelectricity [19]. # 1.3 Methodology Our approach for measuring flexoelectric charges at the nanoscale starts by fabricating cantilevers with a thin (50 nm) layer of hafnium oxide, sandwiched between two platinum electrodes (20 nm). Hafnium oxide is selected due to its high dielectric constant (within microelectronic compatible materials), well-characterized coefficient [20], [21], and its wide availability in cleanroom facilities. e cantilevers measure 20 µm in length and vary in width from 5 µm to 50 µm [$Figure 1.2$](#page-2-0). Even though we favor narrower devices, we include a range of widths in this study to facilitate charge detection. ![<span id="page-2-0"></span>Figure 1.2 SEM images of fabricated cantilevers. a) Cantilever with a length of 20 µm and a width of 5 µm. b) Cantilever with a length of 20 µm and a width of 50 µm.](path) To maximize the flexoelectric response, we excite the cantilevers at their resonance frequency using a piezoelectric shaker under the cantilevers chip. Operating at resonance amplifies the mechanical displacement, improving the signal-to-noise ratio. Measurements are conducted under vacuum (3 · 10−4 ) to reduce air damping, further enhancing displacement amplitude. e generated flexoelectric charges are collected by platinum electrodes and amplified using a low-noise voltage amplifier (Sierra Amps [22]) placed in close proximity to minimize the load capacitance and signal loss. e experimental setup [$Figure 1.3$](#page-3-0) includes a Laser Doppler Vibrometer to monitor cantilever displacement. All signals are processed through a lock-in amplifier (Zurich Instruments UHFLI), which controls the shaker, reads the electrical charges, and monitors the cantilever's displacement. Importantly, the electrical charges and the vibrometer signal are not monitored simultaneously, but subsequently. ![<span id="page-3-0"></span>Figure 1.3 Schematic of the experimental setup to measure flexoelectric charges in nanoscale-thickness materials. e cantilever is actuated by a piezoelectric shaker at its resonance frequency under vacuum. A low-noise amplifier amplifies the generated flexoelectric charges () and parasitic charges ( ). A Laser Doppler Vibrometer monitors the displacement of the cantilever. All signals are processed in a lockin amplifier.](path) e first measurement quantifies the cantilever's displacement for a certain shaker actuation. By sweeping the excitation frequency around the cantilever's resonance frequency, we measure the displacement at the laser's measurement point [$Figure 1.4$](#page-4-0). However, our primary interest is the displacement at the cantilever's tip, which is essential for calculating the strain gradient (beam curvature). To extrapolate the tip displacement from the measured point, we analyze the thermomechanical noise and use a standard cantilever beam model. is method allows us to translate the measurements at any point along the cantilever to the tip [23], [24]. ![<span id="page-4-0"></span>Figure 1.4 Laser Doppler Vibrometer measurements of cantilever displacement around the resonance frequency. e displacement amplitude is linearly proportional to the actuation voltage applied to the shaker. ermomechanical noise measurements, conducted without actuation, are used to verify the laser position on the cantilever and translate the amplitude measurements to cantilever's tip displacement.](path) ![<span id="page-4-1"></span>Figure 1.5 Flexoelectric voltage generated in the cantilever around its resonance frequency. a) Normalized output voltage for a cantilever with a width of 40 µm and a length of 20 µm. We observe the flexoelectric charges superimposed on the background signal. b) Lorentzian signal corresponding to the cantilever's motion, obtained by subtracting the background signal from the output signal.](path) e second measurement evaluates the electrical charges generated by the flexoelectric effect. We perform the measurements without the laser, as its illumination can induce photoelectric effects and interfere with the flexoelectric charges [25]. [Figure 1.5](#page-4-1) shows the electrical measurements obtained from a cantilever with dimensions of 40 µm in width and 20 µm in length. In [Figure 1.5a](#page-4-1), the normalized transfer function is plotted as a function of frequency, revealing the cantilever's resonance peak superimposed on a background signal. is background arises from parasitic capacitance pathways between the shaker's actuation and the cantilever. To isolate the flexoelectric signal, we perform a linear baseline subtraction for both components in phase and in quadrature with the drive, to remove the background contribution, resulting in the cleaned data presented in [Figure 1.5b](#page-4-1). e extracted signal corresponds to the flexoelectric charges generated due to the cantilever's motion, which are linearly dependent on the shaker's actuation. By combining the measured electrical amplitudes () with the cantilever displacements at the tip () at various shaker actuations, we calculate the effective flexoelectric coefficient () using Equation [1.2.](#page-5-0) is calculation also involves the cantilever's width (), the capacitance of the measurement path leading to the low-noise amplifier ( ) and the derivative of the normalized n-mode shape (′ ()). For the cantilever's first mode, 1 ′ () = − 1.3765⁄, where L represents the length of the cantilever. Detailed derivation is provided in the supplementary material. #### 1.4 Results and discussion [Figure 1.6](#page-6-0) presents the effective flexoelectric coefficient for cantilevers of varying widths. We observe that increases with increasing width, reaching a saturation value for widths larger than 40 µm. is trend is attributed to a regime shift, as the cantilevers transition from the < condition to > . is change affects how the flexoelectric tensor components 11 and 12 [4] interact to determine the generated charges. [Table 1.1](#page-6-1) describes the theoretical asymptotic limits of the effective flexoelectric coefficient for both narrow and wide cantilevers (compared to the length). Detailed derivations are provided in the supplementary material. By making the ratio between the limits we obtain ,⁄, = 1⁄(1 − ), which depends solely on the material's Poisson's ratio (), and not on the flexoelectric tensor terms. For hafnium oxide, with = 0.3, the ratio evaluates to approximately 1.42. is theoretical prediction aligns well with the data i[n Figure 1.6,](#page-6-0) where increasing the width results in a 42% increase in . <span id="page-6-1"></span>Table 1.1 Asymptotical effective flexoelectric coefficients. Asymptotical limits of the effective flexoelectric coefficient for narrow and wide cantilevers. e ratio between these limits is independent of the flexoelectric tensor components, depending only on Poisson's ratio of the material. (t = Cantilever's thickness). ![<span id="page-6-0"></span>Figure 1.6 Effective flexoelectric coefficient for cantilevers with different widths. e growth in the effective flexoelectric coefficient is theoretically expected, as the interplay between the tensorial components depend on the width of the cantilever. e figure shows the asymptotical limits for the extreme cases of the width.](path) e direct flexoelectric measurements presented in this work [$Figure 1.6$](#page-6-0) align with the published coefficients we previously obtained from inverse flexoelectric measurements [20]. In that study, we report a value of = 105 ± 10 / for narrow cantilevers, which aligns closely with the direct flexoelectric measurements. # 1.5 Conclusion is study presents the first flexoelectric charge measurements in nanometrically thin materials. We employ microcantilevers with a 50 nm layer of hafnium oxide between platinum electrodes. Electrical signals are captured using a low-noise amplifier and they are compared with the cantilever displacement which is measured with a Laser Doppler Vibrometer. e results align with hafnium oxide coefficient values obtained through inverse methods. Additionally, we demonstrate that the effective flexoelectric coefficient increases with cantilever width (for a fixed length), showing the dependence on cantilever geometry. ese findings could help in the fabrication of next-generation NEMS devices. ## References - [8] P. Zubko, G. Catalan, A. Buckley, P. R. L. Welche, and J. F. Scott, "Strain-Gradient-Induced Polarization in SrTiO 3 Single Crystals," Phys. Rev. Lett., vol. 99, no. 16, p. 167601, Oct. 2007, doi: 10.1103/PhysRevLett.99.167601. - [10] J. Narvaez, S. Saremi, J. Hong, M. Stengel, and G. Catalan, "Large Flexoelectric Anisotropy in Paraelectric Barium Titanate," Phys. Rev. Lett., vol. 115, no. 3, p. 037601, Jul. 2015, doi: 10.1103/PhysRevLett.115.037601. - [12] W. Ma and L. E. Cross, "Flexoelectric polarization of barium strontium titanate in the paraelectric state," Applied Physics Letters, vol. 81, no. 18, pp. 3440–3442, Oct. 2002, doi: 10.1063/1.1518559. - [16] Y. Li, L. Shu, W. Huang, X. Jiang, and H. Wang, "Giant flexoelectricity in Ba0.6Sr0.4TiO3/Ni0.8Zn0.2Fe2O4 composite," Applied Physics Letters, vol. 105, no. 16, p. 162906, Oct. 2014, doi: 10.1063/1.4899060. - [17] W. Huang, K. Kim, S. Zhang, F.-G. Yuan, and X. Jiang, "Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers: Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers," Phys. Status Solidi RRL, vol. 5, no. 9, pp. 350–352, Sep. 2011, doi: 10.1002/pssr.201105326. - [18] S. Huang, T. Kim, D. Hou, D. Cann, J. L. Jones, and X. Jiang, "Flexoelectric characterization of BaTiO3-0.08Bi(Zn1/2Ti1/2)O3," Applied Physics Letters, vol. 110, no. 22, p. 222904, May 2017, doi: 10.1063/1.4984212. - [19] Q. Deng, M. Kammoun, A. Erturk, and P. Sharma, "Nanoscale flexoelectric energy harvesting," International Journal of Solids and Structures, vol. 51, no. 18, pp. 3218– 3225, Sep. 2014, doi: 10.1016/j.ijsolstr.2014.05.018. - [23] L. G. Villanueva, R. B. Karabalin, M. H. Matheny, D. Chi, J. E. Sader, and M. L. Roukes, "Nonlinearity in nanomechanical cantilevers," Phys. Rev. B, vol. 87, no. 2, p. 024304, Jan. 2013, doi: 10.1103/PhysRevB.87.024304. # 1.6 Supplementary Material #### 1.6.1 Flexoelectric formula derivation Derivation of the formula to convert the read voltage (after background subtraction) and cantilever tip displacement into effective flexoelectric coefficient. ![Figure 1.7 Equivalent circuit of the electronic setup. e cantilever is represented as a current source ( ), which has an equivalent parallel resistance () and capacitance ( ). e capacitance of the connector is represented by ( ), the input lownoise amplifier resistance ( ) and capacitance ( ).](path) $$\phi\_n(\mathbf{x}) = \frac{1}{2} \left[ \cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} (\sin(\beta\_n \mathbf{x}) - \sinh(\beta\_n \mathbf{x})) \right] \tag{1.12}$$ $$\phi\_n'(\mathbf{x}) = \frac{1}{2} \left[ -\beta\_n \cdot \sin(\beta\_n \mathbf{x}) - \beta\_n \cdot \sinh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} \cdot \beta\_n \cdot (\cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x})) \right] \tag{1.13}$$ #### 1.6.2 Measurement uncertainty evaluation Extended formula for the direct measurement of flexoelectricity, including all dependencies. $$\mu\_{direct} = \frac{\left(\mathcal{C}\_{device} + \mathcal{C}\_{connect} + \mathcal{C}\_{amplitude}\right) \cdot L}{1.3765 \cdot W} \cdot \frac{V\_{initial}}{Amp\_{sweep} \cdot \sqrt{\frac{4 \cdot Q \cdot K\_B T}{1 \cdot L \cdot W \cdot \Sigma \rho\_l t\_l \cdot (2\pi f\_R)^3 \cdot P \text{SD}/2}}} \tag{1.15}$$ ## 1.6.3 Flexoelectric tensor contributions to the effective flexoelectric coefficient ![∂z](path) ## 1.6.4 Equivalent flexoelectric charges for the measured voltages Here, just for curiosity, we calculate what are the equivalent flexoelectric charges for the measured flexoelectric voltages . Our methodology does not measure flexoelectric charges directly, we measure currents, which is an integration of the charges at a relatively high frequency (200 kHz). We found these calculations interesting to have an estimated electron charge produced by the flexoelectric effect for hafnium oxide.
∂z μeff,<sup>n</sup> = −ν · μ11 + μ12(1 − ν) 1 − ν μeff,<sup>w</sup> <sup>=</sup> μ11 · <sup>ν</sup> ∂z ν − <sup>1</sup> <sup>+</sup>μ12	# Measuring direct flexoelectricity at the nanoscale #### 1.1 Abstract Flexoelectricity is a property of all dielectric materials, where inhomogeneous strain induces electrical polarization. is effect becomes particularly prominent at the nanoscale where larger strain gradients can be obtained. While flexoelectric charges have been measured in mm-scale systems, direct measurements in nanoscale-thickness materials have not yet been achieved. Given that one of the most prominent applications of flexoelectricity is in nanoelectro-mechanical systems (NEMS), confirming the presence and magnitude of the effect at these scales is essential. is study presents the first-ever measurements of flexoelectricgenerated charges (direct effect) in nanoscale-thickness materials, using cantilevers with a 50 nm hafnium oxide layer. We confirm that the estimated flexoelectric coefficient from said measurements aligns with the values obtained from complementary experiments using the flexoelectric inverse effect. Additionally, by changing the cantilever geometry (modifying the width of the cantilevers), we demonstrate a 40% increase in the effective flexoelectric coefficient, explained by the interplay of different flexoelectric tensor components. These findings not only validate the presence of flexoelectric effects at the nanoscale but also open the possibility for full flexoelectric transduction of the motion in NEMS/MEMS devices. ### 1.2 Introduction Flexoelectricity is a property of dielectric materials where an inhomogeneous mechanical deformation, such as bending, induces electrical polarization (direct effect) [1]. Conversely, an applied electric field induces mechanical deformation (inverse effect) [2]. Unlike piezoelectricity, which requires non-centrosymmetric materials, flexoelectricity can occur in all dielectric materials [3]. is effect is particularly significant at the nanoscale, where large strain gradients are easier to achieve [4]. e simplest expression of this relationship is captured in Equation [1.1,](#page-1-0) where the flexoelectric coefficient is represented by , the material strain gradient is ⁄ and the generated polarization is [5]. A conceptual visualization of flexoelectricity can be illustrated using a simple ionic lattice [\(Figure 1.1a](#page-1-1)). When the material bends, the strain gradient causes a relative displacement between positively and negatively charged ions, leading to a local redistribution of charges and inducing a net polarization [\(Figure 1.1c](#page-1-1)). In materials where ionic movement is minimal, flexoelectricity can also be understood as a redistribution of electron density under strain gradients [6], enabling the effect in any type of dielectrics. ## 1.6.4 Equivalent flexoelectric charges for the measured voltages Here, just for curiosity, we calculate what are the equivalent flexoelectric charges for the measured flexoelectric voltages . Our methodology does not measure flexoelectric charges directly, we measure currents, which is an integration of the charges at a relatively high frequency (200 kHz). We found these calculations interesting to have an estimated electron charge produced by the flexoelectric effect for hafnium oxide.
Figure 1.2 SEM images of fabricated cantilevers. a) Cantilever with a length of 20 µm and a width of 5 µm. b) Cantilever with a length of 20 µm and a width of 50 µm.	# Measuring direct flexoelectricity at the nanoscale #### 1.1 Abstract Flexoelectricity is a property of all dielectric materials, where inhomogeneous strain induces electrical polarization. is effect becomes particularly prominent at the nanoscale where larger strain gradients can be obtained. While flexoelectric charges have been measured in mm-scale systems, direct measurements in nanoscale-thickness materials have not yet been achieved. Given that one of the most prominent applications of flexoelectricity is in nanoelectro-mechanical systems (NEMS), confirming the presence and magnitude of the effect at these scales is essential. is study presents the first-ever measurements of flexoelectricgenerated charges (direct effect) in nanoscale-thickness materials, using cantilevers with a 50 nm hafnium oxide layer. We confirm that the estimated flexoelectric coefficient from said measurements aligns with the values obtained from complementary experiments using the flexoelectric inverse effect. Additionally, by changing the cantilever geometry (modifying the width of the cantilevers), we demonstrate a 40% increase in the effective flexoelectric coefficient, explained by the interplay of different flexoelectric tensor components. These findings not only validate the presence of flexoelectric effects at the nanoscale but also open the possibility for full flexoelectric transduction of the motion in NEMS/MEMS devices. ### 1.2 Introduction Flexoelectricity is a property of dielectric materials where an inhomogeneous mechanical deformation, such as bending, induces electrical polarization (direct effect) [1]. Conversely, an applied electric field induces mechanical deformation (inverse effect) [2]. Unlike piezoelectricity, which requires non-centrosymmetric materials, flexoelectricity can occur in all dielectric materials [3]. is effect is particularly significant at the nanoscale, where large strain gradients are easier to achieve [4]. e simplest expression of this relationship is captured in Equation [1.1,](#page-1-0) where the flexoelectric coefficient is represented by , the material strain gradient is ⁄ and the generated polarization is [5]. A conceptual visualization of flexoelectricity can be illustrated using a simple ionic lattice [\(Figure 1.1a](#page-1-1)). When the material bends, the strain gradient causes a relative displacement between positively and negatively charged ions, leading to a local redistribution of charges and inducing a net polarization [\(Figure 1.1c](#page-1-1)). In materials where ionic movement is minimal, flexoelectricity can also be understood as a redistribution of electron density under strain gradients [6], enabling the effect in any type of dielectrics. ![<span id="page-1-1"></span>![](_page_1_Figure_3.jpeg)](path) Direct flexoelectricity in a mechanical structure [\(Figure 1.1b](#page-1-1)) generates surface charges across the beam when it bends along its length. To facilitate the reading of small charges, studies in the literature tend to use samples of large lateral dimensions (millimeters to centimeters). Consequently, to avoid breaking the sample when applying the load, the samples have thicknesses that range from 0.5 to 3 mm. Typically these measurements employ a three- or four-point bending tester, connected to an amplifier [7], [8], [9], [10], [11]. Other authors produce the strain in the material with a loudspeaker and measure charges with a lock-in amplifier. In these cases, the displacement of the loudspeaker is measured using a Displacement Voltage Ratio Transformer (DVRT) [1], [12], [13], [14] or an optical sensor [15], [16]. Additionally, some approaches utilize a piezoelectric shaker to flex the sample, with the displacement controlled by a Laser Doppler Vibrometer (LDV). In these cases, charge measurements are conducted using an amplifier [17], [18]. To date, no method reported in the literature has successfully measured direct flexoelectric effect in materials with nanoscale thicknesses. is is primarily due to the challenges in achieving sufficient sensitivity and accurately controlling strain gradients at such small scales [2]. Given that one of the most promising applications of flexoelectricity lies in Nano-Electro-Mechanical Systems (NEMS), where device dimensions are on the nanometer scale, confirming the presence and magnitude of the flexoelectric effect at these scales is crucial. is could enable the development of novel sensors, actuators, and energy-harvesting devices that use nanoscale flexoelectricity [19]. # 1.3 Methodology Our approach for measuring flexoelectric charges at the nanoscale starts by fabricating cantilevers with a thin (50 nm) layer of hafnium oxide, sandwiched between two platinum electrodes (20 nm). Hafnium oxide is selected due to its high dielectric constant (within microelectronic compatible materials), well-characterized coefficient [20], [21], and its wide availability in cleanroom facilities. e cantilevers measure 20 µm in length and vary in width from 5 µm to 50 µm [$Figure 1.2$](#page-2-0). Even though we favor narrower devices, we include a range of widths in this study to facilitate charge detection. ![<span id="page-2-0"></span>Figure 1.2 SEM images of fabricated cantilevers. a) Cantilever with a length of 20 µm and a width of 5 µm. b) Cantilever with a length of 20 µm and a width of 50 µm.](path) To maximize the flexoelectric response, we excite the cantilevers at their resonance frequency using a piezoelectric shaker under the cantilevers chip. Operating at resonance amplifies the mechanical displacement, improving the signal-to-noise ratio. Measurements are conducted under vacuum (3 · 10−4 ) to reduce air damping, further enhancing displacement amplitude. e generated flexoelectric charges are collected by platinum electrodes and amplified using a low-noise voltage amplifier (Sierra Amps [22]) placed in close proximity to minimize the load capacitance and signal loss. e experimental setup [$Figure 1.3$](#page-3-0) includes a Laser Doppler Vibrometer to monitor cantilever displacement. All signals are processed through a lock-in amplifier (Zurich Instruments UHFLI), which controls the shaker, reads the electrical charges, and monitors the cantilever's displacement. Importantly, the electrical charges and the vibrometer signal are not monitored simultaneously, but subsequently. ![<span id="page-3-0"></span>Figure 1.3 Schematic of the experimental setup to measure flexoelectric charges in nanoscale-thickness materials. e cantilever is actuated by a piezoelectric shaker at its resonance frequency under vacuum. A low-noise amplifier amplifies the generated flexoelectric charges () and parasitic charges ( ). A Laser Doppler Vibrometer monitors the displacement of the cantilever. All signals are processed in a lockin amplifier.](path) e first measurement quantifies the cantilever's displacement for a certain shaker actuation. By sweeping the excitation frequency around the cantilever's resonance frequency, we measure the displacement at the laser's measurement point [$Figure 1.4$](#page-4-0). However, our primary interest is the displacement at the cantilever's tip, which is essential for calculating the strain gradient (beam curvature). To extrapolate the tip displacement from the measured point, we analyze the thermomechanical noise and use a standard cantilever beam model. is method allows us to translate the measurements at any point along the cantilever to the tip [23], [24]. ![<span id="page-4-0"></span>Figure 1.4 Laser Doppler Vibrometer measurements of cantilever displacement around the resonance frequency. e displacement amplitude is linearly proportional to the actuation voltage applied to the shaker. ermomechanical noise measurements, conducted without actuation, are used to verify the laser position on the cantilever and translate the amplitude measurements to cantilever's tip displacement.](path) ![<span id="page-4-1"></span>Figure 1.5 Flexoelectric voltage generated in the cantilever around its resonance frequency. a) Normalized output voltage for a cantilever with a width of 40 µm and a length of 20 µm. We observe the flexoelectric charges superimposed on the background signal. b) Lorentzian signal corresponding to the cantilever's motion, obtained by subtracting the background signal from the output signal.](path) e second measurement evaluates the electrical charges generated by the flexoelectric effect. We perform the measurements without the laser, as its illumination can induce photoelectric effects and interfere with the flexoelectric charges [25]. [Figure 1.5](#page-4-1) shows the electrical measurements obtained from a cantilever with dimensions of 40 µm in width and 20 µm in length. In [Figure 1.5a](#page-4-1), the normalized transfer function is plotted as a function of frequency, revealing the cantilever's resonance peak superimposed on a background signal. is background arises from parasitic capacitance pathways between the shaker's actuation and the cantilever. To isolate the flexoelectric signal, we perform a linear baseline subtraction for both components in phase and in quadrature with the drive, to remove the background contribution, resulting in the cleaned data presented in [Figure 1.5b](#page-4-1). e extracted signal corresponds to the flexoelectric charges generated due to the cantilever's motion, which are linearly dependent on the shaker's actuation. By combining the measured electrical amplitudes () with the cantilever displacements at the tip () at various shaker actuations, we calculate the effective flexoelectric coefficient () using Equation [1.2.](#page-5-0) is calculation also involves the cantilever's width (), the capacitance of the measurement path leading to the low-noise amplifier ( ) and the derivative of the normalized n-mode shape (′ ()). For the cantilever's first mode, 1 ′ () = − 1.3765⁄, where L represents the length of the cantilever. Detailed derivation is provided in the supplementary material. #### 1.4 Results and discussion [Figure 1.6](#page-6-0) presents the effective flexoelectric coefficient for cantilevers of varying widths. We observe that increases with increasing width, reaching a saturation value for widths larger than 40 µm. is trend is attributed to a regime shift, as the cantilevers transition from the < condition to > . is change affects how the flexoelectric tensor components 11 and 12 [4] interact to determine the generated charges. [Table 1.1](#page-6-1) describes the theoretical asymptotic limits of the effective flexoelectric coefficient for both narrow and wide cantilevers (compared to the length). Detailed derivations are provided in the supplementary material. By making the ratio between the limits we obtain ,⁄, = 1⁄(1 − ), which depends solely on the material's Poisson's ratio (), and not on the flexoelectric tensor terms. For hafnium oxide, with = 0.3, the ratio evaluates to approximately 1.42. is theoretical prediction aligns well with the data i[n Figure 1.6,](#page-6-0) where increasing the width results in a 42% increase in . <span id="page-6-1"></span>Table 1.1 Asymptotical effective flexoelectric coefficients. Asymptotical limits of the effective flexoelectric coefficient for narrow and wide cantilevers. e ratio between these limits is independent of the flexoelectric tensor components, depending only on Poisson's ratio of the material. (t = Cantilever's thickness). ![<span id="page-6-0"></span>Figure 1.6 Effective flexoelectric coefficient for cantilevers with different widths. e growth in the effective flexoelectric coefficient is theoretically expected, as the interplay between the tensorial components depend on the width of the cantilever. e figure shows the asymptotical limits for the extreme cases of the width.](path) e direct flexoelectric measurements presented in this work [$Figure 1.6$](#page-6-0) align with the published coefficients we previously obtained from inverse flexoelectric measurements [20]. In that study, we report a value of = 105 ± 10 / for narrow cantilevers, which aligns closely with the direct flexoelectric measurements. # 1.5 Conclusion is study presents the first flexoelectric charge measurements in nanometrically thin materials. We employ microcantilevers with a 50 nm layer of hafnium oxide between platinum electrodes. Electrical signals are captured using a low-noise amplifier and they are compared with the cantilever displacement which is measured with a Laser Doppler Vibrometer. e results align with hafnium oxide coefficient values obtained through inverse methods. Additionally, we demonstrate that the effective flexoelectric coefficient increases with cantilever width (for a fixed length), showing the dependence on cantilever geometry. ese findings could help in the fabrication of next-generation NEMS devices. ## References - [8] P. Zubko, G. Catalan, A. Buckley, P. R. L. Welche, and J. F. Scott, "Strain-Gradient-Induced Polarization in SrTiO 3 Single Crystals," Phys. Rev. Lett., vol. 99, no. 16, p. 167601, Oct. 2007, doi: 10.1103/PhysRevLett.99.167601. - [10] J. Narvaez, S. Saremi, J. Hong, M. Stengel, and G. Catalan, "Large Flexoelectric Anisotropy in Paraelectric Barium Titanate," Phys. Rev. Lett., vol. 115, no. 3, p. 037601, Jul. 2015, doi: 10.1103/PhysRevLett.115.037601. - [12] W. Ma and L. E. Cross, "Flexoelectric polarization of barium strontium titanate in the paraelectric state," Applied Physics Letters, vol. 81, no. 18, pp. 3440–3442, Oct. 2002, doi: 10.1063/1.1518559. - [16] Y. Li, L. Shu, W. Huang, X. Jiang, and H. Wang, "Giant flexoelectricity in Ba0.6Sr0.4TiO3/Ni0.8Zn0.2Fe2O4 composite," Applied Physics Letters, vol. 105, no. 16, p. 162906, Oct. 2014, doi: 10.1063/1.4899060. - [17] W. Huang, K. Kim, S. Zhang, F.-G. Yuan, and X. Jiang, "Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers: Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers," Phys. Status Solidi RRL, vol. 5, no. 9, pp. 350–352, Sep. 2011, doi: 10.1002/pssr.201105326. - [18] S. Huang, T. Kim, D. Hou, D. Cann, J. L. Jones, and X. Jiang, "Flexoelectric characterization of BaTiO3-0.08Bi(Zn1/2Ti1/2)O3," Applied Physics Letters, vol. 110, no. 22, p. 222904, May 2017, doi: 10.1063/1.4984212. - [19] Q. Deng, M. Kammoun, A. Erturk, and P. Sharma, "Nanoscale flexoelectric energy harvesting," International Journal of Solids and Structures, vol. 51, no. 18, pp. 3218– 3225, Sep. 2014, doi: 10.1016/j.ijsolstr.2014.05.018. - [23] L. G. Villanueva, R. B. Karabalin, M. H. Matheny, D. Chi, J. E. Sader, and M. L. Roukes, "Nonlinearity in nanomechanical cantilevers," Phys. Rev. B, vol. 87, no. 2, p. 024304, Jan. 2013, doi: 10.1103/PhysRevB.87.024304. # 1.6 Supplementary Material #### 1.6.1 Flexoelectric formula derivation Derivation of the formula to convert the read voltage (after background subtraction) and cantilever tip displacement into effective flexoelectric coefficient. ![Figure 1.7 Equivalent circuit of the electronic setup. e cantilever is represented as a current source ( ), which has an equivalent parallel resistance () and capacitance ( ). e capacitance of the connector is represented by ( ), the input lownoise amplifier resistance ( ) and capacitance ( ).](path) $$\phi\_n(\mathbf{x}) = \frac{1}{2} \left[ \cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} (\sin(\beta\_n \mathbf{x}) - \sinh(\beta\_n \mathbf{x})) \right] \tag{1.12}$$ $$\phi\_n'(\mathbf{x}) = \frac{1}{2} \left[ -\beta\_n \cdot \sin(\beta\_n \mathbf{x}) - \beta\_n \cdot \sinh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} \cdot \beta\_n \cdot (\cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x})) \right] \tag{1.13}$$ #### 1.6.2 Measurement uncertainty evaluation Extended formula for the direct measurement of flexoelectricity, including all dependencies. $$\mu\_{direct} = \frac{\left(\mathcal{C}\_{device} + \mathcal{C}\_{connect} + \mathcal{C}\_{amplitude}\right) \cdot L}{1.3765 \cdot W} \cdot \frac{V\_{initial}}{Amp\_{sweep} \cdot \sqrt{\frac{4 \cdot Q \cdot K\_B T}{1 \cdot L \cdot W \cdot \Sigma \rho\_l t\_l \cdot (2\pi f\_R)^3 \cdot P \text{SD}/2}}} \tag{1.15}$$ ## 1.6.3 Flexoelectric tensor contributions to the effective flexoelectric coefficient ![∂z](path) ## 1.6.4 Equivalent flexoelectric charges for the measured voltages Here, just for curiosity, we calculate what are the equivalent flexoelectric charges for the measured flexoelectric voltages . Our methodology does not measure flexoelectric charges directly, we measure currents, which is an integration of the charges at a relatively high frequency (200 kHz). We found these calculations interesting to have an estimated electron charge produced by the flexoelectric effect for hafnium oxide.
Direct flexoelectricity in a mechanical structure [\(Figure 1.1b](#page-1-1))	# Measuring direct flexoelectricity at the nanoscale #### 1.1 Abstract Flexoelectricity is a property of all dielectric materials, where inhomogeneous strain induces electrical polarization. is effect becomes particularly prominent at the nanoscale where larger strain gradients can be obtained. While flexoelectric charges have been measured in mm-scale systems, direct measurements in nanoscale-thickness materials have not yet been achieved. Given that one of the most prominent applications of flexoelectricity is in nanoelectro-mechanical systems (NEMS), confirming the presence and magnitude of the effect at these scales is essential. is study presents the first-ever measurements of flexoelectricgenerated charges (direct effect) in nanoscale-thickness materials, using cantilevers with a 50 nm hafnium oxide layer. We confirm that the estimated flexoelectric coefficient from said measurements aligns with the values obtained from complementary experiments using the flexoelectric inverse effect. Additionally, by changing the cantilever geometry (modifying the width of the cantilevers), we demonstrate a 40% increase in the effective flexoelectric coefficient, explained by the interplay of different flexoelectric tensor components. These findings not only validate the presence of flexoelectric effects at the nanoscale but also open the possibility for full flexoelectric transduction of the motion in NEMS/MEMS devices. ### 1.2 Introduction Flexoelectricity is a property of dielectric materials where an inhomogeneous mechanical deformation, such as bending, induces electrical polarization (direct effect) [1]. Conversely, an applied electric field induces mechanical deformation (inverse effect) [2]. Unlike piezoelectricity, which requires non-centrosymmetric materials, flexoelectricity can occur in all dielectric materials [3]. is effect is particularly significant at the nanoscale, where large strain gradients are easier to achieve [4]. e simplest expression of this relationship is captured in Equation [1.1,](#page-1-0) where the flexoelectric coefficient is represented by , the material strain gradient is ⁄ and the generated polarization is [5]. A conceptual visualization of flexoelectricity can be illustrated using a simple ionic lattice [\(Figure 1.1a](#page-1-1)). When the material bends, the strain gradient causes a relative displacement between positively and negatively charged ions, leading to a local redistribution of charges and inducing a net polarization [\(Figure 1.1c](#page-1-1)). In materials where ionic movement is minimal, flexoelectricity can also be understood as a redistribution of electron density under strain gradients [6], enabling the effect in any type of dielectrics. generates surface charges across the beam when it bends along its length. To facilitate the reading of small charges, studies in the literature tend to use samples of large lateral dimensions (millimeters to centimeters). Consequently, to avoid breaking the sample when applying the load, the samples have thicknesses that range from 0.5 to 3 mm. Typically these measurements employ a three- or four-point bending tester, connected to an amplifier [7], [8], [9], [10], [11]. Other authors produce the strain in the material with a loudspeaker and measure charges with a lock-in amplifier. In these cases, the displacement of the loudspeaker is measured using a Displacement Voltage Ratio Transformer (DVRT) [1], [12], [13], [14] or an optical sensor [15], [16]. Additionally, some approaches utilize a piezoelectric shaker to flex the sample, with the displacement controlled by a Laser Doppler Vibrometer (LDV). In these cases, charge measurements are conducted using an amplifier [17], [18]. To date, no method reported in the literature has successfully measured direct flexoelectric effect in materials with nanoscale thicknesses. is is primarily due to the challenges in achieving sufficient sensitivity and accurately controlling strain gradients at such small scales [2]. Given that one of the most promising applications of flexoelectricity lies in Nano-Electro-Mechanical Systems (NEMS), where device dimensions are on the nanometer scale, confirming the presence and magnitude of the flexoelectric effect at these scales is crucial. is could enable the development of novel sensors, actuators, and energy-harvesting devices that use nanoscale flexoelectricity [19]. # 1.3 Methodology Our approach for measuring flexoelectric charges at the nanoscale starts by fabricating cantilevers with a thin (50 nm) layer of hafnium oxide, sandwiched between two platinum electrodes (20 nm). Hafnium oxide is selected due to its high dielectric constant (within microelectronic compatible materials), well-characterized coefficient [20], [21], and its wide availability in cleanroom facilities. e cantilevers measure 20 µm in length and vary in width from 5 µm to 50 µm [$Figure 1.2$](#page-2-0). Even though we favor narrower devices, we include a range of widths in this study to facilitate charge detection. ![<span id="page-2-0"></span>Figure 1.2 SEM images of fabricated cantilevers. a) Cantilever with a length of 20 µm and a width of 5 µm. b) Cantilever with a length of 20 µm and a width of 50 µm.](path) To maximize the flexoelectric response, we excite the cantilevers at their resonance frequency using a piezoelectric shaker under the cantilevers chip. Operating at resonance amplifies the mechanical displacement, improving the signal-to-noise ratio. Measurements are conducted under vacuum (3 · 10−4 ) to reduce air damping, further enhancing displacement amplitude. e generated flexoelectric charges are collected by platinum electrodes and amplified using a low-noise voltage amplifier (Sierra Amps [22]) placed in close proximity to minimize the load capacitance and signal loss. e experimental setup [$Figure 1.3$](#page-3-0) includes a Laser Doppler Vibrometer to monitor cantilever displacement. All signals are processed through a lock-in amplifier (Zurich Instruments UHFLI), which controls the shaker, reads the electrical charges, and monitors the cantilever's displacement. Importantly, the electrical charges and the vibrometer signal are not monitored simultaneously, but subsequently. ![<span id="page-3-0"></span>Figure 1.3 Schematic of the experimental setup to measure flexoelectric charges in nanoscale-thickness materials. e cantilever is actuated by a piezoelectric shaker at its resonance frequency under vacuum. A low-noise amplifier amplifies the generated flexoelectric charges () and parasitic charges ( ). A Laser Doppler Vibrometer monitors the displacement of the cantilever. All signals are processed in a lockin amplifier.](path) e first measurement quantifies the cantilever's displacement for a certain shaker actuation. By sweeping the excitation frequency around the cantilever's resonance frequency, we measure the displacement at the laser's measurement point [$Figure 1.4$](#page-4-0). However, our primary interest is the displacement at the cantilever's tip, which is essential for calculating the strain gradient (beam curvature). To extrapolate the tip displacement from the measured point, we analyze the thermomechanical noise and use a standard cantilever beam model. is method allows us to translate the measurements at any point along the cantilever to the tip [23], [24]. ![<span id="page-4-0"></span>Figure 1.4 Laser Doppler Vibrometer measurements of cantilever displacement around the resonance frequency. e displacement amplitude is linearly proportional to the actuation voltage applied to the shaker. ermomechanical noise measurements, conducted without actuation, are used to verify the laser position on the cantilever and translate the amplitude measurements to cantilever's tip displacement.](path) ![<span id="page-4-1"></span>Figure 1.5 Flexoelectric voltage generated in the cantilever around its resonance frequency. a) Normalized output voltage for a cantilever with a width of 40 µm and a length of 20 µm. We observe the flexoelectric charges superimposed on the background signal. b) Lorentzian signal corresponding to the cantilever's motion, obtained by subtracting the background signal from the output signal.](path) e second measurement evaluates the electrical charges generated by the flexoelectric effect. We perform the measurements without the laser, as its illumination can induce photoelectric effects and interfere with the flexoelectric charges [25]. [Figure 1.5](#page-4-1) shows the electrical measurements obtained from a cantilever with dimensions of 40 µm in width and 20 µm in length. In [Figure 1.5a](#page-4-1), the normalized transfer function is plotted as a function of frequency, revealing the cantilever's resonance peak superimposed on a background signal. is background arises from parasitic capacitance pathways between the shaker's actuation and the cantilever. To isolate the flexoelectric signal, we perform a linear baseline subtraction for both components in phase and in quadrature with the drive, to remove the background contribution, resulting in the cleaned data presented in [Figure 1.5b](#page-4-1). e extracted signal corresponds to the flexoelectric charges generated due to the cantilever's motion, which are linearly dependent on the shaker's actuation. By combining the measured electrical amplitudes () with the cantilever displacements at the tip () at various shaker actuations, we calculate the effective flexoelectric coefficient () using Equation [1.2.](#page-5-0) is calculation also involves the cantilever's width (), the capacitance of the measurement path leading to the low-noise amplifier ( ) and the derivative of the normalized n-mode shape (′ ()). For the cantilever's first mode, 1 ′ () = − 1.3765⁄, where L represents the length of the cantilever. Detailed derivation is provided in the supplementary material. #### 1.4 Results and discussion [Figure 1.6](#page-6-0) presents the effective flexoelectric coefficient for cantilevers of varying widths. We observe that increases with increasing width, reaching a saturation value for widths larger than 40 µm. is trend is attributed to a regime shift, as the cantilevers transition from the < condition to > . is change affects how the flexoelectric tensor components 11 and 12 [4] interact to determine the generated charges. [Table 1.1](#page-6-1) describes the theoretical asymptotic limits of the effective flexoelectric coefficient for both narrow and wide cantilevers (compared to the length). Detailed derivations are provided in the supplementary material. By making the ratio between the limits we obtain ,⁄, = 1⁄(1 − ), which depends solely on the material's Poisson's ratio (), and not on the flexoelectric tensor terms. For hafnium oxide, with = 0.3, the ratio evaluates to approximately 1.42. is theoretical prediction aligns well with the data i[n Figure 1.6,](#page-6-0) where increasing the width results in a 42% increase in . <span id="page-6-1"></span>Table 1.1 Asymptotical effective flexoelectric coefficients. Asymptotical limits of the effective flexoelectric coefficient for narrow and wide cantilevers. e ratio between these limits is independent of the flexoelectric tensor components, depending only on Poisson's ratio of the material. (t = Cantilever's thickness). ![<span id="page-6-0"></span>Figure 1.6 Effective flexoelectric coefficient for cantilevers with different widths. e growth in the effective flexoelectric coefficient is theoretically expected, as the interplay between the tensorial components depend on the width of the cantilever. e figure shows the asymptotical limits for the extreme cases of the width.](path) e direct flexoelectric measurements presented in this work [$Figure 1.6$](#page-6-0) align with the published coefficients we previously obtained from inverse flexoelectric measurements [20]. In that study, we report a value of = 105 ± 10 / for narrow cantilevers, which aligns closely with the direct flexoelectric measurements. # 1.5 Conclusion is study presents the first flexoelectric charge measurements in nanometrically thin materials. We employ microcantilevers with a 50 nm layer of hafnium oxide between platinum electrodes. Electrical signals are captured using a low-noise amplifier and they are compared with the cantilever displacement which is measured with a Laser Doppler Vibrometer. e results align with hafnium oxide coefficient values obtained through inverse methods. Additionally, we demonstrate that the effective flexoelectric coefficient increases with cantilever width (for a fixed length), showing the dependence on cantilever geometry. ese findings could help in the fabrication of next-generation NEMS devices. ## References - [8] P. Zubko, G. Catalan, A. Buckley, P. R. L. Welche, and J. F. Scott, "Strain-Gradient-Induced Polarization in SrTiO 3 Single Crystals," Phys. Rev. Lett., vol. 99, no. 16, p. 167601, Oct. 2007, doi: 10.1103/PhysRevLett.99.167601. - [10] J. Narvaez, S. Saremi, J. Hong, M. Stengel, and G. Catalan, "Large Flexoelectric Anisotropy in Paraelectric Barium Titanate," Phys. Rev. Lett., vol. 115, no. 3, p. 037601, Jul. 2015, doi: 10.1103/PhysRevLett.115.037601. - [12] W. Ma and L. E. Cross, "Flexoelectric polarization of barium strontium titanate in the paraelectric state," Applied Physics Letters, vol. 81, no. 18, pp. 3440–3442, Oct. 2002, doi: 10.1063/1.1518559. - [16] Y. Li, L. Shu, W. Huang, X. Jiang, and H. Wang, "Giant flexoelectricity in Ba0.6Sr0.4TiO3/Ni0.8Zn0.2Fe2O4 composite," Applied Physics Letters, vol. 105, no. 16, p. 162906, Oct. 2014, doi: 10.1063/1.4899060. - [17] W. Huang, K. Kim, S. Zhang, F.-G. Yuan, and X. Jiang, "Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers: Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers," Phys. Status Solidi RRL, vol. 5, no. 9, pp. 350–352, Sep. 2011, doi: 10.1002/pssr.201105326. - [18] S. Huang, T. Kim, D. Hou, D. Cann, J. L. Jones, and X. Jiang, "Flexoelectric characterization of BaTiO3-0.08Bi(Zn1/2Ti1/2)O3," Applied Physics Letters, vol. 110, no. 22, p. 222904, May 2017, doi: 10.1063/1.4984212. - [19] Q. Deng, M. Kammoun, A. Erturk, and P. Sharma, "Nanoscale flexoelectric energy harvesting," International Journal of Solids and Structures, vol. 51, no. 18, pp. 3218– 3225, Sep. 2014, doi: 10.1016/j.ijsolstr.2014.05.018. - [23] L. G. Villanueva, R. B. Karabalin, M. H. Matheny, D. Chi, J. E. Sader, and M. L. Roukes, "Nonlinearity in nanomechanical cantilevers," Phys. Rev. B, vol. 87, no. 2, p. 024304, Jan. 2013, doi: 10.1103/PhysRevB.87.024304. # 1.6 Supplementary Material #### 1.6.1 Flexoelectric formula derivation Derivation of the formula to convert the read voltage (after background subtraction) and cantilever tip displacement into effective flexoelectric coefficient. ![Figure 1.7 Equivalent circuit of the electronic setup. e cantilever is represented as a current source ( ), which has an equivalent parallel resistance () and capacitance ( ). e capacitance of the connector is represented by ( ), the input lownoise amplifier resistance ( ) and capacitance ( ).](path) $$\phi\_n(\mathbf{x}) = \frac{1}{2} \left[ \cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} (\sin(\beta\_n \mathbf{x}) - \sinh(\beta\_n \mathbf{x})) \right] \tag{1.12}$$ $$\phi\_n'(\mathbf{x}) = \frac{1}{2} \left[ -\beta\_n \cdot \sin(\beta\_n \mathbf{x}) - \beta\_n \cdot \sinh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} \cdot \beta\_n \cdot (\cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x})) \right] \tag{1.13}$$ #### 1.6.2 Measurement uncertainty evaluation Extended formula for the direct measurement of flexoelectricity, including all dependencies. $$\mu\_{direct} = \frac{\left(\mathcal{C}\_{device} + \mathcal{C}\_{connect} + \mathcal{C}\_{amplitude}\right) \cdot L}{1.3765 \cdot W} \cdot \frac{V\_{initial}}{Amp\_{sweep} \cdot \sqrt{\frac{4 \cdot Q \cdot K\_B T}{1 \cdot L \cdot W \cdot \Sigma \rho\_l t\_l \cdot (2\pi f\_R)^3 \cdot P \text{SD}/2}}} \tag{1.15}$$ ## 1.6.3 Flexoelectric tensor contributions to the effective flexoelectric coefficient ![∂z](path) ## 1.6.4 Equivalent flexoelectric charges for the measured voltages Here, just for curiosity, we calculate what are the equivalent flexoelectric charges for the measured flexoelectric voltages . Our methodology does not measure flexoelectric charges directly, we measure currents, which is an integration of the charges at a relatively high frequency (200 kHz). We found these calculations interesting to have an estimated electron charge produced by the flexoelectric effect for hafnium oxide.
Figure D.2: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm² scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm² measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ∥ = 40% and ϵ⊥ = 40%, respectively. In each of the maps, the coating direction goes from bottom left to top right.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [$II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5$](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [$1$](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [$5$](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [$3$](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [$B1$](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm² scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm² measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ∥ = 40% and ϵ⊥ = 40%, respectively. In each of the maps, the coating direction goes from bottom left to top right.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [$II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5$](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [$1$](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [$5$](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [$3$](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [$B1$](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [$II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5$](#page-8-1). ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [$1$](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [$5$](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [$3$](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [$B1$](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [$II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5$](#page-8-1). Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [$5$](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [$3$](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [$B1$](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵmax = 40 %), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2.3]). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [$II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5$](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [$1$](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [$5$](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [$3$](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [$B1$](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure 4: Electrical four‐point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB‐silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R∥ and R⊥ (multiplied by film thickness to compensate geometric variations, see eq. [$1$](#page-3-2) in Section [II.2.1](#page-3-0)), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R⊥ were calculated from all local pairs of R∥ and R⊥ of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [$II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5$](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [$1$](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [$5$](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [$3$](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [$B1$](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm² per material state). Scans of the unstretched 7 vol% CB‐film and one scan on the 7 vol% CB‐film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB‐ and matrix-dominated regions.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [$II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5$](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [$1$](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [$5$](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [$3$](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [$B1$](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure D.1: Visualization of PeakForce QNM signals as derived from force‐distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real‐time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [$II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5$](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [$1$](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [$5$](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [$3$](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [$B1$](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [$II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5$](#page-8-1). Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10⁻² s⁻¹ strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [$II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5$](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [$1$](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [$1$](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [$4$](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [$5$](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [$3$](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [$B1$](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.

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FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9\(](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9\)](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [\(11\)](#page-6-3) and [\(12\)](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [\(11\)](#page-6-3) compared to the fully decomposed pathway [\(12\)](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [\(8\)](#page-6-4) − [\(12\)](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 \(2022\).](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 5: (a) The energy profile for H² formation on the kite-like Fe₂O₃ cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. <span id="page-8-0"></span>![FIG. 1: The most stable structures of (Fe2O3)<sup>n</sup> clusters with n = 1 − 4. The total spin S and the binding energy Eb, of the clusters are shown in inserts.](path) ## <span id="page-8-1"></span>B. Ammonia adsorption on (Fe2O3)<sup>n</sup> clusters Adsorption of ammonia on (Fe2O3)<sup>n</sup> clusters is a crucial initial step in the whole dehydrogenation process. Figure [2](#page-9-0) demonstrates the most stable adsorption configurations of the NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters with n = 1−4. The corresponding free energies of adsorption and Fe−N bond distances are shown in Table [I](#page-10-0) at 0 K. Our calculations show that the adsorption of NH<sup>3</sup> on the smallest Fe2O<sup>3</sup> cluster is the most stable among all cluster sizes considered in this study, with an adsorption free energy of -33.68 kcal/mol. This finding is corroborated by Mulliken charge analysis, which shows that more electrons are shared between the lone pair of the N atom and the 3d orbitals of Fe2<sup>+</sup> for <sup>n</sup> <sup>=</sup> 1. On the other hand, for larger cluster sizes with <sup>n</sup> <sup>=</sup> <sup>2</sup>−4, which primarily contain Fe3+, the electron density is more localized over the bonding region, as also reported by Sierka et al.[55](#page-28-0). Therefore, bonding occurs with the nitrogen lone pair. Our theoretical analysis indicates that the adsorption energy ∆Gads of ammonia on (Fe2O3)<sup>n</sup> clusters decreases from n = 1 to n = 3, followed by a slight increase for n = 4. A similar trend in the change of adsorption energy with cluster size was reported by Shulan Zhou et al.[56](#page-28-1) for Run@CNT systems. We also compared the adsorption energy of NH<sup>3</sup> on different metal and metal oxides in Table [I.](#page-10-0) The obtained NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters are about 10 kcal/mol higher than the data reported by Zhang et al. for the Ru(0001) surface[57](#page-28-2). Moreover, the adsorption of NH<sup>3</sup> and NO<sup>x</sup> on the γ-Fe2O3(111) surface was studied by Wei Huang et al.[58](#page-28-3) using periodic density functional calculations. They calculated adsorption energies on octahedral and tetrahedral sites of γ-Fe2O3(111) to be -2.13 kcal/mol and -21.68 kcal/mol, respectively. Similarly, our calculated NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters for n = 3 and n = 4 are close to the data reported by Wei Huang et al.,[58](#page-28-3) as adsorption of NH<sup>3</sup> on the three-coordinated Fe3<sup>+</sup> site resembles the tetrahedral site of γ-Fe2O3(111), while the adsorption on the four-coordinated Fe3<sup>+</sup> site resembles the octahedral site of γ-Fe2O3. <span id="page-9-0"></span>![FIG. 2: The optimized geometries of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> for n = 1−4; N−Fe distances (Å) are shown in parentheses along with the partial atomic charges on neighboring atoms. The total spin S of the clusters is shown in inserts.](path) As mentioned above, the calculated adsorption energies indicate that the adsorption of an NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters (n = 1−4) weakens as the cluster size increases from n = 1 to n = 3. In industrial processes, the dehydrogenation of ammonia typically occurs at high temperatures, often in the range of 400◦C to 700◦C, depending on the specific catalysts and conditions used. Therefore, it is important to determine the range of temperatures at which ammonia adsorption on (Fe2O3)<sup>n</sup> remains stable. Figure S5 demonstrates the temperature dependence of ∆Gads in the range from 0 K to 1200 K for the most stable adsorption configurations of NH<sup>3</sup> on (Fe2O3)n clusters (n = 1 − 4). The negative values of ∆Gads correspond to stable adsorption. As seen in Fig. S5, NH<sup>3</sup> adsorbed on the smallest Fe2O<sup>3</sup> cluster is stable across the entire temperature range of 0 K to 1200 K. However, for larger cluster sizes, ammonia adsorption becomes energetically unfavorable at temperatures of 1107 (K), 961 (K), and 1000 (K) for n = 2,3, and 4, respectively. ## C. NH<sup>3</sup> decomposition on Fe2O<sup>3</sup> In this section, we discuss the complete NH<sup>3</sup> decomposition and H<sup>2</sup> formation reactions [\(7\)](#page-6-0) - [\(12\)](#page-7-0) on the smallest considered cluster, Fe2O3, at room temperature, T=298.15 K, explored by the AFIR method. This method allows for the automatic exploration of the full reaction path network, systematically accounting for the variety of possible isomer structures and adsorption sites. This is an important approach in nanocatalysis because it has been demonstrated that the most stable structures are not always the most reactive. Therefore, a systematic search for reaction pathways that accounts for the contributions of low-energy isomers is required to accurately describe the catalytic properties of clusters at finite temperatures.[49](#page-27-10) To illustrate the isomer and reaction-site effects, we explicitly consider two different isomers of the Fe2O<sup>3</sup> cluster: the most stable kite-like structure with one terminal oxygen atom, and the linear structure isomer with two terminal oxygen atoms which is 6.24 kcal/mol less stable (see Fig. S1). The kite-like structure possess two type of catalytically active metal centers - two-coordinated and three-coordinated Fe sites. Therefore we consider adsorption and decomposition of NH<sup>3</sup> molecule on both of them. Figure [3\(](#page-13-0)a) demonstrates that the adsorption of NH<sup>3</sup> on the kite-like Fe2O<sup>3</sup> cluster is an exothermic reaction, occurring at both the two-coordinated and three-coordinated Fe sites. The adsorption free energies are -26.98 kcal/mol for the two-coordinated Fe site (intermediate I′ 1 1) and -11.29 kcal/mol for the three-coordinated Fe site (intermediate I′′ 1 1), respectively. The optimized structures of all intermediates (I) and transition satates (T) along the reaction pathways are shown in Fig. [3\(](#page-13-0)b) and [4\(](#page-14-0)b), for the kite-like and linear clusters, respectively. Here the lower index corresponds to the cluster size n, while the numbering corresponds to the order of intermedeates (transition states) along the reaction path. As discussed in the previous section, the most stable adsorption site for NH<sup>3</sup> is the two-coordinated Fe site, with an Fe−N bond length of 2.11 Å. In contrast, the Fe−N bond length at the three-coordinated Fe site is 2.16 Å. These findings are supported by the fact that NH<sup>3</sup> adsorption highly depends on the local geometry and electronic structure of the catalyst. In the case of the Fe2O<sup>3</sup> kite-like structure, the first dehydrogenation reaction is the second step in the reaction mechanism, occurring after adsorption with activation barriers of 26.98 kcal/mol and 22.12 kcal/mol through the reaction paths I′ 1 1-T′ 1 1-I′ 1 2 and I′′ 1 1-T′′ 1 1-I′′ 1 2, respectively. The reactions on these two-coodrinated and three-coordinated active sites are exothermic by 16.31 kcal/mol and 7.53 kcal/mol, respectively. However, the first dehydrogenation of NH<sup>3</sup> on the lineartype structure Fig. [4\(](#page-14-0)a) occurs with smaller activation barrier of 16.22 kcal/mol via the reaction path I<sup>L</sup> 1 1 - T<sup>L</sup> 1 1 - I<sup>L</sup> 1 2, demonstrating that the less stable linear isomer is more reactive. The role of Fe2O<sup>3</sup> isomer structure on NH<sup>3</sup> adsorption and first hydrogen atom transfer was previousely studied by Chaoyue Xie et al.[60](#page-28-5) They performed DFT-D3 calculations on the adsorption mechanisms of different molecules (NH3, NO, O2) on activated carbon (AC) supported iron-based catalysts FexOy/AC. The calculated adsorption electronic energies of NH<sup>3</sup> were -37.4 kcal/mol and -53.7 kcal/mol on different isomers of Fe2O3/AC, and the first hydrogen atom transfer had an activation barrier of 15.5 kcal/mol. Similarly, the adsorption and dehydrogenation of ammonia on different metal oxides were investigated by Erdtman and co-workers[62](#page-28-7) for the application of gas sensors. They reported that the adsorption energy of NH<sup>3</sup> on the RuO2(110) surface is -38.24 kcal/mol, and the first N−H bond cleavage had an activation energy barrier of 17.45 kcal/mol. The third step of the NH<sup>3</sup> dehydrogenation reaction [\(9\)](#page-6-1) involves the dissociation of the adsorbed NH∗ 2 intermediate into NH∗ and H∗ species. In this step, the abstracted hydrogen atom transfers to one of the oxygen atoms in the cluster. Figure [3\(](#page-13-0)a) demonstrates, that in the case of the kite-like structure the energy barriers for this step are 43.91 kcal/mol and 34.51 kcal/mol, corresponding to the reaction paths I′ 1 2 - T′ 1 2 - I′ 1 3 and I′′ 1 2 - T′′ 1 2 - I′′ 1 3, respectively. In the fourth step [\(10\)](#page-6-2), the adsorbed NH∗ intermediate further dissociates into N∗ and H∗ species as shown in Fig. [3\(](#page-13-0)a). The reaction barriers associated with this step are 46.98 kcal/mol and 8.95 kcal/mol for the two-coordinated and three-coordinated reaction paths, respectively. The decomposition of NH<sup>3</sup> on kite-like structures becomes endothermic starting from the third step [\(9\)](#page-6-1). Our calculations reveal that NH<sup>3</sup> dehydrogenation has a high energy barrier when the NH<sup>3</sup> molecule is adsorbed at a two-coordinated Fe site, which is the most stable adsorption site. On the other hand, dehydrogenation of the adsorbed NH<sup>3</sup> at a three-coordinated Fe site has a considerably lower activation barrier of 8.95 kcal/mol for the reaction step [\(10\)](#page-6-2). Overall, for the NH<sup>3</sup> decomposition reaction on the kite-like Fe2O<sup>3</sup> structure, with initial NH<sup>3</sup> adsorption on the two-coordinated Fe atom, the rate-limiting step is the fourth reaction [\(10\)](#page-6-2), with a barrier of 46.98 kcal/mol. Alternatively, for the less favorable NH<sup>3</sup> adsorption on the threecoordinated Fe atom, the rate-limiting step is the third reaction step [\(9\)](#page-6-1), with a barrier of 34.51 kcal/mol. The reaction pathway calculated for NH<sup>3</sup> decomposition on the linear-type Fe2O<sup>3</sup> isomer is shown in Fig. [4\(](#page-14-0)a), and respective intermediate and transition state structures are shown in Fig. [4\(](#page-14-0)b). Since this structure consists of two iron atoms connected through a central oxygen, each containing a terminal oxygen, the reaction mechanism differs slightly from that of the kite-like isomer. For instance, in the third step of the reaction, the second hydrogen from the adsorbed NH∗ 2 intermediate is transferred to the second terminal oxygen. The energy barrier for this step on the linear-type structure is 23.8 kcal/mol, as shown in the reaction path (I<sup>L</sup> 1 2 - T<sup>L</sup> 1 2 - I<sup>L</sup> 1 3) in Fig. [4a](#page-14-0). The fourth step on this isomer is not straightforward, involving the central oxygen atom breaking its bond with one of the neighboring iron atoms while forming an Fe − N − Fe bridge. This process leads to two different intermediates: the formation of the adsorbed H2O ∗ and the transfer of a hydrogen atom from one side of the Fe − N − Fe bridge to the other. Subsequently, the final dehydrogenation step from the NH∗ intermediate occurs, with an activation energy barrier of 34.76 kcal/mol. <span id="page-13-0"></span>![FIG. 3: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ reaction path on the kite-like isomer of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-14-0"></span>![FIG. 4: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> + → N ∗ + 3H∗ reaction path on the linear-type isomet of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) As a next step we consider possible H<sup>2</sup> formation via reactions [\(11\)](#page-6-3) and [\(12\)](#page-7-0) on the kite-like and linear isomers of Fe2O<sup>3</sup> cluster. The possible pathways for H<sup>2</sup> formation in the case of the most stable ammonia adsorption on the two-coordinated site (I′ intermediates) of the kite-like Fe2O<sup>3</sup> isomer are shown in Fig. [5\(](#page-15-0)a), while the corresponding structures of the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [5\(](#page-15-0)b). Note that H<sup>2</sup> formation can occur after partial decomposition of ammonia in reaction [\(11\)](#page-6-3), starting from intermediate (I<sup>I</sup> 1 3) via the path I′ 1 3 - T′ 1 6 - I′ 1 7 - T′ 1 7 - I′ 1 8. On the other hand, H<sup>2</sup> formation can occur via full decomposition of ammonia in reaction [\(12\)](#page-7-0), through the intermediate (II 1 4) via the path I′ 1 4 - T′ 1 4 - I′ 1 5 - T′ 1 5 - I′ 1 6. In both cases, the reaction pathways include breaking one O−H bond and forming an Fe−H bond. The H<sup>2</sup> formation barriers through these intermediates are 89.74 kcal/mol and 92.49 kcal/mol, respectively. From these results, we conclude that H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> structure is more favorable via reaction [\(11\)](#page-6-3), with the NH<sup>∗</sup> intermediate remaining adsorbed on the cluster. The H<sup>2</sup> formation reaction, starting from (I<sup>I</sup> 1 4), is the rate-limiting step in molecular hydrogen formation on the kite Fe2O<sup>3</sup> cluster. Similarly, the H<sup>2</sup> formation reaction pathways on the linear-type structure of Fe2O<sup>3</sup> are shown in Fig. [6\(](#page-16-0)a), while the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [6\(](#page-16-0)b). The H<sup>2</sup> formation through the NH<sup>∗</sup> intermediate (I<sup>L</sup> 1 4) via the reaction path I<sup>L</sup> 1 4 - T<sup>L</sup> 1 8 - I<sup>L</sup> 1 9 - T<sup>L</sup> 1 9 - I<sup>L</sup> 1 10 has an energy barrier of 79.99 kcal/mol. On the other hand H<sup>2</sup> formation through intermediate (I<sup>L</sup> 1 6) via reaction path I<sup>L</sup> 1 6 - T<sup>L</sup> 1 6 - I<sup>L</sup> 1 7 - T<sup>L</sup> 1 7 - I<sup>L</sup> 1 8 has an activation energy of 70.84 kcal/mol, which is about 10 kcal/mol lower energy than reaction path through intermediate (I<sup>L</sup> 1 4). Overall, on the basis of our calculated reaction pathways for H<sup>2</sup> formation show similar pattern for both kite-type and linear-type Fe2O3, where H<sup>2</sup> formation in reactions [\(11\)](#page-6-3) and [\(12\)](#page-7-0) take place via breaking one of O−H bond and forming intermediate Fe−H bond. However, from both thermodynamic and kinetic perspectives, H<sup>2</sup> formation on the two types of Fe2O<sup>3</sup> structures varies. Reaction [\(11\)](#page-6-3) is more favorable on the kite-like structure, while reaction [\(12\)](#page-7-0) is more favorable on the linear structure. This highlights that the rate-limiting step for H<sup>2</sup> formation is highly dependent on the catalyst's structure. <span id="page-15-0"></span>![FIG. 5: (a) The energy profile for H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-16-0"></span>![FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [\(11\)](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [\(11\)](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [\(12\)](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [\(10\)](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8\(](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8\)](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [\(10\)](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [\(11](#page-6-3) and [12\)](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [\(11\)](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [\(11\)](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9\(](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9\)](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [\(11\)](#page-6-3) and [\(12\)](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [\(11\)](#page-6-3) compared to the fully decomposed pathway [\(12\)](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [\(8\)](#page-6-4) − [\(12\)](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 \(2022\).](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 10: Reaction barrier (∆G ‡ ) for NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions on (Fe2O3)<sup>n</sup> (n = 1−4) clusters.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 \(2022\).](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 2: The optimized geometries of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> for n = 1−4; N−Fe distances (Å) are shown in parentheses along with the partial atomic charges on neighboring atoms. The total spin S of the clusters is shown in inserts.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. As mentioned above, the calculated adsorption energies indicate that the adsorption of an NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters (n = 1−4) weakens as the cluster size increases from n = 1 to n = 3. In industrial processes, the dehydrogenation of ammonia typically occurs at high temperatures, often in the range of 400◦C to 700◦C, depending on the specific catalysts and conditions used. Therefore, it is important to determine the range of temperatures at which ammonia adsorption on (Fe2O3)<sup>n</sup> remains stable. Figure S5 demonstrates the temperature dependence of ∆Gads in the range from 0 K to 1200 K for the most stable adsorption configurations of NH<sup>3</sup> on (Fe2O3)n clusters (n = 1 − 4). The negative values of ∆Gads correspond to stable adsorption. As seen in Fig. S5, NH<sup>3</sup> adsorbed on the smallest Fe2O<sup>3</sup> cluster is stable across the entire temperature range of 0 K to 1200 K. However, for larger cluster sizes, ammonia adsorption becomes energetically unfavorable at temperatures of 1107 (K), 961 (K), and 1000 (K) for n = 2,3, and 4, respectively. ## C. NH<sup>3</sup> decomposition on Fe2O<sup>3</sup> In this section, we discuss the complete NH<sup>3</sup> decomposition and H<sup>2</sup> formation reactions [\(7\)](#page-6-0) - [\(12\)](#page-7-0) on the smallest considered cluster, Fe2O3, at room temperature, T=298.15 K, explored by the AFIR method. This method allows for the automatic exploration of the full reaction path network, systematically accounting for the variety of possible isomer structures and adsorption sites. This is an important approach in nanocatalysis because it has been demonstrated that the most stable structures are not always the most reactive. Therefore, a systematic search for reaction pathways that accounts for the contributions of low-energy isomers is required to accurately describe the catalytic properties of clusters at finite temperatures.[49](#page-27-10) To illustrate the isomer and reaction-site effects, we explicitly consider two different isomers of the Fe2O<sup>3</sup> cluster: the most stable kite-like structure with one terminal oxygen atom, and the linear structure isomer with two terminal oxygen atoms which is 6.24 kcal/mol less stable (see Fig. S1). The kite-like structure possess two type of catalytically active metal centers - two-coordinated and three-coordinated Fe sites. Therefore we consider adsorption and decomposition of NH<sup>3</sup> molecule on both of them. Figure [3\(](#page-13-0)a) demonstrates that the adsorption of NH<sup>3</sup> on the kite-like Fe2O<sup>3</sup> cluster is an exothermic reaction, occurring at both the two-coordinated and three-coordinated Fe sites. The adsorption free energies are -26.98 kcal/mol for the two-coordinated Fe site (intermediate I′ 1 1) and -11.29 kcal/mol for the three-coordinated Fe site (intermediate I′′ 1 1), respectively. The optimized structures of all intermediates (I) and transition satates (T) along the reaction pathways are shown in Fig. [3\(](#page-13-0)b) and [4\(](#page-14-0)b), for the kite-like and linear clusters, respectively. Here the lower index corresponds to the cluster size n, while the numbering corresponds to the order of intermedeates (transition states) along the reaction path. As discussed in the previous section, the most stable adsorption site for NH<sup>3</sup> is the two-coordinated Fe site, with an Fe−N bond length of 2.11 Å. In contrast, the Fe−N bond length at the three-coordinated Fe site is 2.16 Å. These findings are supported by the fact that NH<sup>3</sup> adsorption highly depends on the local geometry and electronic structure of the catalyst. In the case of the Fe2O<sup>3</sup> kite-like structure, the first dehydrogenation reaction is the second step in the reaction mechanism, occurring after adsorption with activation barriers of 26.98 kcal/mol and 22.12 kcal/mol through the reaction paths I′ 1 1-T′ 1 1-I′ 1 2 and I′′ 1 1-T′′ 1 1-I′′ 1 2, respectively. The reactions on these two-coodrinated and three-coordinated active sites are exothermic by 16.31 kcal/mol and 7.53 kcal/mol, respectively. However, the first dehydrogenation of NH<sup>3</sup> on the lineartype structure Fig. [4\(](#page-14-0)a) occurs with smaller activation barrier of 16.22 kcal/mol via the reaction path I<sup>L</sup> 1 1 - T<sup>L</sup> 1 1 - I<sup>L</sup> 1 2, demonstrating that the less stable linear isomer is more reactive. The role of Fe2O<sup>3</sup> isomer structure on NH<sup>3</sup> adsorption and first hydrogen atom transfer was previousely studied by Chaoyue Xie et al.[60](#page-28-5) They performed DFT-D3 calculations on the adsorption mechanisms of different molecules (NH3, NO, O2) on activated carbon (AC) supported iron-based catalysts FexOy/AC. The calculated adsorption electronic energies of NH<sup>3</sup> were -37.4 kcal/mol and -53.7 kcal/mol on different isomers of Fe2O3/AC, and the first hydrogen atom transfer had an activation barrier of 15.5 kcal/mol. Similarly, the adsorption and dehydrogenation of ammonia on different metal oxides were investigated by Erdtman and co-workers[62](#page-28-7) for the application of gas sensors. They reported that the adsorption energy of NH<sup>3</sup> on the RuO2(110) surface is -38.24 kcal/mol, and the first N−H bond cleavage had an activation energy barrier of 17.45 kcal/mol. The third step of the NH<sup>3</sup> dehydrogenation reaction [\(9\)](#page-6-1) involves the dissociation of the adsorbed NH∗ 2 intermediate into NH∗ and H∗ species. In this step, the abstracted hydrogen atom transfers to one of the oxygen atoms in the cluster. Figure [3\(](#page-13-0)a) demonstrates, that in the case of the kite-like structure the energy barriers for this step are 43.91 kcal/mol and 34.51 kcal/mol, corresponding to the reaction paths I′ 1 2 - T′ 1 2 - I′ 1 3 and I′′ 1 2 - T′′ 1 2 - I′′ 1 3, respectively. In the fourth step [\(10\)](#page-6-2), the adsorbed NH∗ intermediate further dissociates into N∗ and H∗ species as shown in Fig. [3\(](#page-13-0)a). The reaction barriers associated with this step are 46.98 kcal/mol and 8.95 kcal/mol for the two-coordinated and three-coordinated reaction paths, respectively. The decomposition of NH<sup>3</sup> on kite-like structures becomes endothermic starting from the third step [\(9\)](#page-6-1). Our calculations reveal that NH<sup>3</sup> dehydrogenation has a high energy barrier when the NH<sup>3</sup> molecule is adsorbed at a two-coordinated Fe site, which is the most stable adsorption site. On the other hand, dehydrogenation of the adsorbed NH<sup>3</sup> at a three-coordinated Fe site has a considerably lower activation barrier of 8.95 kcal/mol for the reaction step [\(10\)](#page-6-2). Overall, for the NH<sup>3</sup> decomposition reaction on the kite-like Fe2O<sup>3</sup> structure, with initial NH<sup>3</sup> adsorption on the two-coordinated Fe atom, the rate-limiting step is the fourth reaction [\(10\)](#page-6-2), with a barrier of 46.98 kcal/mol. Alternatively, for the less favorable NH<sup>3</sup> adsorption on the threecoordinated Fe atom, the rate-limiting step is the third reaction step [\(9\)](#page-6-1), with a barrier of 34.51 kcal/mol. The reaction pathway calculated for NH<sup>3</sup> decomposition on the linear-type Fe2O<sup>3</sup> isomer is shown in Fig. [4\(](#page-14-0)a), and respective intermediate and transition state structures are shown in Fig. [4\(](#page-14-0)b). Since this structure consists of two iron atoms connected through a central oxygen, each containing a terminal oxygen, the reaction mechanism differs slightly from that of the kite-like isomer. For instance, in the third step of the reaction, the second hydrogen from the adsorbed NH∗ 2 intermediate is transferred to the second terminal oxygen. The energy barrier for this step on the linear-type structure is 23.8 kcal/mol, as shown in the reaction path (I<sup>L</sup> 1 2 - T<sup>L</sup> 1 2 - I<sup>L</sup> 1 3) in Fig. [4a](#page-14-0). The fourth step on this isomer is not straightforward, involving the central oxygen atom breaking its bond with one of the neighboring iron atoms while forming an Fe − N − Fe bridge. This process leads to two different intermediates: the formation of the adsorbed H2O ∗ and the transfer of a hydrogen atom from one side of the Fe − N − Fe bridge to the other. Subsequently, the final dehydrogenation step from the NH∗ intermediate occurs, with an activation energy barrier of 34.76 kcal/mol. <span id="page-13-0"></span>![FIG. 3: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ reaction path on the kite-like isomer of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-14-0"></span>![FIG. 4: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> + → N ∗ + 3H∗ reaction path on the linear-type isomet of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) As a next step we consider possible H<sup>2</sup> formation via reactions [\(11\)](#page-6-3) and [\(12\)](#page-7-0) on the kite-like and linear isomers of Fe2O<sup>3</sup> cluster. The possible pathways for H<sup>2</sup> formation in the case of the most stable ammonia adsorption on the two-coordinated site (I′ intermediates) of the kite-like Fe2O<sup>3</sup> isomer are shown in Fig. [5\(](#page-15-0)a), while the corresponding structures of the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [5\(](#page-15-0)b). Note that H<sup>2</sup> formation can occur after partial decomposition of ammonia in reaction [\(11\)](#page-6-3), starting from intermediate (I<sup>I</sup> 1 3) via the path I′ 1 3 - T′ 1 6 - I′ 1 7 - T′ 1 7 - I′ 1 8. On the other hand, H<sup>2</sup> formation can occur via full decomposition of ammonia in reaction [\(12\)](#page-7-0), through the intermediate (II 1 4) via the path I′ 1 4 - T′ 1 4 - I′ 1 5 - T′ 1 5 - I′ 1 6. In both cases, the reaction pathways include breaking one O−H bond and forming an Fe−H bond. The H<sup>2</sup> formation barriers through these intermediates are 89.74 kcal/mol and 92.49 kcal/mol, respectively. From these results, we conclude that H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> structure is more favorable via reaction [\(11\)](#page-6-3), with the NH<sup>∗</sup> intermediate remaining adsorbed on the cluster. The H<sup>2</sup> formation reaction, starting from (I<sup>I</sup> 1 4), is the rate-limiting step in molecular hydrogen formation on the kite Fe2O<sup>3</sup> cluster. Similarly, the H<sup>2</sup> formation reaction pathways on the linear-type structure of Fe2O<sup>3</sup> are shown in Fig. [6\(](#page-16-0)a), while the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [6\(](#page-16-0)b). The H<sup>2</sup> formation through the NH<sup>∗</sup> intermediate (I<sup>L</sup> 1 4) via the reaction path I<sup>L</sup> 1 4 - T<sup>L</sup> 1 8 - I<sup>L</sup> 1 9 - T<sup>L</sup> 1 9 - I<sup>L</sup> 1 10 has an energy barrier of 79.99 kcal/mol. On the other hand H<sup>2</sup> formation through intermediate (I<sup>L</sup> 1 6) via reaction path I<sup>L</sup> 1 6 - T<sup>L</sup> 1 6 - I<sup>L</sup> 1 7 - T<sup>L</sup> 1 7 - I<sup>L</sup> 1 8 has an activation energy of 70.84 kcal/mol, which is about 10 kcal/mol lower energy than reaction path through intermediate (I<sup>L</sup> 1 4). Overall, on the basis of our calculated reaction pathways for H<sup>2</sup> formation show similar pattern for both kite-type and linear-type Fe2O3, where H<sup>2</sup> formation in reactions [\(11\)](#page-6-3) and [\(12\)](#page-7-0) take place via breaking one of O−H bond and forming intermediate Fe−H bond. However, from both thermodynamic and kinetic perspectives, H<sup>2</sup> formation on the two types of Fe2O<sup>3</sup> structures varies. Reaction [\(11\)](#page-6-3) is more favorable on the kite-like structure, while reaction [\(12\)](#page-7-0) is more favorable on the linear structure. This highlights that the rate-limiting step for H<sup>2</sup> formation is highly dependent on the catalyst's structure. <span id="page-15-0"></span>![FIG. 5: (a) The energy profile for H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-16-0"></span>![FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [\(11\)](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [\(11\)](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [\(12\)](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [\(10\)](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8\(](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8\)](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [\(10\)](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [\(11](#page-6-3) and [12\)](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [\(11\)](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [\(11\)](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9\(](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9\)](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [\(11\)](#page-6-3) and [\(12\)](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [\(11\)](#page-6-3) compared to the fully decomposed pathway [\(12\)](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [\(8\)](#page-6-4) − [\(12\)](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 \(2022\).](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. <span id="page-8-0"></span>![FIG. 1: The most stable structures of (Fe2O3)<sup>n</sup> clusters with n = 1 − 4. The total spin S and the binding energy Eb, of the clusters are shown in inserts.](path) ## <span id="page-8-1"></span>B. Ammonia adsorption on (Fe2O3)<sup>n</sup> clusters Adsorption of ammonia on (Fe2O3)<sup>n</sup> clusters is a crucial initial step in the whole dehydrogenation process. Figure [2](#page-9-0) demonstrates the most stable adsorption configurations of the NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters with n = 1−4. The corresponding free energies of adsorption and Fe−N bond distances are shown in Table [I](#page-10-0) at 0 K. Our calculations show that the adsorption of NH<sup>3</sup> on the smallest Fe2O<sup>3</sup> cluster is the most stable among all cluster sizes considered in this study, with an adsorption free energy of -33.68 kcal/mol. This finding is corroborated by Mulliken charge analysis, which shows that more electrons are shared between the lone pair of the N atom and the 3d orbitals of Fe2<sup>+</sup> for <sup>n</sup> <sup>=</sup> 1. On the other hand, for larger cluster sizes with <sup>n</sup> <sup>=</sup> <sup>2</sup>−4, which primarily contain Fe3+, the electron density is more localized over the bonding region, as also reported by Sierka et al.[55](#page-28-0). Therefore, bonding occurs with the nitrogen lone pair. Our theoretical analysis indicates that the adsorption energy ∆Gads of ammonia on (Fe2O3)<sup>n</sup> clusters decreases from n = 1 to n = 3, followed by a slight increase for n = 4. A similar trend in the change of adsorption energy with cluster size was reported by Shulan Zhou et al.[56](#page-28-1) for Run@CNT systems. We also compared the adsorption energy of NH<sup>3</sup> on different metal and metal oxides in Table [I.](#page-10-0) The obtained NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters are about 10 kcal/mol higher than the data reported by Zhang et al. for the Ru(0001) surface[57](#page-28-2). Moreover, the adsorption of NH<sup>3</sup> and NO<sup>x</sup> on the γ-Fe2O3(111) surface was studied by Wei Huang et al.[58](#page-28-3) using periodic density functional calculations. They calculated adsorption energies on octahedral and tetrahedral sites of γ-Fe2O3(111) to be -2.13 kcal/mol and -21.68 kcal/mol, respectively. Similarly, our calculated NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters for n = 3 and n = 4 are close to the data reported by Wei Huang et al.,[58](#page-28-3) as adsorption of NH<sup>3</sup> on the three-coordinated Fe3<sup>+</sup> site resembles the tetrahedral site of γ-Fe2O3(111), while the adsorption on the four-coordinated Fe3<sup>+</sup> site resembles the octahedral site of γ-Fe2O3. <span id="page-9-0"></span>![FIG. 2: The optimized geometries of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> for n = 1−4; N−Fe distances (Å) are shown in parentheses along with the partial atomic charges on neighboring atoms. The total spin S of the clusters is shown in inserts.](path) As mentioned above, the calculated adsorption energies indicate that the adsorption of an NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters (n = 1−4) weakens as the cluster size increases from n = 1 to n = 3. In industrial processes, the dehydrogenation of ammonia typically occurs at high temperatures, often in the range of 400◦C to 700◦C, depending on the specific catalysts and conditions used. Therefore, it is important to determine the range of temperatures at which ammonia adsorption on (Fe2O3)<sup>n</sup> remains stable. Figure S5 demonstrates the temperature dependence of ∆Gads in the range from 0 K to 1200 K for the most stable adsorption configurations of NH<sup>3</sup> on (Fe2O3)n clusters (n = 1 − 4). The negative values of ∆Gads correspond to stable adsorption. As seen in Fig. S5, NH<sup>3</sup> adsorbed on the smallest Fe2O<sup>3</sup> cluster is stable across the entire temperature range of 0 K to 1200 K. However, for larger cluster sizes, ammonia adsorption becomes energetically unfavorable at temperatures of 1107 (K), 961 (K), and 1000 (K) for n = 2,3, and 4, respectively. ## C. NH<sup>3</sup> decomposition on Fe2O<sup>3</sup> In this section, we discuss the complete NH<sup>3</sup> decomposition and H<sup>2</sup> formation reactions [\(7\)](#page-6-0) - [\(12\)](#page-7-0) on the smallest considered cluster, Fe2O3, at room temperature, T=298.15 K, explored by the AFIR method. This method allows for the automatic exploration of the full reaction path network, systematically accounting for the variety of possible isomer structures and adsorption sites. This is an important approach in nanocatalysis because it has been demonstrated that the most stable structures are not always the most reactive. Therefore, a systematic search for reaction pathways that accounts for the contributions of low-energy isomers is required to accurately describe the catalytic properties of clusters at finite temperatures.[49](#page-27-10) To illustrate the isomer and reaction-site effects, we explicitly consider two different isomers of the Fe2O<sup>3</sup> cluster: the most stable kite-like structure with one terminal oxygen atom, and the linear structure isomer with two terminal oxygen atoms which is 6.24 kcal/mol less stable (see Fig. S1). The kite-like structure possess two type of catalytically active metal centers - two-coordinated and three-coordinated Fe sites. Therefore we consider adsorption and decomposition of NH<sup>3</sup> molecule on both of them. Figure [3\(](#page-13-0)a) demonstrates that the adsorption of NH<sup>3</sup> on the kite-like Fe2O<sup>3</sup> cluster is an exothermic reaction, occurring at both the two-coordinated and three-coordinated Fe sites. The adsorption free energies are -26.98 kcal/mol for the two-coordinated Fe site (intermediate I′ 1 1) and -11.29 kcal/mol for the three-coordinated Fe site (intermediate I′′ 1 1), respectively. The optimized structures of all intermediates (I) and transition satates (T) along the reaction pathways are shown in Fig. [3\(](#page-13-0)b) and [4\(](#page-14-0)b), for the kite-like and linear clusters, respectively. Here the lower index corresponds to the cluster size n, while the numbering corresponds to the order of intermedeates (transition states) along the reaction path. As discussed in the previous section, the most stable adsorption site for NH<sup>3</sup> is the two-coordinated Fe site, with an Fe−N bond length of 2.11 Å. In contrast, the Fe−N bond length at the three-coordinated Fe site is 2.16 Å. These findings are supported by the fact that NH<sup>3</sup> adsorption highly depends on the local geometry and electronic structure of the catalyst. In the case of the Fe2O<sup>3</sup> kite-like structure, the first dehydrogenation reaction is the second step in the reaction mechanism, occurring after adsorption with activation barriers of 26.98 kcal/mol and 22.12 kcal/mol through the reaction paths I′ 1 1-T′ 1 1-I′ 1 2 and I′′ 1 1-T′′ 1 1-I′′ 1 2, respectively. The reactions on these two-coodrinated and three-coordinated active sites are exothermic by 16.31 kcal/mol and 7.53 kcal/mol, respectively. However, the first dehydrogenation of NH<sup>3</sup> on the lineartype structure Fig. [4\(](#page-14-0)a) occurs with smaller activation barrier of 16.22 kcal/mol via the reaction path I<sup>L</sup> 1 1 - T<sup>L</sup> 1 1 - I<sup>L</sup> 1 2, demonstrating that the less stable linear isomer is more reactive. The role of Fe2O<sup>3</sup> isomer structure on NH<sup>3</sup> adsorption and first hydrogen atom transfer was previousely studied by Chaoyue Xie et al.[60](#page-28-5) They performed DFT-D3 calculations on the adsorption mechanisms of different molecules (NH3, NO, O2) on activated carbon (AC) supported iron-based catalysts FexOy/AC. The calculated adsorption electronic energies of NH<sup>3</sup> were -37.4 kcal/mol and -53.7 kcal/mol on different isomers of Fe2O3/AC, and the first hydrogen atom transfer had an activation barrier of 15.5 kcal/mol. Similarly, the adsorption and dehydrogenation of ammonia on different metal oxides were investigated by Erdtman and co-workers[62](#page-28-7) for the application of gas sensors. They reported that the adsorption energy of NH<sup>3</sup> on the RuO2(110) surface is -38.24 kcal/mol, and the first N−H bond cleavage had an activation energy barrier of 17.45 kcal/mol. The third step of the NH<sup>3</sup> dehydrogenation reaction [\(9\)](#page-6-1) involves the dissociation of the adsorbed NH∗ 2 intermediate into NH∗ and H∗ species. In this step, the abstracted hydrogen atom transfers to one of the oxygen atoms in the cluster. Figure [3\(](#page-13-0)a) demonstrates, that in the case of the kite-like structure the energy barriers for this step are 43.91 kcal/mol and 34.51 kcal/mol, corresponding to the reaction paths I′ 1 2 - T′ 1 2 - I′ 1 3 and I′′ 1 2 - T′′ 1 2 - I′′ 1 3, respectively. In the fourth step [\(10\)](#page-6-2), the adsorbed NH∗ intermediate further dissociates into N∗ and H∗ species as shown in Fig. [3\(](#page-13-0)a). The reaction barriers associated with this step are 46.98 kcal/mol and 8.95 kcal/mol for the two-coordinated and three-coordinated reaction paths, respectively. The decomposition of NH<sup>3</sup> on kite-like structures becomes endothermic starting from the third step [\(9\)](#page-6-1). Our calculations reveal that NH<sup>3</sup> dehydrogenation has a high energy barrier when the NH<sup>3</sup> molecule is adsorbed at a two-coordinated Fe site, which is the most stable adsorption site. On the other hand, dehydrogenation of the adsorbed NH<sup>3</sup> at a three-coordinated Fe site has a considerably lower activation barrier of 8.95 kcal/mol for the reaction step [\(10\)](#page-6-2). Overall, for the NH<sup>3</sup> decomposition reaction on the kite-like Fe2O<sup>3</sup> structure, with initial NH<sup>3</sup> adsorption on the two-coordinated Fe atom, the rate-limiting step is the fourth reaction [\(10\)](#page-6-2), with a barrier of 46.98 kcal/mol. Alternatively, for the less favorable NH<sup>3</sup> adsorption on the threecoordinated Fe atom, the rate-limiting step is the third reaction step [\(9\)](#page-6-1), with a barrier of 34.51 kcal/mol. The reaction pathway calculated for NH<sup>3</sup> decomposition on the linear-type Fe2O<sup>3</sup> isomer is shown in Fig. [4\(](#page-14-0)a), and respective intermediate and transition state structures are shown in Fig. [4\(](#page-14-0)b). Since this structure consists of two iron atoms connected through a central oxygen, each containing a terminal oxygen, the reaction mechanism differs slightly from that of the kite-like isomer. For instance, in the third step of the reaction, the second hydrogen from the adsorbed NH∗ 2 intermediate is transferred to the second terminal oxygen. The energy barrier for this step on the linear-type structure is 23.8 kcal/mol, as shown in the reaction path (I<sup>L</sup> 1 2 - T<sup>L</sup> 1 2 - I<sup>L</sup> 1 3) in Fig. [4a](#page-14-0). The fourth step on this isomer is not straightforward, involving the central oxygen atom breaking its bond with one of the neighboring iron atoms while forming an Fe − N − Fe bridge. This process leads to two different intermediates: the formation of the adsorbed H2O ∗ and the transfer of a hydrogen atom from one side of the Fe − N − Fe bridge to the other. Subsequently, the final dehydrogenation step from the NH∗ intermediate occurs, with an activation energy barrier of 34.76 kcal/mol. <span id="page-13-0"></span>![FIG. 3: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ reaction path on the kite-like isomer of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-14-0"></span>![FIG. 4: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> + → N ∗ + 3H∗ reaction path on the linear-type isomet of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) As a next step we consider possible H<sup>2</sup> formation via reactions [\(11\)](#page-6-3) and [\(12\)](#page-7-0) on the kite-like and linear isomers of Fe2O<sup>3</sup> cluster. The possible pathways for H<sup>2</sup> formation in the case of the most stable ammonia adsorption on the two-coordinated site (I′ intermediates) of the kite-like Fe2O<sup>3</sup> isomer are shown in Fig. [5\(](#page-15-0)a), while the corresponding structures of the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [5\(](#page-15-0)b). Note that H<sup>2</sup> formation can occur after partial decomposition of ammonia in reaction [\(11\)](#page-6-3), starting from intermediate (I<sup>I</sup> 1 3) via the path I′ 1 3 - T′ 1 6 - I′ 1 7 - T′ 1 7 - I′ 1 8. On the other hand, H<sup>2</sup> formation can occur via full decomposition of ammonia in reaction [\(12\)](#page-7-0), through the intermediate (II 1 4) via the path I′ 1 4 - T′ 1 4 - I′ 1 5 - T′ 1 5 - I′ 1 6. In both cases, the reaction pathways include breaking one O−H bond and forming an Fe−H bond. The H<sup>2</sup> formation barriers through these intermediates are 89.74 kcal/mol and 92.49 kcal/mol, respectively. From these results, we conclude that H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> structure is more favorable via reaction [\(11\)](#page-6-3), with the NH<sup>∗</sup> intermediate remaining adsorbed on the cluster. The H<sup>2</sup> formation reaction, starting from (I<sup>I</sup> 1 4), is the rate-limiting step in molecular hydrogen formation on the kite Fe2O<sup>3</sup> cluster. Similarly, the H<sup>2</sup> formation reaction pathways on the linear-type structure of Fe2O<sup>3</sup> are shown in Fig. [6\(](#page-16-0)a), while the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [6\(](#page-16-0)b). The H<sup>2</sup> formation through the NH<sup>∗</sup> intermediate (I<sup>L</sup> 1 4) via the reaction path I<sup>L</sup> 1 4 - T<sup>L</sup> 1 8 - I<sup>L</sup> 1 9 - T<sup>L</sup> 1 9 - I<sup>L</sup> 1 10 has an energy barrier of 79.99 kcal/mol. On the other hand H<sup>2</sup> formation through intermediate (I<sup>L</sup> 1 6) via reaction path I<sup>L</sup> 1 6 - T<sup>L</sup> 1 6 - I<sup>L</sup> 1 7 - T<sup>L</sup> 1 7 - I<sup>L</sup> 1 8 has an activation energy of 70.84 kcal/mol, which is about 10 kcal/mol lower energy than reaction path through intermediate (I<sup>L</sup> 1 4). Overall, on the basis of our calculated reaction pathways for H<sup>2</sup> formation show similar pattern for both kite-type and linear-type Fe2O3, where H<sup>2</sup> formation in reactions [\(11\)](#page-6-3) and [\(12\)](#page-7-0) take place via breaking one of O−H bond and forming intermediate Fe−H bond. However, from both thermodynamic and kinetic perspectives, H<sup>2</sup> formation on the two types of Fe2O<sup>3</sup> structures varies. Reaction [\(11\)](#page-6-3) is more favorable on the kite-like structure, while reaction [\(12\)](#page-7-0) is more favorable on the linear structure. This highlights that the rate-limiting step for H<sup>2</sup> formation is highly dependent on the catalyst's structure. <span id="page-15-0"></span>![FIG. 5: (a) The energy profile for H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-16-0"></span>![FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [\(11\)](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [\(11\)](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [\(12\)](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [\(10\)](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8\(](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8\)](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [\(10\)](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [\(11](#page-6-3) and [12\)](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [\(11\)](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [\(11\)](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9\(](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9\)](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [\(11\)](#page-6-3) and [\(12\)](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [\(11\)](#page-6-3) compared to the fully decomposed pathway [\(12\)](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [\(8\)](#page-6-4) − [\(12\)](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 \(2022\).](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 4: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> + → N ∗ + 3H∗ reaction path on the linear-type isomet of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. As a next step we consider possible H<sup>2</sup> formation via reactions [\(11\)](#page-6-3) and [\(12\)](#page-7-0) on the kite-like and linear isomers of Fe2O<sup>3</sup> cluster. The possible pathways for H<sup>2</sup> formation in the case of the most stable ammonia adsorption on the two-coordinated site (I′ intermediates) of the kite-like Fe2O<sup>3</sup> isomer are shown in Fig. [5\(](#page-15-0)a), while the corresponding structures of the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [5\(](#page-15-0)b). Note that H<sup>2</sup> formation can occur after partial decomposition of ammonia in reaction [\(11\)](#page-6-3), starting from intermediate (I<sup>I</sup> 1 3) via the path I′ 1 3 - T′ 1 6 - I′ 1 7 - T′ 1 7 - I′ 1 8. On the other hand, H<sup>2</sup> formation can occur via full decomposition of ammonia in reaction [\(12\)](#page-7-0), through the intermediate (II 1 4) via the path I′ 1 4 - T′ 1 4 - I′ 1 5 - T′ 1 5 - I′ 1 6. In both cases, the reaction pathways include breaking one O−H bond and forming an Fe−H bond. The H<sup>2</sup> formation barriers through these intermediates are 89.74 kcal/mol and 92.49 kcal/mol, respectively. From these results, we conclude that H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> structure is more favorable via reaction [\(11\)](#page-6-3), with the NH<sup>∗</sup> intermediate remaining adsorbed on the cluster. The H<sup>2</sup> formation reaction, starting from (I<sup>I</sup> 1 4), is the rate-limiting step in molecular hydrogen formation on the kite Fe2O<sup>3</sup> cluster. Similarly, the H<sup>2</sup> formation reaction pathways on the linear-type structure of Fe2O<sup>3</sup> are shown in Fig. [6\(](#page-16-0)a), while the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [6\(](#page-16-0)b). The H<sup>2</sup> formation through the NH<sup>∗</sup> intermediate (I<sup>L</sup> 1 4) via the reaction path I<sup>L</sup> 1 4 - T<sup>L</sup> 1 8 - I<sup>L</sup> 1 9 - T<sup>L</sup> 1 9 - I<sup>L</sup> 1 10 has an energy barrier of 79.99 kcal/mol. On the other hand H<sup>2</sup> formation through intermediate (I<sup>L</sup> 1 6) via reaction path I<sup>L</sup> 1 6 - T<sup>L</sup> 1 6 - I<sup>L</sup> 1 7 - T<sup>L</sup> 1 7 - I<sup>L</sup> 1 8 has an activation energy of 70.84 kcal/mol, which is about 10 kcal/mol lower energy than reaction path through intermediate (I<sup>L</sup> 1 4). Overall, on the basis of our calculated reaction pathways for H<sup>2</sup> formation show similar pattern for both kite-type and linear-type Fe2O3, where H<sup>2</sup> formation in reactions [\(11\)](#page-6-3) and [\(12\)](#page-7-0) take place via breaking one of O−H bond and forming intermediate Fe−H bond. However, from both thermodynamic and kinetic perspectives, H<sup>2</sup> formation on the two types of Fe2O<sup>3</sup> structures varies. Reaction [\(11\)](#page-6-3) is more favorable on the kite-like structure, while reaction [\(12\)](#page-7-0) is more favorable on the linear structure. This highlights that the rate-limiting step for H<sup>2</sup> formation is highly dependent on the catalyst's structure. <span id="page-15-0"></span>![FIG. 5: (a) The energy profile for H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-16-0"></span>![FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [\(11\)](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [\(11\)](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [\(12\)](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [\(10\)](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8\(](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8\)](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [\(10\)](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [\(11](#page-6-3) and [12\)](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [\(11\)](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [\(11\)](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9\(](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9\)](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [\(11\)](#page-6-3) and [\(12\)](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [\(11\)](#page-6-3) compared to the fully decomposed pathway [\(12\)](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [\(8\)](#page-6-4) − [\(12\)](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 \(2022\).](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [\(11\)](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [\(11\)](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [\(12\)](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [\(10\)](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8\(](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8\)](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [\(10\)](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [\(11](#page-6-3) and [12\)](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [\(11\)](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [\(11\)](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9\(](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9\)](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [\(11\)](#page-6-3) and [\(12\)](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [\(11\)](#page-6-3) compared to the fully decomposed pathway [\(12\)](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [\(8\)](#page-6-4) − [\(12\)](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 \(2022\).](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8\(](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8\)](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [\(10\)](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [\(11](#page-6-3) and [12\)](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [\(11\)](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [\(11\)](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9\(](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9\)](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [\(11\)](#page-6-3) and [\(12\)](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [\(11\)](#page-6-3) compared to the fully decomposed pathway [\(12\)](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [\(8\)](#page-6-4) − [\(12\)](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 \(2022\).](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [\(8\)](#page-6-4) − [\(12\)](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 \(2022\).](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 3: (a) The energy profile for NH<sup>3</sup> → NH<sup></sup> 2 + H<sup></sup> → NH<sup></sup> + 2H<sup></sup> → N * + 3H* reaction path on the kite-like isomer of Fe<sub>2</sub>O<sub>3</sub> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. <span id="page-8-0"></span>![FIG. 1: The most stable structures of (Fe2O3)<sup>n</sup> clusters with n = 1 − 4. The total spin S and the binding energy Eb, of the clusters are shown in inserts.](path) ## <span id="page-8-1"></span>B. Ammonia adsorption on (Fe2O3)<sup>n</sup> clusters Adsorption of ammonia on (Fe2O3)<sup>n</sup> clusters is a crucial initial step in the whole dehydrogenation process. Figure [2](#page-9-0) demonstrates the most stable adsorption configurations of the NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters with n = 1−4. The corresponding free energies of adsorption and Fe−N bond distances are shown in Table [I](#page-10-0) at 0 K. Our calculations show that the adsorption of NH<sup>3</sup> on the smallest Fe2O<sup>3</sup> cluster is the most stable among all cluster sizes considered in this study, with an adsorption free energy of -33.68 kcal/mol. This finding is corroborated by Mulliken charge analysis, which shows that more electrons are shared between the lone pair of the N atom and the 3d orbitals of Fe2<sup>+</sup> for <sup>n</sup> <sup>=</sup> 1. On the other hand, for larger cluster sizes with <sup>n</sup> <sup>=</sup> <sup>2</sup>−4, which primarily contain Fe3+, the electron density is more localized over the bonding region, as also reported by Sierka et al.[55](#page-28-0). Therefore, bonding occurs with the nitrogen lone pair. Our theoretical analysis indicates that the adsorption energy ∆Gads of ammonia on (Fe2O3)<sup>n</sup> clusters decreases from n = 1 to n = 3, followed by a slight increase for n = 4. A similar trend in the change of adsorption energy with cluster size was reported by Shulan Zhou et al.[56](#page-28-1) for Run@CNT systems. We also compared the adsorption energy of NH<sup>3</sup> on different metal and metal oxides in Table [I.](#page-10-0) The obtained NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters are about 10 kcal/mol higher than the data reported by Zhang et al. for the Ru(0001) surface[57](#page-28-2). Moreover, the adsorption of NH<sup>3</sup> and NO<sup>x</sup> on the γ-Fe2O3(111) surface was studied by Wei Huang et al.[58](#page-28-3) using periodic density functional calculations. They calculated adsorption energies on octahedral and tetrahedral sites of γ-Fe2O3(111) to be -2.13 kcal/mol and -21.68 kcal/mol, respectively. Similarly, our calculated NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters for n = 3 and n = 4 are close to the data reported by Wei Huang et al.,[58](#page-28-3) as adsorption of NH<sup>3</sup> on the three-coordinated Fe3<sup>+</sup> site resembles the tetrahedral site of γ-Fe2O3(111), while the adsorption on the four-coordinated Fe3<sup>+</sup> site resembles the octahedral site of γ-Fe2O3. <span id="page-9-0"></span>![FIG. 2: The optimized geometries of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> for n = 1−4; N−Fe distances (Å) are shown in parentheses along with the partial atomic charges on neighboring atoms. The total spin S of the clusters is shown in inserts.](path) As mentioned above, the calculated adsorption energies indicate that the adsorption of an NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters (n = 1−4) weakens as the cluster size increases from n = 1 to n = 3. In industrial processes, the dehydrogenation of ammonia typically occurs at high temperatures, often in the range of 400◦C to 700◦C, depending on the specific catalysts and conditions used. Therefore, it is important to determine the range of temperatures at which ammonia adsorption on (Fe2O3)<sup>n</sup> remains stable. Figure S5 demonstrates the temperature dependence of ∆Gads in the range from 0 K to 1200 K for the most stable adsorption configurations of NH<sup>3</sup> on (Fe2O3)n clusters (n = 1 − 4). The negative values of ∆Gads correspond to stable adsorption. As seen in Fig. S5, NH<sup>3</sup> adsorbed on the smallest Fe2O<sup>3</sup> cluster is stable across the entire temperature range of 0 K to 1200 K. However, for larger cluster sizes, ammonia adsorption becomes energetically unfavorable at temperatures of 1107 (K), 961 (K), and 1000 (K) for n = 2,3, and 4, respectively. ## C. NH<sup>3</sup> decomposition on Fe2O<sup>3</sup> In this section, we discuss the complete NH<sup>3</sup> decomposition and H<sup>2</sup> formation reactions [\(7\)](#page-6-0) - [\(12\)](#page-7-0) on the smallest considered cluster, Fe2O3, at room temperature, T=298.15 K, explored by the AFIR method. This method allows for the automatic exploration of the full reaction path network, systematically accounting for the variety of possible isomer structures and adsorption sites. This is an important approach in nanocatalysis because it has been demonstrated that the most stable structures are not always the most reactive. Therefore, a systematic search for reaction pathways that accounts for the contributions of low-energy isomers is required to accurately describe the catalytic properties of clusters at finite temperatures.[49](#page-27-10) To illustrate the isomer and reaction-site effects, we explicitly consider two different isomers of the Fe2O<sup>3</sup> cluster: the most stable kite-like structure with one terminal oxygen atom, and the linear structure isomer with two terminal oxygen atoms which is 6.24 kcal/mol less stable (see Fig. S1). The kite-like structure possess two type of catalytically active metal centers - two-coordinated and three-coordinated Fe sites. Therefore we consider adsorption and decomposition of NH<sup>3</sup> molecule on both of them. Figure [3\(](#page-13-0)a) demonstrates that the adsorption of NH<sup>3</sup> on the kite-like Fe2O<sup>3</sup> cluster is an exothermic reaction, occurring at both the two-coordinated and three-coordinated Fe sites. The adsorption free energies are -26.98 kcal/mol for the two-coordinated Fe site (intermediate I′ 1 1) and -11.29 kcal/mol for the three-coordinated Fe site (intermediate I′′ 1 1), respectively. The optimized structures of all intermediates (I) and transition satates (T) along the reaction pathways are shown in Fig. [3\(](#page-13-0)b) and [4\(](#page-14-0)b), for the kite-like and linear clusters, respectively. Here the lower index corresponds to the cluster size n, while the numbering corresponds to the order of intermedeates (transition states) along the reaction path. As discussed in the previous section, the most stable adsorption site for NH<sup>3</sup> is the two-coordinated Fe site, with an Fe−N bond length of 2.11 Å. In contrast, the Fe−N bond length at the three-coordinated Fe site is 2.16 Å. These findings are supported by the fact that NH<sup>3</sup> adsorption highly depends on the local geometry and electronic structure of the catalyst. In the case of the Fe2O<sup>3</sup> kite-like structure, the first dehydrogenation reaction is the second step in the reaction mechanism, occurring after adsorption with activation barriers of 26.98 kcal/mol and 22.12 kcal/mol through the reaction paths I′ 1 1-T′ 1 1-I′ 1 2 and I′′ 1 1-T′′ 1 1-I′′ 1 2, respectively. The reactions on these two-coodrinated and three-coordinated active sites are exothermic by 16.31 kcal/mol and 7.53 kcal/mol, respectively. However, the first dehydrogenation of NH<sup>3</sup> on the lineartype structure Fig. [4\(](#page-14-0)a) occurs with smaller activation barrier of 16.22 kcal/mol via the reaction path I<sup>L</sup> 1 1 - T<sup>L</sup> 1 1 - I<sup>L</sup> 1 2, demonstrating that the less stable linear isomer is more reactive. The role of Fe2O<sup>3</sup> isomer structure on NH<sup>3</sup> adsorption and first hydrogen atom transfer was previousely studied by Chaoyue Xie et al.[60](#page-28-5) They performed DFT-D3 calculations on the adsorption mechanisms of different molecules (NH3, NO, O2) on activated carbon (AC) supported iron-based catalysts FexOy/AC. The calculated adsorption electronic energies of NH<sup>3</sup> were -37.4 kcal/mol and -53.7 kcal/mol on different isomers of Fe2O3/AC, and the first hydrogen atom transfer had an activation barrier of 15.5 kcal/mol. Similarly, the adsorption and dehydrogenation of ammonia on different metal oxides were investigated by Erdtman and co-workers[62](#page-28-7) for the application of gas sensors. They reported that the adsorption energy of NH<sup>3</sup> on the RuO2(110) surface is -38.24 kcal/mol, and the first N−H bond cleavage had an activation energy barrier of 17.45 kcal/mol. The third step of the NH<sup>3</sup> dehydrogenation reaction [\(9\)](#page-6-1) involves the dissociation of the adsorbed NH∗ 2 intermediate into NH∗ and H∗ species. In this step, the abstracted hydrogen atom transfers to one of the oxygen atoms in the cluster. Figure [3\(](#page-13-0)a) demonstrates, that in the case of the kite-like structure the energy barriers for this step are 43.91 kcal/mol and 34.51 kcal/mol, corresponding to the reaction paths I′ 1 2 - T′ 1 2 - I′ 1 3 and I′′ 1 2 - T′′ 1 2 - I′′ 1 3, respectively. In the fourth step [\(10\)](#page-6-2), the adsorbed NH∗ intermediate further dissociates into N∗ and H∗ species as shown in Fig. [3\(](#page-13-0)a). The reaction barriers associated with this step are 46.98 kcal/mol and 8.95 kcal/mol for the two-coordinated and three-coordinated reaction paths, respectively. The decomposition of NH<sup>3</sup> on kite-like structures becomes endothermic starting from the third step [\(9\)](#page-6-1). Our calculations reveal that NH<sup>3</sup> dehydrogenation has a high energy barrier when the NH<sup>3</sup> molecule is adsorbed at a two-coordinated Fe site, which is the most stable adsorption site. On the other hand, dehydrogenation of the adsorbed NH<sup>3</sup> at a three-coordinated Fe site has a considerably lower activation barrier of 8.95 kcal/mol for the reaction step [\(10\)](#page-6-2). Overall, for the NH<sup>3</sup> decomposition reaction on the kite-like Fe2O<sup>3</sup> structure, with initial NH<sup>3</sup> adsorption on the two-coordinated Fe atom, the rate-limiting step is the fourth reaction [\(10\)](#page-6-2), with a barrier of 46.98 kcal/mol. Alternatively, for the less favorable NH<sup>3</sup> adsorption on the threecoordinated Fe atom, the rate-limiting step is the third reaction step [\(9\)](#page-6-1), with a barrier of 34.51 kcal/mol. The reaction pathway calculated for NH<sup>3</sup> decomposition on the linear-type Fe2O<sup>3</sup> isomer is shown in Fig. [4\(](#page-14-0)a), and respective intermediate and transition state structures are shown in Fig. [4\(](#page-14-0)b). Since this structure consists of two iron atoms connected through a central oxygen, each containing a terminal oxygen, the reaction mechanism differs slightly from that of the kite-like isomer. For instance, in the third step of the reaction, the second hydrogen from the adsorbed NH∗ 2 intermediate is transferred to the second terminal oxygen. The energy barrier for this step on the linear-type structure is 23.8 kcal/mol, as shown in the reaction path (I<sup>L</sup> 1 2 - T<sup>L</sup> 1 2 - I<sup>L</sup> 1 3) in Fig. [4a](#page-14-0). The fourth step on this isomer is not straightforward, involving the central oxygen atom breaking its bond with one of the neighboring iron atoms while forming an Fe − N − Fe bridge. This process leads to two different intermediates: the formation of the adsorbed H2O ∗ and the transfer of a hydrogen atom from one side of the Fe − N − Fe bridge to the other. Subsequently, the final dehydrogenation step from the NH∗ intermediate occurs, with an activation energy barrier of 34.76 kcal/mol. <span id="page-13-0"></span>![FIG. 3: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ reaction path on the kite-like isomer of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-14-0"></span>![FIG. 4: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> + → N ∗ + 3H∗ reaction path on the linear-type isomet of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) As a next step we consider possible H<sup>2</sup> formation via reactions [\(11\)](#page-6-3) and [\(12\)](#page-7-0) on the kite-like and linear isomers of Fe2O<sup>3</sup> cluster. The possible pathways for H<sup>2</sup> formation in the case of the most stable ammonia adsorption on the two-coordinated site (I′ intermediates) of the kite-like Fe2O<sup>3</sup> isomer are shown in Fig. [5\(](#page-15-0)a), while the corresponding structures of the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [5\(](#page-15-0)b). Note that H<sup>2</sup> formation can occur after partial decomposition of ammonia in reaction [\(11\)](#page-6-3), starting from intermediate (I<sup>I</sup> 1 3) via the path I′ 1 3 - T′ 1 6 - I′ 1 7 - T′ 1 7 - I′ 1 8. On the other hand, H<sup>2</sup> formation can occur via full decomposition of ammonia in reaction [\(12\)](#page-7-0), through the intermediate (II 1 4) via the path I′ 1 4 - T′ 1 4 - I′ 1 5 - T′ 1 5 - I′ 1 6. In both cases, the reaction pathways include breaking one O−H bond and forming an Fe−H bond. The H<sup>2</sup> formation barriers through these intermediates are 89.74 kcal/mol and 92.49 kcal/mol, respectively. From these results, we conclude that H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> structure is more favorable via reaction [\(11\)](#page-6-3), with the NH<sup>∗</sup> intermediate remaining adsorbed on the cluster. The H<sup>2</sup> formation reaction, starting from (I<sup>I</sup> 1 4), is the rate-limiting step in molecular hydrogen formation on the kite Fe2O<sup>3</sup> cluster. Similarly, the H<sup>2</sup> formation reaction pathways on the linear-type structure of Fe2O<sup>3</sup> are shown in Fig. [6\(](#page-16-0)a), while the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [6\(](#page-16-0)b). The H<sup>2</sup> formation through the NH<sup>∗</sup> intermediate (I<sup>L</sup> 1 4) via the reaction path I<sup>L</sup> 1 4 - T<sup>L</sup> 1 8 - I<sup>L</sup> 1 9 - T<sup>L</sup> 1 9 - I<sup>L</sup> 1 10 has an energy barrier of 79.99 kcal/mol. On the other hand H<sup>2</sup> formation through intermediate (I<sup>L</sup> 1 6) via reaction path I<sup>L</sup> 1 6 - T<sup>L</sup> 1 6 - I<sup>L</sup> 1 7 - T<sup>L</sup> 1 7 - I<sup>L</sup> 1 8 has an activation energy of 70.84 kcal/mol, which is about 10 kcal/mol lower energy than reaction path through intermediate (I<sup>L</sup> 1 4). Overall, on the basis of our calculated reaction pathways for H<sup>2</sup> formation show similar pattern for both kite-type and linear-type Fe2O3, where H<sup>2</sup> formation in reactions [\(11\)](#page-6-3) and [\(12\)](#page-7-0) take place via breaking one of O−H bond and forming intermediate Fe−H bond. However, from both thermodynamic and kinetic perspectives, H<sup>2</sup> formation on the two types of Fe2O<sup>3</sup> structures varies. Reaction [\(11\)](#page-6-3) is more favorable on the kite-like structure, while reaction [\(12\)](#page-7-0) is more favorable on the linear structure. This highlights that the rate-limiting step for H<sup>2</sup> formation is highly dependent on the catalyst's structure. <span id="page-15-0"></span>![FIG. 5: (a) The energy profile for H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-16-0"></span>![FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [\(11\)](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [\(11\)](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [\(12\)](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [\(10\)](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8\(](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8\)](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [\(10\)](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [\(11](#page-6-3) and [12\)](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [\(11\)](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [\(11\)](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9\(](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9\)](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [\(11\)](#page-6-3) and [\(12\)](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [\(11\)](#page-6-3) compared to the fully decomposed pathway [\(12\)](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [\(8\)](#page-6-4) − [\(12\)](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 \(2022\).](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 1: The most stable structures of (Fe2O3)<sup>n</sup> clusters with n = 1 − 4. The total spin S and the binding energy Eb, of the clusters are shown in inserts.	## Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for hydrogen production from ammonia The catalytic activities of high-spin small Fe(III) oxides have been investigated for efficient hydrogen production through ammonia decomposition, using the Artificial Force Induced Reaction (AFIR) method within the framework of density functional theory (DFT) with the B3LYP hybrid exchange-correlation functional. Our results reveal that the adsorption free energy of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n = 1 − 4) decreases with increasing cluster size up to n = 3, followed by a slight increase at n = 4. The strongest NH<sup>3</sup> adsorption energy, 33.68 kcal/mol, was found for Fe2O3, where NH<sup>3</sup> interacts with a two-coordinated Fe site, forming an Fe-N bond with a length of 2.11 Å. A comparative analysis of NH<sup>3</sup> decomposition and H<sup>2</sup> formation on various Fe(III) oxide sizes identifies the rate-determining steps for each reaction. We found that the rate-determining step for the full NH<sup>3</sup> decomposition on (Fe2O3)<sup>n</sup> (n = 1−4) is size-dependent, with the NH<sup>∗</sup> ⇌ N ∗ + 3H∗ reaction acting as the limiting step for n = 1−3. Additionally, our findings indicate that H<sup>2</sup> formation is favored following the partial decomposition of NH<sup>3</sup> on Fe(III) oxides. ### I. INTRODUCTION: The ammonia decomposition reaction has recently received extensive attention due to its potential use as an alternative green energy source[1](#page-25-0)[–5](#page-25-1). One of the key advantages of ammonia as a green energy source is its ability to be liquefied at low pressure and a relatively low temperature of 20 ◦C, making it an attractive candidate for hydrogen storage and transportation. As with many other chemical processes, catalysts play a crucial role in ammonia decomposition to achieve fast and efficient H<sup>2</sup> production. Experimental and theoretical studies have demonstrated that Rubased catalysts are the most active for ammonia decomposition[6–](#page-25-2)[8](#page-25-3). However, ruthenium's high cost and limited availability pose challenges for its large-scale industrial application. Therefore, developing new types of cost-effective catalysts for NH<sup>3</sup> decomposition, based on non-noble metals or metal oxides, has become a significant area of research for effective hydrogen generation[9](#page-25-4) . Numerous studies have focused on the activity of catalysts involving various metals and alloys[10](#page-25-5) . Among the most studied non-noble metals, iron (Fe) stands out as a leading catalyst due to its low cost and availability. While the reactivity of Fe is lower compared to other transition metals, it can be enhanced by using nanoparticles instead of extended surfaces. Indeed, it is well known that the reactivity of small-size clusters can be finely tuned by adjusting their size, geometry, and electronic structure, making them promising catalysts in various catalytic processes[11–](#page-25-6)[15](#page-25-7). For example, Nishimaki, et al.[16](#page-25-8) experimentally studied ammonia decomposition on Fe nanoparticles of various grain sizes (20 nm to 1 µm) in an ammonia steam environment. Their findings indicated that the highly reactive surface of nanoparticles enhances NH<sup>3</sup> dissociation without increasing the nitrogen content in the gas phase, resulting in nitride phases that depend on the grain size and morphology. As an alternative approach, ammonia decomposition reactions on small nanosized Fe clusters are frequently investigated using density functional theory (DFT) methods. Theoretical studies suggest that the mechanisms of ammonia decomposition involve stepwise dehydrogenation, where the rate-limiting step can vary depending on the size, type, and shape of the catalysts. Thus, G. Lanzani and K. Laasonen employed spin-polarized DFT to examine the adsorption and dissociation of NH<sup>3</sup> on a single nanosized icosahedral Fe<sup>55</sup> cluster[17](#page-25-9). Their research indicated that the overall reaction barrier for stepwise dehydrogenation was 1.48 eV, with different active sites on the Fe<sup>55</sup> cluster (facets and vertices), where the rate-limiting step was the initial hydrogen dissociation. Similarly, G.S. Otero et al.[18](#page-25-10) conducted a comprehensive comparative study on various sizes of Fe clusters (Fe16, Fe22, Fe32, Fe59, Fe80, Fe113, Fe190) and Fe(111) surfaces with additional adatoms. Their findings indicated that the reaction kinetics were influenced more by the strength of NH<sup>3</sup> adsorption rather than the activation energy barrier. Stronger NH<sup>3</sup> adsorption led to enhanced dissociation compared to desorption. The studies mentioned above primarily focus on the catalytic activities of large Fe clusters and Fe surfaces in the ammonia decomposition reaction. However, Xilin Zhang et al.[19](#page-25-11) specifically investigated the activities of relatively small Fe clusters, ranging from single Fe atoms to Fe<sup>4</sup> clusters. They found that the highest catalytic activity for stepwise NH<sup>3</sup> dehydrogenation was observed with nonatomic iron clusters. Interestingly, they observed that the rate-limiting steps differed: co-absorbate NH dissociation for Fe and Fe3, and co-absorbate NH<sup>2</sup> dissociation for Fe<sup>2</sup> and Fe4. The NH<sup>3</sup> decomposition reaction can be enhanced in the presence of oxygen, where it can proceed through various pathways, including ammonia oxidation and hydrogen evolution reactions. Moreover, metal oxides are commonly employed as catalyst supports in ammonia decomposition to enhance dispersion and catalytic stability. Among these supports, widely used materials include Al2O3, TiO2, as well as carbon nanotubes and nanofibers[7](#page-25-12)[,20](#page-26-0)[–24](#page-26-1). However, metal oxides not only serve as support but also play a crucial role in hydrogen evolution reactions in electrocatalysis, where the oxidation state of metals significantly influences the catalytic activity of ammonia decomposition. In particular, iron-based oxides, such as Fe2O3, are extensively studied forms of iron oxide due to their low cost and abundance, although their activity and stability can vary depending on their structure and size[25](#page-26-2)[–31](#page-26-3) . In this work, we elucidate the role of the size- and structural effects on the catalytic activity of iron-oxide-based nano-catalysts toward efficient ammonia decomposition. In particular, we investigated the theoretical mechanisms of stepwise ammonia decomposition on (Fe2O3)<sup>n</sup> clusters with n = 1−4 to compare the reactivity of different-sized Fe(III) oxides using the Artificial Force Induced Reaction (AFIR) method[32](#page-26-4)[,33](#page-26-5). Additionally, we examined the NH<sup>3</sup> adsorption and various energy barriers for NH<sup>3</sup> dehydrogenation on different active sites of Fe(III) oxides. Our investigation aims to contribute to the design of nanocatalysts based on Fe2O<sup>3</sup> by exploring the activity of small-sized Fe(III) oxide clusters. ### II. COMPUTATIONAL DETAILS All calculations were performed using spin-unrestricted Kohn-Sham DFT with Becke's threeparameter hybrid functional combined with the Lee, Yang, and Parr correlation functional, denoted as B3LYP[34–](#page-26-6)[36](#page-26-7). In our calculations we have employed the LANL2DZ[37](#page-26-8)[–39](#page-26-9) basis set with effective core potentials (ECP), as well as the Pople-style 6-31+G\* basis set, equivalent to 6- 31+G(d), which includes polarization (d) and diffuse (sp) functions, as it is implemented in the Gaussian 16 program[40](#page-26-10). These methods have been successfully applied to metals and metal oxide systems in previous studies. Thus, Glukhovtsev et al.[41](#page-27-0) reported that the performance of the B3LYP/ECP method for systems containing iron with various types of bonding showed good agreement with experimental data and high-level theoretical methods (CCSD(T), MCPE, CASSCF). Similarly, Taguchi, et al.[42](#page-27-1) studied Fe6O2(NO3)4(hmp)8(H2O)22, [Fe4(N3)6(hmp)6], and Fe8O3(OMe)(pdm)4(pdmH)4(MeOH)<sup>25</sup> clusters using the B3LYP/LANL2DZ level of theory, obtaining results that were consistent with experimental data. At the initial stage, the most stable isomers of iron trioxide for each selected size were investigated using the DFT method. A single iron trioxide molecule contains two Fe3<sup>+</sup> ions; therefore, there are often several energetically accessible spin states (0, 1, 2, 3, 4, 5). For the starting cluster Fe2O3, the lowest energy structure corresponds to the nonet state with a total spin S=4. For (Fe2O3)2, the lowest energy solution was found with a total spin S=10, indicating an increase in the number of Fe3<sup>+</sup> ions, which raises the total spin projection. For (Fe2O3)3, the lowest energy structure was found with a total spin S=15, and lastly, in the case of (Fe2O3)4, the lowest energy structure had a total spin S=20. Therefore, all clusters considered in our study were in a ferromagnetic configuration. We confirmed that spin contamination in the low-lying energy structures was negligible. To analyze the most favorable pathways of NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions catalyzed by small (Fe2O3)<sup>n</sup> (n = 1−4) clusters, we applied the SC-AFIR and DS-AFIR methods implemented in the Global Reaction Route Mapping (GRRM) strategy[32](#page-26-4)[,43–](#page-27-2)[46](#page-27-3). These automated reaction path search methods have been successfully applied to many catalytic reactions in combination with DFT methods[33,](#page-26-5)[47–](#page-27-4)[50](#page-27-5). The basic idea in the AFIR strategy is to push fragments (reactants) A and B of the whole system together or pull them apart by minimizing the following AFIR function[32](#page-26-4): The external force term in [\(1\)](#page-4-0) perturbs the given adiabatic Potential Energy Surface (PES), E(Q), with geometrical parameters Q in the AFIR function. Here, α defines the strength of the artificial force which depends on the weighted sum of the interatomic distances ri j between atoms i and j, with the weighths ωi j defined as This perturbation of the PES facilitates the exploration of additional approximate transition states and local minima on the surface. The model collision energy parameter γ in [\(3\)](#page-5-0) serves as an approximate upper limit for the barrier height that the system can be affected by the AFIR function[32](#page-26-4). In our calculations, γ was set to 300 kJ/mol for the entire system. During the initial reaction path search, the LANL2DZ basis set was applied with an artificial force to yield approximate products and transition states (TS). Subsequently, we utilized the 6-31+G\* basis set to optimize these approximate transition states and local minima without the artificial force, employing the Locally Updated Planes (LUP) method. The vibrational frequency calculations have been performed to confirm the nature of the stationary points, whether they are minima or transition states. The results presented in this paper include reaction route mapping at the B3LYP/LANL2DZ level and reaction pathways at the B3LYP/6-31+G(d) level. where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. where G(NH3@(Fe2O3)n) is the free energy of the most stable structure of the (Fe2O3)n cluster with the adsorbed ammonia molecule, G(Fe2O3)n) is the free energy of the bare (Fe2O3)n cluster, and G(NH3) is the free energy of a single ammonia molecule. The values of free energy G in [\(5\)](#page-5-1) can be calculated as follows: ### III. RESULTS AND DISCUSSION In the present work we systematically investigated the ammonia decomposition reaction mechanisms on (Fe2O3)<sup>n</sup> clusters of various sizes n, where n = 1−4. Firstly, we identified approximate reaction pathways for the interactions between NH<sup>3</sup> molecules and the most stable isomers of (Fe2O3)<sup>n</sup> clusters using the AFIR technique. The obtained AFIR pathways were subsequently re-optimized along the minimum energy path using the Locally Updated Plane (LUP) method, without applying artificial forces. We calculated various reaction mechanisms and the stepwise dissociation[51](#page-27-6) of hydrogen atoms from nitrogen-containing compounds on Fe(III) oxide clusters, following the elementary steps: Here <sup>∗</sup> denotes the adsorbed intermedeates on the (Fe2O3)<sup>n</sup> cluster's surface. Finally, the adsorbed hydrogen atoms on the (Fe2O3)<sup>n</sup> clusters can combine to produce molecular hydrogen (H2): The paper is organized as follows. We first discuss the structures of free clusters, followed by the adsorption of NH<sup>3</sup> on the most stable isomers of (Fe2O3)n, n = 1 − 4, clusters. We then examine the complete dehydrogenation and H<sup>2</sup> formation processes for each cluster size. ## A. Structure of (Fe2O3)<sup>n</sup> clusters with n = 1−4 Figure [1](#page-8-0) demonstrates the most stable structures of small (Fe2O3)<sup>n</sup> clusters with n = 1 − 4, as obtained in the present work using automated GRRM approach. A total of up to 60 isomer structures have been obtained for each cluster size n. The low-energy isomers for each cluster size, along with their relative binding energies are presented in Figs. S1-S4. The most stable structures are consistent with those obtained in our previous DFT study, which employed B3LYP functional and five different types of basis sets (LANL2DZ, 6-31+G∗ , 6-311+G∗ , Sapporo-DZP, and augcc-pVTZ).[52](#page-27-7) We found that the most stable structure of the smallest Fe2O<sup>3</sup> cluster is a nonet kite-like type with a binding energy Eb=362.7 kcal/mol. The kite-like structure is a commonly studied configuration[53](#page-27-8)[,54](#page-27-9) and was previously investigated by Sierka et al.[55](#page-28-0), who observed the most stable spin configuration for this structure to be S=0. In contrast, we found that the lowest energy structure corresponds to a nonet state with S=4, while the singlet kite-like structure is 0.62 kcal/mol less stable as shown in Table S1. The results of our calculations show that the absolute binding energy of (Fe2O3)<sup>n</sup> rapidly increases with increasing cluster size n from 1 to 2 by 60.4 kcal/mol. However, further growth in binding energy with cluster size slows down, demonstrating a tendency for saturation as n increases. ## <span id="page-8-1"></span>B. Ammonia adsorption on (Fe2O3)<sup>n</sup> clusters Adsorption of ammonia on (Fe2O3)<sup>n</sup> clusters is a crucial initial step in the whole dehydrogenation process. Figure [2](#page-9-0) demonstrates the most stable adsorption configurations of the NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters with n = 1−4. The corresponding free energies of adsorption and Fe−N bond distances are shown in Table [I](#page-10-0) at 0 K. Our calculations show that the adsorption of NH<sup>3</sup> on the smallest Fe2O<sup>3</sup> cluster is the most stable among all cluster sizes considered in this study, with an adsorption free energy of -33.68 kcal/mol. This finding is corroborated by Mulliken charge analysis, which shows that more electrons are shared between the lone pair of the N atom and the 3d orbitals of Fe2<sup>+</sup> for <sup>n</sup> <sup>=</sup> 1. On the other hand, for larger cluster sizes with <sup>n</sup> <sup>=</sup> <sup>2</sup>−4, which primarily contain Fe3+, the electron density is more localized over the bonding region, as also reported by Sierka et al.[55](#page-28-0). Therefore, bonding occurs with the nitrogen lone pair. Our theoretical analysis indicates that the adsorption energy ∆Gads of ammonia on (Fe2O3)<sup>n</sup> clusters decreases from n = 1 to n = 3, followed by a slight increase for n = 4. A similar trend in the change of adsorption energy with cluster size was reported by Shulan Zhou et al.[56](#page-28-1) for Run@CNT systems. We also compared the adsorption energy of NH<sup>3</sup> on different metal and metal oxides in Table [I.](#page-10-0) The obtained NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters are about 10 kcal/mol higher than the data reported by Zhang et al. for the Ru(0001) surface[57](#page-28-2). Moreover, the adsorption of NH<sup>3</sup> and NO<sup>x</sup> on the γ-Fe2O3(111) surface was studied by Wei Huang et al.[58](#page-28-3) using periodic density functional calculations. They calculated adsorption energies on octahedral and tetrahedral sites of γ-Fe2O3(111) to be -2.13 kcal/mol and -21.68 kcal/mol, respectively. Similarly, our calculated NH<sup>3</sup> adsorption energies on (Fe2O3)<sup>n</sup> clusters for n = 3 and n = 4 are close to the data reported by Wei Huang et al.,[58](#page-28-3) as adsorption of NH<sup>3</sup> on the three-coordinated Fe3<sup>+</sup> site resembles the tetrahedral site of γ-Fe2O3(111), while the adsorption on the four-coordinated Fe3<sup>+</sup> site resembles the octahedral site of γ-Fe2O3. <span id="page-9-0"></span>![FIG. 2: The optimized geometries of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> for n = 1−4; N−Fe distances (Å) are shown in parentheses along with the partial atomic charges on neighboring atoms. The total spin S of the clusters is shown in inserts.](path) As mentioned above, the calculated adsorption energies indicate that the adsorption of an NH<sup>3</sup> molecule on (Fe2O3)<sup>n</sup> clusters (n = 1−4) weakens as the cluster size increases from n = 1 to n = 3. In industrial processes, the dehydrogenation of ammonia typically occurs at high temperatures, often in the range of 400◦C to 700◦C, depending on the specific catalysts and conditions used. Therefore, it is important to determine the range of temperatures at which ammonia adsorption on (Fe2O3)<sup>n</sup> remains stable. Figure S5 demonstrates the temperature dependence of ∆Gads in the range from 0 K to 1200 K for the most stable adsorption configurations of NH<sup>3</sup> on (Fe2O3)n clusters (n = 1 − 4). The negative values of ∆Gads correspond to stable adsorption. As seen in Fig. S5, NH<sup>3</sup> adsorbed on the smallest Fe2O<sup>3</sup> cluster is stable across the entire temperature range of 0 K to 1200 K. However, for larger cluster sizes, ammonia adsorption becomes energetically unfavorable at temperatures of 1107 (K), 961 (K), and 1000 (K) for n = 2,3, and 4, respectively. ## C. NH<sup>3</sup> decomposition on Fe2O<sup>3</sup> In this section, we discuss the complete NH<sup>3</sup> decomposition and H<sup>2</sup> formation reactions [\(7\)](#page-6-0) - [\(12\)](#page-7-0) on the smallest considered cluster, Fe2O3, at room temperature, T=298.15 K, explored by the AFIR method. This method allows for the automatic exploration of the full reaction path network, systematically accounting for the variety of possible isomer structures and adsorption sites. This is an important approach in nanocatalysis because it has been demonstrated that the most stable structures are not always the most reactive. Therefore, a systematic search for reaction pathways that accounts for the contributions of low-energy isomers is required to accurately describe the catalytic properties of clusters at finite temperatures.[49](#page-27-10) To illustrate the isomer and reaction-site effects, we explicitly consider two different isomers of the Fe2O<sup>3</sup> cluster: the most stable kite-like structure with one terminal oxygen atom, and the linear structure isomer with two terminal oxygen atoms which is 6.24 kcal/mol less stable (see Fig. S1). The kite-like structure possess two type of catalytically active metal centers - two-coordinated and three-coordinated Fe sites. Therefore we consider adsorption and decomposition of NH<sup>3</sup> molecule on both of them. Figure [3\(](#page-13-0)a) demonstrates that the adsorption of NH<sup>3</sup> on the kite-like Fe2O<sup>3</sup> cluster is an exothermic reaction, occurring at both the two-coordinated and three-coordinated Fe sites. The adsorption free energies are -26.98 kcal/mol for the two-coordinated Fe site (intermediate I′ 1 1) and -11.29 kcal/mol for the three-coordinated Fe site (intermediate I′′ 1 1), respectively. The optimized structures of all intermediates (I) and transition satates (T) along the reaction pathways are shown in Fig. [3\(](#page-13-0)b) and [4\(](#page-14-0)b), for the kite-like and linear clusters, respectively. Here the lower index corresponds to the cluster size n, while the numbering corresponds to the order of intermedeates (transition states) along the reaction path. As discussed in the previous section, the most stable adsorption site for NH<sup>3</sup> is the two-coordinated Fe site, with an Fe−N bond length of 2.11 Å. In contrast, the Fe−N bond length at the three-coordinated Fe site is 2.16 Å. These findings are supported by the fact that NH<sup>3</sup> adsorption highly depends on the local geometry and electronic structure of the catalyst. In the case of the Fe2O<sup>3</sup> kite-like structure, the first dehydrogenation reaction is the second step in the reaction mechanism, occurring after adsorption with activation barriers of 26.98 kcal/mol and 22.12 kcal/mol through the reaction paths I′ 1 1-T′ 1 1-I′ 1 2 and I′′ 1 1-T′′ 1 1-I′′ 1 2, respectively. The reactions on these two-coodrinated and three-coordinated active sites are exothermic by 16.31 kcal/mol and 7.53 kcal/mol, respectively. However, the first dehydrogenation of NH<sup>3</sup> on the lineartype structure Fig. [4\(](#page-14-0)a) occurs with smaller activation barrier of 16.22 kcal/mol via the reaction path I<sup>L</sup> 1 1 - T<sup>L</sup> 1 1 - I<sup>L</sup> 1 2, demonstrating that the less stable linear isomer is more reactive. The role of Fe2O<sup>3</sup> isomer structure on NH<sup>3</sup> adsorption and first hydrogen atom transfer was previousely studied by Chaoyue Xie et al.[60](#page-28-5) They performed DFT-D3 calculations on the adsorption mechanisms of different molecules (NH3, NO, O2) on activated carbon (AC) supported iron-based catalysts FexOy/AC. The calculated adsorption electronic energies of NH<sup>3</sup> were -37.4 kcal/mol and -53.7 kcal/mol on different isomers of Fe2O3/AC, and the first hydrogen atom transfer had an activation barrier of 15.5 kcal/mol. Similarly, the adsorption and dehydrogenation of ammonia on different metal oxides were investigated by Erdtman and co-workers[62](#page-28-7) for the application of gas sensors. They reported that the adsorption energy of NH<sup>3</sup> on the RuO2(110) surface is -38.24 kcal/mol, and the first N−H bond cleavage had an activation energy barrier of 17.45 kcal/mol. The third step of the NH<sup>3</sup> dehydrogenation reaction [\(9\)](#page-6-1) involves the dissociation of the adsorbed NH∗ 2 intermediate into NH∗ and H∗ species. In this step, the abstracted hydrogen atom transfers to one of the oxygen atoms in the cluster. Figure [3\(](#page-13-0)a) demonstrates, that in the case of the kite-like structure the energy barriers for this step are 43.91 kcal/mol and 34.51 kcal/mol, corresponding to the reaction paths I′ 1 2 - T′ 1 2 - I′ 1 3 and I′′ 1 2 - T′′ 1 2 - I′′ 1 3, respectively. In the fourth step [\(10\)](#page-6-2), the adsorbed NH∗ intermediate further dissociates into N∗ and H∗ species as shown in Fig. [3\(](#page-13-0)a). The reaction barriers associated with this step are 46.98 kcal/mol and 8.95 kcal/mol for the two-coordinated and three-coordinated reaction paths, respectively. The decomposition of NH<sup>3</sup> on kite-like structures becomes endothermic starting from the third step [\(9\)](#page-6-1). Our calculations reveal that NH<sup>3</sup> dehydrogenation has a high energy barrier when the NH<sup>3</sup> molecule is adsorbed at a two-coordinated Fe site, which is the most stable adsorption site. On the other hand, dehydrogenation of the adsorbed NH<sup>3</sup> at a three-coordinated Fe site has a considerably lower activation barrier of 8.95 kcal/mol for the reaction step [\(10\)](#page-6-2). Overall, for the NH<sup>3</sup> decomposition reaction on the kite-like Fe2O<sup>3</sup> structure, with initial NH<sup>3</sup> adsorption on the two-coordinated Fe atom, the rate-limiting step is the fourth reaction [\(10\)](#page-6-2), with a barrier of 46.98 kcal/mol. Alternatively, for the less favorable NH<sup>3</sup> adsorption on the threecoordinated Fe atom, the rate-limiting step is the third reaction step [\(9\)](#page-6-1), with a barrier of 34.51 kcal/mol. The reaction pathway calculated for NH<sup>3</sup> decomposition on the linear-type Fe2O<sup>3</sup> isomer is shown in Fig. [4\(](#page-14-0)a), and respective intermediate and transition state structures are shown in Fig. [4\(](#page-14-0)b). Since this structure consists of two iron atoms connected through a central oxygen, each containing a terminal oxygen, the reaction mechanism differs slightly from that of the kite-like isomer. For instance, in the third step of the reaction, the second hydrogen from the adsorbed NH∗ 2 intermediate is transferred to the second terminal oxygen. The energy barrier for this step on the linear-type structure is 23.8 kcal/mol, as shown in the reaction path (I<sup>L</sup> 1 2 - T<sup>L</sup> 1 2 - I<sup>L</sup> 1 3) in Fig. [4a](#page-14-0). The fourth step on this isomer is not straightforward, involving the central oxygen atom breaking its bond with one of the neighboring iron atoms while forming an Fe − N − Fe bridge. This process leads to two different intermediates: the formation of the adsorbed H2O ∗ and the transfer of a hydrogen atom from one side of the Fe − N − Fe bridge to the other. Subsequently, the final dehydrogenation step from the NH∗ intermediate occurs, with an activation energy barrier of 34.76 kcal/mol. <span id="page-13-0"></span>![FIG. 3: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ reaction path on the kite-like isomer of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-14-0"></span>![FIG. 4: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> + → N ∗ + 3H∗ reaction path on the linear-type isomet of Fe2O<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) As a next step we consider possible H<sup>2</sup> formation via reactions [\(11\)](#page-6-3) and [\(12\)](#page-7-0) on the kite-like and linear isomers of Fe2O<sup>3</sup> cluster. The possible pathways for H<sup>2</sup> formation in the case of the most stable ammonia adsorption on the two-coordinated site (I′ intermediates) of the kite-like Fe2O<sup>3</sup> isomer are shown in Fig. [5\(](#page-15-0)a), while the corresponding structures of the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [5\(](#page-15-0)b). Note that H<sup>2</sup> formation can occur after partial decomposition of ammonia in reaction [\(11\)](#page-6-3), starting from intermediate (I<sup>I</sup> 1 3) via the path I′ 1 3 - T′ 1 6 - I′ 1 7 - T′ 1 7 - I′ 1 8. On the other hand, H<sup>2</sup> formation can occur via full decomposition of ammonia in reaction [\(12\)](#page-7-0), through the intermediate (II 1 4) via the path I′ 1 4 - T′ 1 4 - I′ 1 5 - T′ 1 5 - I′ 1 6. In both cases, the reaction pathways include breaking one O−H bond and forming an Fe−H bond. The H<sup>2</sup> formation barriers through these intermediates are 89.74 kcal/mol and 92.49 kcal/mol, respectively. From these results, we conclude that H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> structure is more favorable via reaction [\(11\)](#page-6-3), with the NH<sup>∗</sup> intermediate remaining adsorbed on the cluster. The H<sup>2</sup> formation reaction, starting from (I<sup>I</sup> 1 4), is the rate-limiting step in molecular hydrogen formation on the kite Fe2O<sup>3</sup> cluster. Similarly, the H<sup>2</sup> formation reaction pathways on the linear-type structure of Fe2O<sup>3</sup> are shown in Fig. [6\(](#page-16-0)a), while the optimized equilibrium and transition states along the reaction path are illustrated in Fig. [6\(](#page-16-0)b). The H<sup>2</sup> formation through the NH<sup>∗</sup> intermediate (I<sup>L</sup> 1 4) via the reaction path I<sup>L</sup> 1 4 - T<sup>L</sup> 1 8 - I<sup>L</sup> 1 9 - T<sup>L</sup> 1 9 - I<sup>L</sup> 1 10 has an energy barrier of 79.99 kcal/mol. On the other hand H<sup>2</sup> formation through intermediate (I<sup>L</sup> 1 6) via reaction path I<sup>L</sup> 1 6 - T<sup>L</sup> 1 6 - I<sup>L</sup> 1 7 - T<sup>L</sup> 1 7 - I<sup>L</sup> 1 8 has an activation energy of 70.84 kcal/mol, which is about 10 kcal/mol lower energy than reaction path through intermediate (I<sup>L</sup> 1 4). Overall, on the basis of our calculated reaction pathways for H<sup>2</sup> formation show similar pattern for both kite-type and linear-type Fe2O3, where H<sup>2</sup> formation in reactions [\(11\)](#page-6-3) and [\(12\)](#page-7-0) take place via breaking one of O−H bond and forming intermediate Fe−H bond. However, from both thermodynamic and kinetic perspectives, H<sup>2</sup> formation on the two types of Fe2O<sup>3</sup> structures varies. Reaction [\(11\)](#page-6-3) is more favorable on the kite-like structure, while reaction [\(12\)](#page-7-0) is more favorable on the linear structure. This highlights that the rate-limiting step for H<sup>2</sup> formation is highly dependent on the catalyst's structure. <span id="page-15-0"></span>![FIG. 5: (a) The energy profile for H<sup>2</sup> formation on the kite-like Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) <span id="page-16-0"></span>![FIG. 6: (a) The energy profile for H<sup>2</sup> formation on the linear isomer of the Fe2O<sup>3</sup> cluster at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## D. NH<sup>3</sup> decomposition on Fe4O<sup>6</sup> In the following subsection, we discuss the catalytic activity of (Fe2O3)<sup>2</sup> towards NH<sup>3</sup> dehydrogenation and H<sup>2</sup> formation reactions. On the basis of adsorption characteristics discussed in [III B,](#page-8-1) the threefold coordinate Fe3<sup>+</sup> site of the Fe4O<sup>6</sup> cluster is the most stable site for NH<sup>3</sup> adsorption. Complete reaction pathway for stepwise decomposition of NH<sup>3</sup> and formation of H<sup>2</sup> reactions on (Fe2O3)<sup>2</sup> cluster are depicted in Fig. [7\(](#page-18-0)a), and the corresponding intermediate and transition state structures are shown in Fig. [7\(](#page-18-0)b). From this point forward, the first dehydrogenation step follows starting from the intermediate (I21) where NH<sup>3</sup> molecule interacting with three-coordinated Fe site of (Fe2O3)<sup>2</sup> cluster by transferring a hydrogen to its one of neighboring oxygen via reaction pathway (I21 - T11 - I22) and reaction barrier of this step is 21.47 kcal/mol which is 5.51 kcal/mol lower energy barrier than first hydrogen transfer on kite-like Fe2O<sup>3</sup> cluster. This reaction also involves different isomer of (Fe2O3)2, where decompostion takes place on the second minima isomer of (Fe2O3)<sup>2</sup> shown in Fig. S2. Relative binding energy of second minima isomer is 2.35 kcal/mol. The second dehydrogenation step follows from adsorbate NH∗ 2 intermediate (I22) further dissociate to NH∗ + 2H∗ which dissociated hydrogen atom subsequently transferred to another neighboring oxygen as shown in the reaction path (I22 - T22 - I23). This reaction occurs with energy barrier of 38.57 kcal/mol. The ultimate dehydrogenation step is the formation of N∗ + 3H<sup>∗</sup> where N is bound to the central top Fe3<sup>+</sup> and all the hydrogen atoms interact with three neighboring oxygens. The last dehydrogenation step occurs with the energy barrier 3.86 kcal/mol higher than the energy barrier of the second dehydrogenation step and it is shown in the reaction pathway (I23 - T23 - I24). It suggests that dehydrogenation of adsorbate NH<sup>∗</sup> is rate-determining step on (Fe2O3)<sup>2</sup> cluster. Moreover, from a thermodynamic viewpoint calculated dehydrogenation steps of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster is endothermic by 6.24, 18.6, and 23.78 kcal/mol. Considering H<sup>2</sup> formation reactions via two reaction pathways. First H<sup>2</sup> formation reaction [\(11\)](#page-6-3) occurs with partial decomposition of NH<sup>3</sup> starting from intermediate (I23) through (I29). The first stage through this reaction path starting from (I23), the transition state (T26) was found that the H atom adsorbed on the Fe atom and formed a Fe−H bond. In the second stage of the reaction, the transition state (T27) was the one that splits the adsorbed H atom from the adjacent O atom to adsorbed NH∗ . Then, the dissociated H atom was adsorbed in the O atom which is an adjacent atom to the Fe−H bond, and at the final stage, the dissociative molecular H<sup>2</sup> formed, and barrier of this reaction is 91.1 kcal/mol. Completed reaction pathway for reaction [\(11\)](#page-6-3) is (I23 - T26 - I27 - T27 - I28 - T28 - I29). The second H<sup>2</sup> formation reaction [\(12\)](#page-7-0) is that occurs with fully decomposed NH<sup>3</sup> molecule starting from intermediate (I24) through intermediate (I26). It is important to note that last dehydrogenation reaction [\(10\)](#page-6-2) is the one which has the highest barrier on the (Fe2O3)<sup>2</sup> cluster. So dissociative molecular hydrogen formation through this reaction path cost an energy as shown in reaction path (I24 - T24 - I25 - T25 - I26). Overall, as it seen from depicted reaction pathways in Fig. [7,](#page-18-0) H<sup>2</sup> formation reaction is kinetically and energetically costly in reaction N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2, and it is more favorable via reaction NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> which is partial decomposition of NH<sup>3</sup> on (Fe2O3)<sup>2</sup> cluster. <span id="page-18-0"></span>![FIG. 7: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>2</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## E. NH<sup>3</sup> decomposition on Fe6O<sup>9</sup> The energy profile for the stepwise dehydrogenation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is presented in Fig. [8\(](#page-20-0)a), while the intermediate and transition state structures along this reaction pathway are shown in Fig. [8\(](#page-20-0)b). The dissociation of NH<sup>3</sup> on the (Fe2O3)<sup>3</sup> cluster is more complex compared to smaller Fe(III) oxide structures, as NH<sup>3</sup> can adsorb at various sites on the (Fe2O3)<sup>3</sup> surface. We identified the most favorable adsorption configuration, I31, with an adsorption energy of ∆G = −21.51 kcal/mol, from which the stepwise decomposition reaction proceeds. The first dehydrogenation reaction, as described in [\(8\)](#page-6-4), begins with NH∗ 3 adsorbed on the (Fe2O3)<sup>3</sup> cluster as I31 and proceeds through the transition state T31. The energy barrier along this pathway is 22.75 kcal/mol, which is slightly higher than the barrier for the first H abstraction from NH<sup>3</sup> on the (Fe2O3)<sup>2</sup> cluster. Although the first dehydrogenation reaction on the (Fe2O3)<sup>3</sup> cluster is endothermic, we observed that when the NH∗ 2 species migrates to a bridging position between two Fe atoms (Fe − N − Fe), the reaction becomes exothermic by 11.44 kcal/mol, as shown in the reaction pathways I32 − T32 − I33 and I33 − T33 − I34. The second H abstraction involves the further dehydrogenation of NH∗ 2 into NH∗ and H∗ , with an energy barrier of 35.97 kcal/mol along the pathway I34 − T34 − I35. This barrier is 10 kcal/mol higher than that of the first dehydrogenation step. Additionally, this reaction is endothermic, with a reaction energy of 15.74 kcal/mol. Similarly, in the third step [\(10\)](#page-6-2), the remaining NH∗ dissociates into N∗ and H∗ , with an energy barrier 17.94 kcal/mol higher than that of the second dissociation step. This is the largest barrier encountered in the decomposition of NH3. The calculated reaction pathway indicates that this process is endothermic, with a reaction energy of 25.76 kcal/mol. Lastly, the possible H<sup>2</sup> formation reactions [\(11](#page-6-3) and [12\)](#page-7-0) on the (Fe2O3)<sup>3</sup> cluster were calculated, as shown in Fig. [8.](#page-20-0) The first H<sup>2</sup> formation reaction [\(11\)](#page-6-3) begins with one adsorbed NH<sup>∗</sup> and two H∗ species on the (Fe2O3)<sup>3</sup> cluster. The reaction proceeds in a manner similar to that discussed in the previous subsection: the adsorbed H∗ on oxygen, adjacent to the NH∗ adsorbed on Fe, migrates away by forming Fe−H bonds through transition states T37 and T38. The overall energy barrier for H<sup>2</sup> formation via reaction [\(11\)](#page-6-3) is 100.74 kcal/mol. The second possible H<sup>2</sup> formation pathway starts from fully decomposed NH<sup>3</sup> (I36) and proceeds through the transition state T36. This pathway has a significantly high energy barrier, calculated to be 116.89 kcal/mol, as shown in the reaction path I36 − T36 − I37. These results suggest that, from both a thermodynamic and kinetic perspective, H<sup>2</sup> formation after full dehydrogenation of NH<sup>3</sup> is less favorable. <span id="page-20-0"></span>![FIG. 8: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction paths on the (Fe2O3)<sup>3</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ## F. NH<sup>3</sup> decomposition on Fe8O<sup>12</sup> Finally, the decomposition of NH<sup>3</sup> and the H<sup>2</sup> formation pathways on the (Fe2O3)<sup>4</sup> cluster are illustrated in Fig. [9\(](#page-21-0)a), with the intermediate and transition state structures shown in Fig. [9\(](#page-21-0)b). As discussed in previous subsections, increasing the number of units n in (Fe2O3)<sup>n</sup> increases the number of active sites that interact with NH3. However, similar to the reactions on (Fe2O3)<sup>n</sup> (n = 2,3), the most stable adsorption site for NH<sup>3</sup> on (Fe2O3)<sup>4</sup> is a three-coordinated Fe site, with an adsorption energy of -21.94 kcal/mol at room temperature, slightly higher than that on (Fe2O3)3. The dehydrogenation of NH<sup>3</sup> begins with the adsorption of NH<sup>∗</sup> 3 , as shown in the intermediate state I41. The first dehydrogenation step involves breaking one N−H bond and forming an O−H bond, with an energy barrier of 22.48 kcal/mol, as shown in the reaction pathway I41 − T41 − I42. The second dehydrogenation step [\(9\)](#page-6-1) involves the dissociation of NH∗ 2 + H∗ to form NH∗ + 2H∗ , proceeding through the transition state T42. The energy barrier for this step is 43.96 kcal/mol, which is higher than the corresponding second dehydrogenation steps on (Fe2O3)<sup>n</sup> (n = 1 − 3). The final dehydrogenation step occurs along the pathway I43 − T43 − I44, with a barrier of 42.24 kcal/mol. All NH<sup>3</sup> dehydrogenation steps on (Fe2O3)<sup>4</sup> are endothermic, with reaction energies of 3.85 kcal/mol, 15.39 kcal/mol, and 41.47 kcal/mol, respectively. The final reaction pathway on the (Fe2O3)<sup>4</sup> cluster involves H<sup>2</sup> formation from both partially and fully decomposed NH3, as described in [\(11\)](#page-6-3) and [\(12\)](#page-7-0). As observed for all sizes of (Fe2O3)<sup>n</sup> clusters, H<sup>2</sup> formation is energetically more favorable after the partial decomposition of NH<sup>3</sup> in reaction [\(11\)](#page-6-3) compared to the fully decomposed pathway [\(12\)](#page-7-0). However, this pathway also presents the highest energy barrier on this cluster. <span id="page-21-0"></span>![FIG. 9: (a) The energy profile for NH∗ <sup>3</sup> → NH<sup>∗</sup> 2 + H<sup>∗</sup> → NH<sup>∗</sup> + 2H<sup>∗</sup> → N ∗ + 3H∗ and H<sup>2</sup> formation reaction path on the (Fe2O3)<sup>4</sup> at T=298.15 K. (b) Geometries of the optimized equilibrium and transition states along the reaction path.](path) ### IV. COMPARISON AND CONCLUSION Our results, illustrated in Fig. [3,](#page-13-0) Fig. [4,](#page-14-0) Fig. [7,](#page-18-0) Fig. [8,](#page-20-0) and Fig. [9,](#page-21-0) indicate that NH<sup>3</sup> dehydrogenation can be a thermodynamically favorable reaction on (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the favorability depends on the size and geometry of the cluster, as well as the specific reaction steps described in [\(8\)](#page-6-4) − [\(12\)](#page-7-0). culated the change in Gibbs free energy (∆G) as a function of temperature at 1 bar pressure, as shown in Fig. S6. Across all reactions studied, we observed that ∆G increases with temperature. This suggests that NH<sup>3</sup> dehydrogenation on (Fe2O3)<sup>n</sup> (n = 2,4) can be energetically favorable at moderate temperatures, depending on the specific reaction step. However, as the temperature rises beyond a certain threshold, the reaction becomes unfavorable. For example, as shown in Fig. S6 (a), (b), and (c), all dehydrogenation reactions on (Fe2O3)<sup>n</sup> (n = 1) are energetically favorable within the temperature range of 0−1000 K. In contrast, on (Fe2O3)<sup>n</sup> (n = 2,4), only the last dehydrogenation step is limiting. Since the ∆G of the third dehydrogenation reaction is already greater than zero at 0 K, this step is not favorable at any temperature. Another larger cluster considered in this study, (Fe2O3)<sup>n</sup> (n = 3), exhibits better stability of the reaction intermedeates during the second dehydrogenation step, remaining favorable up to 800 K. On the other hand, the second dehydrogenation reaction on (Fe2O3)<sup>n</sup> (n = 4) is favorable only up to 400 K. The most endothermic dehydrogenation reaction on this cluster is the step NH∗ ⇌ N ∗ + 3H∗ . The first and second dehydrogenation steps are favorable up to 1100 K and 700 K, respectively. Moreover, we observed the variation of ∆G with temperature for the H<sup>2</sup> formation reaction on (Fe2O3)<sup>n</sup> (n = 1 − 4). Our results indicate that the formation of molecular hydrogen is not thermodynamically favorable at any temperature. However, temperature is not the only factor determining whether the reaction occurs. If sufficient energy is available to overcome the activation barrier, the reaction can still proceed. The effective production of molecular hydrogen from ammonia is determined by the stepwise dehydrogenation of adsorbed ammonia on the catalyst. Catalytic reaction mechanisms are analyzed by identifying the rate-determining step in the dehydrogenation of NH3, which corresponds to the step requiring the highest energy to activate the N−H bond. However, it is important to note that in catalysis, the overall energy barrier is more significant than the barrier for any single intermediate reaction step. Several studies have reported different rate-determining steps depending on the type of catalyst used[63](#page-28-8). Xiuyuan Lu et al. found that the rate-determining step in NH<sup>3</sup> decomposition on different phases of Ru surface catalysts is the formation of molecular nitrogen[64](#page-28-9). In contrast, studies by Xilin Zhang et al.[19](#page-25-11) on ammonia decomposition on small iron clusters showed that the ratedetermining step on single Fe and Fe<sup>3</sup> is the NH → N + H step, whereas for Fe<sup>2</sup> and Fe4, the rate-determining step is the NH<sup>2</sup> → NH + H step. Similarly, a detailed comparison of the energy barriers for each elementary step in NH<sup>3</sup> decomposition and H<sup>2</sup> formation on different sizes and shapes of (Fe2O3)<sup>n</sup> (n = 1 − 4) is shown in Fig. [10.](#page-23-0) Based on the results from our calculations, the rate-determining step in ammonia decomposition and H<sup>2</sup> formation varies with the size of the (Fe2O3)<sup>n</sup> (n = 1−4) oxide clusters. In general, the final step of H<sup>2</sup> formation represents the highest energy barrier on all (Fe2O3)<sup>n</sup> (n = 1−4) clusters. However, the analysis of NH<sup>3</sup> decomposition shows that the NH → N + H step is typically the rate-determining step, except in the case of (Fe2O3)4, where the rate-determining step is the second H dissociation step. Furthermore, the first dehydrogenation step exhibits an energy barrier that is nearly identical across all clusters, with the process being exothermic for clusters n = 1 and n = 3, and endothermic for clusters n = 2 and n = 4. For the second dehydrogenation step, (Fe2O3)<sup>3</sup> demonstrates significantly higher activity compared to the other cluster sizes. It is also important to note that n = 1 (linear) is the only special configuration of Fe2O<sup>3</sup> containing two terminal O−<sup>2</sup> ions, unlike the other types of Fe2O3, which may promote a potentially high activity for NH<sup>3</sup> dehydrogenation and molecular hydrogen formation. Overall, the lowest energy barrier observed for H<sup>2</sup> formation is associated with the largest cluster considered in this study. In this research, various structures of (Fe2O3)<sup>n</sup> (n = 1−4) were obtained using the SC-AFIR method, and we investigated the ammonia decomposition and molecular hydrogen formation reaction pathways on the most stable isomers of (Fe2O3)<sup>n</sup> (n = 1−4) clusters. This analysis employed the SC-AFIR and DS-AFIR methods within the Global Reaction Route Mapping (GRRM) strategy, utilizing the B3LYP exchange-correlation functional in Kohn-Sham DFT. The results indicate that the catalytic activity in ammonia decomposition varies depending on the size and shape of the high-spin iron trioxides. The adsorption analysis reveals that the NH<sup>3</sup> molecule preferentially adsorbs at two-coordinated Fe sites in n = 1, and at three-coordinated Fe sites in n = 2 − 4 clusters. Furthermore, the adsorption energy tends to decrease from n = 1 to n = 3 of the (Fe2O3)<sup>n</sup> clusters, then slightly increases for the (Fe2O3)<sup>4</sup> cluster. From a thermodynamic perspective, the adsorption of the NH<sup>3</sup> molecule on (Fe2O3)<sup>1</sup> is favorable across the entire temperature range of 0 K to 1200 K. In contrast, for the larger clusters (Fe2O3)<sup>n</sup> (n = 2,4), ammonia adsorption becomes energetically unfavorable at temperatures of 1107 K, 961 K, and 1000 K for n = 2,3, and 4, respectively. A comparison of the rate-determining steps in the ammonia dehydrogenation reaction reveals a dependency on the size of the iron trioxide clusters. Thus, the reaction step NH<sup>∗</sup> → N ∗ +H ∗ is the rate-determining step for the smaller iron trioxide clusters (Fe2O3)<sup>n</sup> (n = 1 − 3). In contrast, the reaction step NH∗ <sup>2</sup> → NH<sup>∗</sup> +H ∗ is identified as the rate-determining step for the (Fe2O3)<sup>n</sup> (n = 4) cluster. Additionally, we observed that the energy barrier for molecular hydrogen formation increases with the size of the clusters (Fe2O3)<sup>n</sup> (n = 1−3) but then experiences a drastic decrease for the (Fe2O3)<sup>4</sup> cluster. We have investigated the catalytic activity of high-spin (Fe2O3)<sup>n</sup> (n = 1 − 4) clusters for decomposition of NH3. We believe that the results are valuable for designing iron trioxide-based nanosized catalysts by regulating the size of the (Fe2O3)<sup>n</sup> clusters to enhance H<sup>2</sup> production from the catalytic decomposition of ammonia. ### ACKNOWLEDGMENTS This work was partly supported by MEXT Program: Data Creation and Utilization-Type Material Research and Development Project Grant Number JPMXP1122712807, and partially supported by NAWA "STE(E)R-ING towards International Doctoral School" Calculations were performed using computational resources of the Institute for Solid State Physics, the University of Tokyo, Japan, and the Research Center for Computational Science, Okazaki, Japan (Project: 23- IMS-C016). S.I. is grateful to the MANABIYA system of the Institute for Chemical Reaction Design and Discovery (ICReDD) of Hokkaido University, which was established by the World Premier International Research Initiative (WPI), MEXT, Japan, to support the learning of the GRRM program techniques for DFT calculations. ### REFERENCES - Q. Shi, Y. Zhou, J. Cheng, Y. Pan, Y. Wu, L. Zhu, and Z. Yuan, [Microporous and Mesoporous](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) Materials 332[, 111681 \(2022\).](https://doi.org/https://doi.org/10.1016/j.micromeso.2022.111681) - M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Iz- maylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, and D. J. Fox, "Gaussian 16 Revision C.01," (2016), gaussian Inc. Wallingford CT. - <sup>46</sup>S. Maeda, Y. Harabuchi, M. Takagi, K. Saita, K. Suzuki, T. Ichino, Y. Sumiya, K. Sugiyama, and Y. Ono, "Implementation and performance of the artificial force induced reaction method in the grrm17 program," (2018). - <span id="page-27-4"></span><sup>47</sup>A. K. Sharma, W. Sameera, M. Jin, L. Adak, C. Okuzono, T. Iwamoto, M. Kato, M. Nakamura, and K. Morokuma, Journal of the American Chemical Society 139, 16117 (2017). ## Supporting Information for # "Theoretical design of nanocatalysts based on (Fe2O3)<sup>n</sup> clusters for ## hydrogen production from ammonia" where Eel((Fe2O3)n) and EZPE((Fe2O3)n) are the electronic and zero-point energies of a cluster (Fe2O3)<sup>n</sup> with a number of units n, while E(Fe) and E(O) are the energis of free Fe and O atoms. ![FIG. S1: Five the most stable isomers of Fe2O<sup>3</sup> with total spin S=4. The binding energies, Eb, are shown in inserts.](path) ![FIG. S2: Five the most stable isomers of (Fe2O3)<sup>2</sup> with total spin S=10. The binding energies, Eb, are shown in inserts.](path) ![FIG. S3: Five the most stable isomers of (Fe2O3)<sup>3</sup> with total spin S=15. The binding energies, Eb, are shown in inserts.](path) ![FIG. S4: Five the most stable isomers of (Fe2O3)<sup>4</sup> with total spin S=20. The binding energies, Eb, are shown in inserts.](path) ![FIG. S5: The temperature dependence of adsorption free energy for NH<sup>3</sup> adsorption on (Fe2O3)<sup>n</sup> n=1-4 at 1 atm.](path) ![FIG. S6: Variation of Gibbs free energy (∆G) with temperature for each dehydrogenation step of NH<sup>3</sup> on (Fe2O3)<sup>n</sup> (n=1−4) clusters, along with the H<sup>2</sup> formation reactions NH<sup>∗</sup> + 2H<sup>∗</sup> ⇌ NH<sup>∗</sup> + H<sup>2</sup> and N<sup>∗</sup> + 3H<sup>∗</sup> ⇌ N <sup>∗</sup> + H<sup>∗</sup> + H2.](path)
FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL^z (a) and spin δS^z (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.	# Light-induced Orbital and Spin Magnetism in 3d, 4d, and 5d Transition Metals Understanding the coherent interplay of light with the magnetization in metals has been a long-standing problem in ultrafast magnetism. While it is known that when laser light acts on a metal it can induce magnetization via the process known as the inverse Faraday effect (IFE), the most basic ingredients of this phenomenon are still largely unexplored. In particular, given a strong recent interest in orbital non-equilibrium dynamics and its role in mediating THz emission in transition metals, the exploration of distinct features in spin and orbital IFE is pertinent. Here, we present a first complete study of the spin and orbital IFE in 3d, 4d and 5d transition metals of groups IV−XI from first-principles. By examining the dependence on the light polarization and frequency, we show that the laser-induced spin and orbital moments may vary significantly both in magnitude and sign. We underpin the interplay between the crystal field splitting and spin-orbit interaction as the key factor which determines the magnitude and key differences between the spin and orbital response. Additionally, we highlight the anisotropy of the effect with respect to the ferromagnetic magnetization and to the crystal structure. The provided complete map of IFE in transition metals is a key reference point in the field of optical magnetism. ### Introduction The demonstration of ultrafast demagnetization in ferromagnets by the application of femtosecond laser pulses [\[1\]](#page-6-0) gave rise to the field of ultrafast spintronics and set the stage for efficient manipulation of magnetism by light. By now it has been rigorously demonstrated that all-optical helicitydependent magnetization switching can be achieved in wide classes of magnetic materials [\[2](#page-6-1)[–10\]](#page-7-0), thus paving the way to contactless ultrafast magnetic recording and information processing. In interpretation of the switching experiments, the inverse Faraday effect (IFE) − i.e. the phenomenon of magnetization induced by a coherent interaction with light acting on a material − has been considered as one of the major underlying mechanisms for the magnetization reversal. Although theoretically predicted and experimentally observed decades ago [\[11–](#page-7-1)[13\]](#page-7-2), a consensus in the theoretical understanding of IFE is still lacking [\[14](#page-7-3)[–19\]](#page-7-4) – however, several microscopic methods have been recently developed for the calculation of this phenomenon in diverse setups [\[20–](#page-7-5)[27\]](#page-7-6). Apart from its established role in the magnetization switching of ferromagnets and ferrimagnents, the impact of IFE in the THz regime has also been studied in antiferromagnets like CrPt [\[28\]](#page-7-7) and Mn2Au [\[29\]](#page-7-8). Moreover, it has been shown that the component of IFE which is perpendicular to the magnetization direction behaves differently with respect to the light helicity than the parallel component [\[21\]](#page-7-9), giving rise to helicity-dependent optical torques [\[30–](#page-7-10)[32\]](#page-7-11) and THz emission [\[33\]](#page-7-12). The impact of the optical torques has been demonstrated to be significant in antiferromagnetic Mn2Au [\[34\]](#page-7-13) where transverse IFE may be responsible for the THz emission [\[35,](#page-7-14) [36\]](#page-7-15), provide an alternative way to switching the magnetization [\[37\]](#page-7-16) as well as to drive domain wall motion [\[38\]](#page-7-17). Recently it has also been shown that light can induce colossal magnetic moments in altermagnets [\[39\]](#page-7-18). Although IFE is conventionally associated with an excitation by circularly polarized light, it can also be activated by linearly-polarized laser pulses [\[25,](#page-7-19) [29,](#page-7-8) [39–](#page-7-18)[42\]](#page-7-20). In solids, two contributions to the magnetization exist: due to spin and orbital moment of electrons. And while the discussion of IFE is normally restricted to the response of spin, since recently, non-equilibrium dynamics of orbital angular momentum started to attract significant attention [\[43](#page-7-21)[–45\]](#page-7-22). Namely, the emergence of orbital currents in the context of the orbital Hall effect [\[46–](#page-7-23)[49\]](#page-8-0), orbital nature of current-induced torques on the magnetization [\[50](#page-8-1)[–52\]](#page-8-2), and current-induced orbital accumulation sizeable even in light materials [\[53–](#page-8-3)[56\]](#page-8-4) have been addressed theoretically and demonstrated experimentally. Notably, in the context of light-induced magnetism, Berritta and co-workers have predicted that the IFE in selected transition metals can exhibit a sizeable orbital component [\[20\]](#page-7-5), with the generality of this observation reaching even into the realm of altermagnets [\[39\]](#page-7-18). At the same time it is also known that within the context of plasmonic IFE the orbital magnetic moment due to electrons excited by laser pulses in small nanoparticles of noble and simple metals can reach atomic values [\[57–](#page-8-5)[59\]](#page-8-6). Despite the fact that very little is known about the interplay of spin and orbital IFE in real materials, it is believed that IFE is a very promising effect in the context of orbitronics − a field which deals with manipulation of the orbital degree of freedom by external perturbations. Although the importance of IFE in mediating the lightinduced magnetism is beyond doubt, a comprehensive indepth material-specific knowledge acquired from microscopic theoretical analysis of this effect is still missing. While acquiring this knowledge is imperative for the field of ultrafast spintronics, since the initial seminal work by Berritta and coworkers [\[20\]](#page-7-5), very little effort has been dedicated to the explorations of this phenomenon from first principles. Here, we ![<span id="page-1-0"></span>FIG. 1. Map of light-induced magnetism in transition metals. (a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) fill this gap by providing a detailed first-principles study of the light-induced spin and orbital magnetism for the 3d, 4d and 5d transition metals of groups IV−XI. By exploring the dependence on the frequency and the polarization of light, we acquire insights into the origin and differences between spin and orbital flavors of IFE, as well as key features of their behavior. Our work provide a solid foundation for further advances in the field of optical magnetism and interaction of light and matter. # Results # Light-induced magnetism in transition metals We begin our discussion by exploring the IFE in a series of 3d, 4d and 5d transition metals of groups IV−XI. IFE is a non-linear optomagnetic effect in which magnetization δO ∝ E × E ∗ is induced as a second-order response to the electric field E of a laser pulse. We calculate the light-induced spin δS and orbital δL magnetic moments by means of the Keldysh formalism, see the section Methods [\[21,](#page-7-9) [39\]](#page-7-18). We focus on light energies hω̵ of 0.25 eV and 1.55 eV which have been routinely used before in our studies within the same formalism [\[29,](#page-7-8) [39,](#page-7-18) [60\]](#page-8-7). Additionally, we choose a lifetime broadening Γ of 25 meV that corresponds to room temperature in order to account for effects of disorder on the electronic states and to provide realistic estimations of the effect. In all calculations we choose the light to be circularly polarized in the xy or yz planes with a light intensity I of 10 GW/cm<sup>2</sup> , while for magnetic materials ferromagnetic magnetization is considered to be along the z-axis. In Fig. [1](#page-1-0) we present the calculated light-induced δL<sup>z</sup> [Fig. [1\(](#page-1-0)a-c)] and δS<sup>z</sup> [Fig. [1\(](#page-1-0)b-d)] moments of the considered transition metals at the Fermi energy. Light is circularly polarized in the xy plane, with energies hω̵ of 0.25 eV [Fig. [1\(](#page-1-0)ab)] and 1.55 eV [Fig. [1\(](#page-1-0)c-d)]. Regarding the 3d magnetic elements Fe, Co, Ni, we analyze both right and left handed circular polarizations of light. The exact values of δL<sup>z</sup> and δS<sup>z</sup> are listed in Tables [II](#page-14-0) and [III](#page-15-0) of the Supplemental Material (SM). Not shown are computed transverse x and y compo- ![<span id="page-2-1"></span>FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.](path) At first sight we notice a strong dependence of the magnitude and sign of both δL<sup>z</sup> and δS<sup>z</sup> on the light frequency. We point out the case of fcc Rh where an colossal δL<sup>z</sup> = −56 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> is predicted at hω̵ = 0.25 eV and which is drastically reduced at the higher frequency, while the spin component remains suppressed in both cases. For the case of magnetic hcp Co we notice a helicity-dependent change of sign for δL<sup>z</sup> at hω̵ = 0.25 eV which is not present for δSz. In general, δL<sup>z</sup> varies stronger with the light helicity than δSz. Moreover, δL<sup>z</sup> is one to two orders of magnitude larger than δS<sup>z</sup> for the non-magnetic elements, while they are of the same magnitude for the magnetic elemental materials. The last two observations are in agreement with the findings of Ref. [\[20\]](#page-7-5) and indicate how the spontaneous magnetization strongly influences the effect in ferromagnets by the time-reversal symmetry breaking. For comparison, in Table [I](#page-2-0) we list the values of the total light-induced moments, defined as the sum of δL<sup>z</sup> and δSz, for the transitional metals studied in [\[20\]](#page-7-5) with hω̵ = 1.55 eV. Our calculated values are of the same order of magnitude, with the exceptions of Au, and of Co for the case of left-handed polarization, which can be attributed to the difference in the computational methods. From Fig. [1\(](#page-1-0)c-d) we get a clear picture of how the induced moments scale with the strength of spin-orbit coupling (SOC). The effect is overall larger for the heavier 5d elements, both in the orbital δL<sup>z</sup> and spin δS<sup>z</sup> channels. However, at the lower frequency of hω̵ = 0.25 eV it is more difficult to draw such a conclusion since the corresponding photon energy falls into the range of SOC strength, which promotes the role of electronic transitions among spin-orbit split bands occurring within a narrow range around the Fermi energy and limited regions in k-space. In contrast, the use of a much larger frequency involves transitions among manifestly orbitally-distinct states which take place over larger portions <span id="page-2-0"></span>TABLE I. Total light-induced magnetic moments for transition metals previously studied in [\[20\]](#page-7-5) in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV. of the reciprocal space with a more uniform impact of the spin-orbit strength on the magnitude of the transition probabilities. From this discussion we have to exclude the ferromagnetic elements since the spontaneous magnetization induces bands splittings on the scale of exchange strength therefore making the effect much more complex. We study the dependence on the light polarization by performing additional calculations for light circularly polarized in the yz-plane, presenting the results for the z and x components of δL and δS respectively in Figs. [6](#page-10-0) and [7](#page-11-0) of the SM for hω̵ = 0.25 eV and hω̵ = 1.55 eV. Our results for δS<sup>z</sup> and δS<sup>x</sup> for the magnetic transition metals reproduce exactly the values of Ref. [\[21\]](#page-7-9), computed with the same method. We further present the values of δL<sup>z</sup> and δLx, where we find the orbital response along the magnetization, δLz, to be by a factor of two larger than the corresponding spin response, when averaged over all magnetic elements. Generally, for the z-component the effect at the magnetic elements is sim- ![<span id="page-3-0"></span>FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level E<sup>F</sup> (dashed gray line). The horizontal dotted red lines at E<sup>F</sup> ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.](path) ilar in magnitude with the case of polarization in the xy-plane shown in Fig. [1,](#page-1-0) and even in the light helicity, while it vanishes for the non-magnetic elements. On the other hand, for the x-component the effect at the magnetic elements becomes one-two orders of magnitude smaller and is odd in the light helicity, although for the non-magnetic elements the values are comparable to the case shown in Fig. [1.](#page-1-0) It is important to note that for light circularly-polarized in the plane containing ferromagnetic magnetization, the light-induced transverse to magnetization induced moments, even though being smaller than the longitudinal ones, are related to light-induced torques that are experimentally demonstrated to lead to helicity-dependent THz emission [\[21,](#page-7-9) [33\]](#page-7-12). In order to get a better understanding of the impact of the time-reversal symmetry breaking on IFE, we consider the cases of ferromagnetic, non-relativistic (i.e. computed without SOC) ferromagnetic, and antiferromagnetic fcc Ni. In Figs. [8](#page-12-0) and [9](#page-13-0) of the SM we present the band filling dependence of the Cartesian components of the light-induced orbital moments δL, for hω̵ = 0.25 eV and hω̵ = 1.55 eV, respectively. In the non-relativistic case only a δL<sup>i</sup> parallel to the light propagation axis is induced which is odd in the helicity, and no δS<sup>i</sup> is induced, exemplifying that the orbital response is the primary non-relativistic one, whereas the spin response is generated through SOC, as was also shown in [\[20\]](#page-7-5). Similarly, for the antiferromagnetic case, only a component parallel to the light propagation axis is induced which is odd in the helicity and remains unchanged under different polarization flavors. While this is the case also for the induced components δL<sup>x</sup> and δL<sup>y</sup> which are transverse to the magnetization in the ferromagnetic case, the situation drastically changes for the induced component δL<sup>z</sup> parallel to the magnetization, as discussed earlier. We note that a tiny, odd in the helicity, δL<sup>x</sup> or δL<sup>y</sup> is additionally induced when light is rotating in a plane containing ferromagnetic magnetization. On the other hand, the induced δL<sup>x</sup> and δLy, developing normal to the polarization plane, serve as a non-relativistic "background" which is independent of the magnetization, with features due to the crystal structure driving the effect over larger regions in energy. The additional band-splittings induced by SOC result in the IFE exhibiting more features with band filling in the relativistic scenario. Remarkably, while the relativistic antiferromagnetic and non-relativistic ferromagnetic cases in principle have similar to each other behavior in energy, a larger signal arises in the antiferromagnetic case by the virtue of flatter bands (see also the discussion for Hf and Pt below). Lastly, we note that a similar behavior has been observed for the in-plane spin IFE in PT-symmetric Mn2Au, however, in the latter case additional out-of-plane moments arise due to linearly polarized light as a result of broken by the magnetization inversion symmetry [\[29\]](#page-7-8). Next, we focus on the case of 5d transition metals where in Fig. [2\(](#page-2-1)a-b) we explore the relation of δL<sup>z</sup> and δSz, respectively, to the band filling, for light circularly polarized in the xy plane and the frequency of hω̵ = 1.55 eV. When going from group IV (hcp Hf) to XI (fcc Au) we observe a smooth ![<span id="page-4-0"></span>FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.](path) variation of δL<sup>z</sup> from positive to negative values, as well as nicely shaped plateaus, where δL<sup>z</sup> remains relatively robust in a wide energy region, for hcp Hf, bcc Ta, bcc W, fcc Ir, and fcc Pt. We note that such plateaus are often characteristic of orbital effects, as witnessed for example in orbital Hall insulators [\[61](#page-8-8)[–63\]](#page-8-9) and orbital Rashba systems [\[60\]](#page-8-7). On the contrary, δS<sup>z</sup> exhibits a very erratic behavior with strong variations for each material, at the same time being much smaller in magnitude than δLz. The above observation is a clear manifestation of how differently the orbital and spin degrees of freedom behave under light excitation, with the orbital angular momentum having its origin in intrinsic structural parameters as manifested in the crystal field splitting, whereas the spin angular momentum is more sensitive to finer details of the electronic structure mediated by SOC. Indeed, the light-induced spin and orbital moments exhibit similarly erratic behavior with band filling once the frequency of the light is drastically reduced to reach the range of spin-orbit interaction, see Fig. [10](#page-13-1) of the SM for hω̵ = 0.25 eV. # Anatomy of IFE in k-space Among the 5d transition metals, hcp Hf and fcc Pt exhibit the largest computed moments at the Fermi energy for hω̵ = 1.55 eV, with the corresponding values of δL<sup>z</sup> = 9.1 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −0.8 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Hf, and δL<sup>z</sup> = −7.4 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −1.1 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Pt. Therefore, we select these two materials and explore the behavior of their light-induced moments in reciprocal space. We present the band-resolved δL<sup>z</sup> and δS<sup>z</sup> for hcp Hf in Fig. [3\(](#page-3-0)a-b), as well as band-resolved δL<sup>z</sup> and δS<sup>z</sup> for fcc Pt in Fig. [3\(](#page-3-0)c-d). For the case of Hf, transitions along the bands near the A-point are the main source of δL<sup>z</sup> and δSz. Light-induced δL<sup>z</sup> consists of hotspot-like negative contributions and secondary, but extended over energy and k-space consistently positive contributions, which are hardly visible. Overall, however, the latter lead to a large positive orbital integrated response. On the other hand, δS<sup>z</sup> For the case of Pt, δL<sup>z</sup> and δS<sup>z</sup> arise from transitions close to X and L high-symmetry points, with both originating in roughly the same regions of (E, k)-space, but often having an opposite sign to each other. Note that the bands in Pt are much more dispersive in the considered energy window, which results in an effective reduction of the regions in (E, k)-space which contribute to the spin and orbital response alike. This is in contrast to Hf, where much flatter bands reside above and below the Fermi energy within the energy window of the laser pulse, providing significant integrated, albeit very small locally, contributions. Moreover, in fcc Pt the Fermi energy cuts through the band edges of the d-states, where the effect of spin-orbit interaction is the strongest, which explains the emergence of strong hotspot-like contributions with a clear correlation in the magnitude of spin and orbital response and the resultant similar behavior of δL<sup>z</sup> and δS<sup>z</sup> with band filling around the Fermi energy. We further scrutinize the reciprocal space distribution of δL<sup>z</sup> and δSz, shown for hcp Hf and fcc Pt in Fig. [3\(](#page-3-0)e-h). For both materials δL<sup>z</sup> distributions consist of large uniform areas of either positive or negative sign. On the contrary, δS<sup>z</sup> distributions are much finer and richer in details with more areas of opposite sign, consistent with the picture we drew above from the band-resolved analysis. A similar behavior of the light-induced magnetism in reciprocal space has been recently reported for rutile altermagnets [\[39\]](#page-7-18). We also observe that, as discussed above, while for Hf the contributions are well spread throughout the Brillouin zone, for Pt the spin and orbital contributions are located at the edges of the considered k<sup>z</sup> − k<sup>y</sup> plane. Overall, this fact indicates that the microscopic behavior of light-induced magnetism varies strongly among transition metals and crucially depends on the crystal structure and position of the Fermi level with respect to the states ![<span id="page-5-0"></span>FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ### Anisotropy of light-induced magnetism We first address the anisotropy of light-induced magnetism with respect to the magnetization by examining the response of magnetic elements under different flavors of circular polarization. In Fig. [4\(](#page-4-0)a-b) we present the computed δL<sup>z</sup> and δS<sup>z</sup> in relation to scattering lifetime Γ, for the cases of magnetic bcc Fe and hcp Co under excitation by light which is circularly polarized in the xy or yz planes at hω̵ = 1.55 eV. As we have already seen in Fig. [1](#page-1-0) for light polarized in the xyplane, the responses behave differently for right (solid lines) or left (dashed lines) polarization. This difference is more pronounced for δLz, for which we even observe a change of sign for Co. Surprisingly, the situation is drastically different for the case of yz-polarization. In this case, for bcc Fe the response is even in the helicity, while for hcp Co δL<sup>z</sup> is almost even and δS<sup>z</sup> is perfectly even in the helicity. Such different behavior with respect to the light helicity between the two magnetic elements can be traced back to the additional anisotropy originating in the crystal structure itself − an effect which we discuss below. Notably, we witness a highly nonlinear behavior with respect to Γ, with the response reaching colossal values in the clean limit. For example, δL<sup>z</sup> and δS<sup>z</sup> can reach as much as 40 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> and 70 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> in Co, respectively, given the scattering lifetime of 1 meV. Besides the potential tunability of the inverse Faraday effect by the degree of the disorder of the samples, one key message that we extract from our observations is that when comparing the values for laser-induced magnetic moments obtained with different methods, special care has to be taken since the implementations of disorder effects, even within a simple constant broadening model, may differ among various approaches. Finally, we analyze the anisotropy that the crystal structure induces in the light-induced magnetism. We select the case of hcp Re and present in Fig. [5\(](#page-5-0)a-b) the x and z components of δL and δS, respectively, in relation to Γ. Due to the inequivalence of x and z axes in the hcp structure, the δL<sup>z</sup> which originates in light circularly polarized in the xy-plane (light blue line) differs from the δL<sup>x</sup> for the case of the yz polarization (brown line). The situation is similar for the case of δS<sup>z</sup> and δSx. On the contrary, as presented in Fig. [11](#page-14-1) of the SM, in the case of fcc Pt we have a perfect match between δL<sup>z</sup> and δL<sup>x</sup> (or δS<sup>z</sup> and δSx) when changing the plane of circular polarization because the x and z axes are equivalent in the fcc structure. The situation is similar for the y component when light is circularly polarized in the xz-plane. As also expected from the symmetry of the crystal structure, we confirm that the light-induced moments are perfectly odd in the helicity for the nonmagnetic elements, which is not the case for the magnetic elements, see Figs. [5](#page-5-0) and [11](#page-14-1) of the SM [\[20\]](#page-7-5). ### Discussion The main goal of our work is to showcase the importance of the orbital component of IFE and its distinct behavior from the spin counterpart. While so far it was mainly the interaction of light with the spin magnetization that was taken into consideration for the interpretation of IFE-related effects, we speculate that the orbital IFE may provide a novel way to coherently induce magnetization and manipulate the magnetic order. For example, in a recent study, different types of optical torques that may arise in ferromagnetic layers were interpreted in terms of the light-induced orbital moment and its interaction with the magnetization through the spin-orbit interaction [\[64\]](#page-8-10). Since it is known that current-induced orbital accumulation and orbital torques exhibit a long-range behavior [\[49,](#page-8-0) [65](#page-8-11)[–67\]](#page-8-12) due to a characteristic small orbital decay, a question arises whether similar behavior can be exhibited by optical torques caused by the orbital IFE. The emergence of long-ranged orbital IFE would come as no surprise given the fact that several recent experiments reported that laser excitation can drive long-range ballistic orbital currents resulting in THz emission [\[68](#page-8-13)[–71\]](#page-8-14). a perturbation by the electric field of the original ground state Hamiltonian, it is common to utilize the angular momentum of light in order to understand the interaction with the magnetic order by the means of transfer of angular momentum. The spin of light through the helicity of circularly polarized pulses plays a crucial role in helicity-dependent all-optical switching scenarios. On the other hand, there is a strong recent interest in utilizing the orbital angular momentum of light via irradiation of matter with e.g. vortex beams or twisted light, in order to probe the magnetization [\[72\]](#page-8-15), generate photocurrents [\[73\]](#page-8-16), drive ultrafast demagnetization [\[74\]](#page-8-17), and induce IFE [\[40,](#page-7-24) [75\]](#page-8-18). Therefore, it is imperative to treat both spin and orbital degrees of freedom on equal footing when exploring the lightmatter interaction. This will not only trigger further advances in the field of ultrafast magnetism and THz spintronics, but also enable a transition to the novel field of attosecond spintronics [\[76,](#page-8-19) [77\]](#page-8-20). Method. In this work we calculate the first-principles electronic structures of 3d, 4d and 5d transition metals by using the full-potential linearized augmented plane wave FLEUR code [\[78\]](#page-8-21). We describe exchange and correlation effects by using the non-relativistic PBE [\[79\]](#page-8-22) functional, while relativistic effects are described by the second-variation scheme [\[80\]](#page-9-0). The parameters of our first-principles calculations, i.e. lattice constants, muffin-tin radii, plane-wave cutoffs, etc. are taken from Table I of Ref. [\[81\]](#page-9-1). Next, we construct maximally-localized Wannier functions (MLWFs) by employing the Wannier90 code [\[82\]](#page-9-2) and its interface with the FLEUR code [\[83\]](#page-9-3). Similarly to [\[81\]](#page-9-1), we choose s, p and d orbitals for the initial projections and disentagle 18 MLWFs out of 36 Bloch states within a frozen window of 5.0 eV above the Fermi energy for each atom in the unit cell. where O<sup>i</sup> is the i-th component of either the orbital angular momentum operator L<sup>i</sup> or of the spin operator S<sup>i</sup> . Moreover, a<sup>0</sup> = 4πϵ0h̵<sup>2</sup> /(mee 2 ) is the Bohr's radius, I = ϵ0cE<sup>2</sup> 0 /2 is the intensity of the pulse, ϵ<sup>0</sup> is the vacuum permittivity, m<sup>e</sup> is the electron mass, e is the elementary charge, h̵ is the reduced Planck constant, c is the light velocity, E<sup>H</sup> = e 2 /(4πϵ0a0) is the Hartree energy, and ϵ<sup>j</sup> is the j-th component of the polarization vector of the pulse. For example, we describe right/left-handedly polarized light in the xy-plane as ϵ = (1,±i, <sup>0</sup>)/<sup>√</sup> 2, respectively, and define its propagation vector to lie along the normal to the polarization plane. A detailed form of the tensor φijk can be seen in Eq.(14) of Ref. [\[21\]](#page-7-9). For the orbital response, the prefactor in Eq. [\(1\)](#page-6-2) must be multiplied by an additional factor of 2. A 128×128×128 interpolation k-mesh is sufficient to obtain well-converged results. In all calculations the lifetime broadening Γ was set at 25 meV, the light frequency hω̵ at 0.25 eV and 1.55 eV, the intensity of light at 10 GW/cm<sup>2</sup> , and we covered an energy region of [−2.5, 2.5] eV around the Fermi energy level E<sup>F</sup> . Acknowledgements. We thank Frank Freimuth and Maximilian Merte for discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR 173/3 − 268565370 (project A11) and by the K. and A. Wallenberg Foundation (Grants No. 2022.0079 and 2023.0336). We acknowledge support from the EIC Pathfinder OPEN grant 101129641 "OBELIX". We also gratefully acknowledge the Julich Supercomputing Cen- ¨ tre and RWTH Aachen University for providing computational resources under projects jiff40 and jara0062. Author Contributions. T. A. performed numerical calculations and analysed the results. T. A. and Y. M. wrote the manuscript. 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(a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-11-0"></span>FIG. 7. (a-d) Light-induced orbital δL<sup>x</sup> (a, c) and spin δS<sup>x</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-12-0"></span>FIG. 8. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-13-0"></span>FIG. 9. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ![<span id="page-13-1"></span>FIG. 10. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-14-1"></span>FIG. 11. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic fcc Pt. The induced magnetic moments along the x-axis and along the z-axis are presented. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) <span id="page-14-0"></span>TABLE II. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 0.25 eV. <span id="page-15-0"></span>TABLE III. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV.
FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.	# Light-induced Orbital and Spin Magnetism in 3d, 4d, and 5d Transition Metals Understanding the coherent interplay of light with the magnetization in metals has been a long-standing problem in ultrafast magnetism. While it is known that when laser light acts on a metal it can induce magnetization via the process known as the inverse Faraday effect (IFE), the most basic ingredients of this phenomenon are still largely unexplored. In particular, given a strong recent interest in orbital non-equilibrium dynamics and its role in mediating THz emission in transition metals, the exploration of distinct features in spin and orbital IFE is pertinent. Here, we present a first complete study of the spin and orbital IFE in 3d, 4d and 5d transition metals of groups IV−XI from first-principles. By examining the dependence on the light polarization and frequency, we show that the laser-induced spin and orbital moments may vary significantly both in magnitude and sign. We underpin the interplay between the crystal field splitting and spin-orbit interaction as the key factor which determines the magnitude and key differences between the spin and orbital response. Additionally, we highlight the anisotropy of the effect with respect to the ferromagnetic magnetization and to the crystal structure. The provided complete map of IFE in transition metals is a key reference point in the field of optical magnetism. ### Introduction The demonstration of ultrafast demagnetization in ferromagnets by the application of femtosecond laser pulses [\[1\]](#page-6-0) gave rise to the field of ultrafast spintronics and set the stage for efficient manipulation of magnetism by light. By now it has been rigorously demonstrated that all-optical helicitydependent magnetization switching can be achieved in wide classes of magnetic materials [\[2](#page-6-1)[–10\]](#page-7-0), thus paving the way to contactless ultrafast magnetic recording and information processing. In interpretation of the switching experiments, the inverse Faraday effect (IFE) − i.e. the phenomenon of magnetization induced by a coherent interaction with light acting on a material − has been considered as one of the major underlying mechanisms for the magnetization reversal. Although theoretically predicted and experimentally observed decades ago [\[11–](#page-7-1)[13\]](#page-7-2), a consensus in the theoretical understanding of IFE is still lacking [\[14](#page-7-3)[–19\]](#page-7-4) – however, several microscopic methods have been recently developed for the calculation of this phenomenon in diverse setups [\[20–](#page-7-5)[27\]](#page-7-6). Apart from its established role in the magnetization switching of ferromagnets and ferrimagnents, the impact of IFE in the THz regime has also been studied in antiferromagnets like CrPt [\[28\]](#page-7-7) and Mn2Au [\[29\]](#page-7-8). Moreover, it has been shown that the component of IFE which is perpendicular to the magnetization direction behaves differently with respect to the light helicity than the parallel component [\[21\]](#page-7-9), giving rise to helicity-dependent optical torques [\[30–](#page-7-10)[32\]](#page-7-11) and THz emission [\[33\]](#page-7-12). The impact of the optical torques has been demonstrated to be significant in antiferromagnetic Mn2Au [\[34\]](#page-7-13) where transverse IFE may be responsible for the THz emission [\[35,](#page-7-14) [36\]](#page-7-15), provide an alternative way to switching the magnetization [\[37\]](#page-7-16) as well as to drive domain wall motion [\[38\]](#page-7-17). Recently it has also been shown that light can induce colossal magnetic moments in altermagnets [\[39\]](#page-7-18). Although IFE is conventionally associated with an excitation by circularly polarized light, it can also be activated by linearly-polarized laser pulses [\[25,](#page-7-19) [29,](#page-7-8) [39–](#page-7-18)[42\]](#page-7-20). In solids, two contributions to the magnetization exist: due to spin and orbital moment of electrons. And while the discussion of IFE is normally restricted to the response of spin, since recently, non-equilibrium dynamics of orbital angular momentum started to attract significant attention [\[43](#page-7-21)[–45\]](#page-7-22). Namely, the emergence of orbital currents in the context of the orbital Hall effect [\[46–](#page-7-23)[49\]](#page-8-0), orbital nature of current-induced torques on the magnetization [\[50](#page-8-1)[–52\]](#page-8-2), and current-induced orbital accumulation sizeable even in light materials [\[53–](#page-8-3)[56\]](#page-8-4) have been addressed theoretically and demonstrated experimentally. Notably, in the context of light-induced magnetism, Berritta and co-workers have predicted that the IFE in selected transition metals can exhibit a sizeable orbital component [\[20\]](#page-7-5), with the generality of this observation reaching even into the realm of altermagnets [\[39\]](#page-7-18). At the same time it is also known that within the context of plasmonic IFE the orbital magnetic moment due to electrons excited by laser pulses in small nanoparticles of noble and simple metals can reach atomic values [\[57–](#page-8-5)[59\]](#page-8-6). Despite the fact that very little is known about the interplay of spin and orbital IFE in real materials, it is believed that IFE is a very promising effect in the context of orbitronics − a field which deals with manipulation of the orbital degree of freedom by external perturbations. Although the importance of IFE in mediating the lightinduced magnetism is beyond doubt, a comprehensive indepth material-specific knowledge acquired from microscopic theoretical analysis of this effect is still missing. While acquiring this knowledge is imperative for the field of ultrafast spintronics, since the initial seminal work by Berritta and coworkers [\[20\]](#page-7-5), very little effort has been dedicated to the explorations of this phenomenon from first principles. Here, we ![<span id="page-1-0"></span>FIG. 1. Map of light-induced magnetism in transition metals. (a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) fill this gap by providing a detailed first-principles study of the light-induced spin and orbital magnetism for the 3d, 4d and 5d transition metals of groups IV−XI. By exploring the dependence on the frequency and the polarization of light, we acquire insights into the origin and differences between spin and orbital flavors of IFE, as well as key features of their behavior. Our work provide a solid foundation for further advances in the field of optical magnetism and interaction of light and matter. # Results # Light-induced magnetism in transition metals We begin our discussion by exploring the IFE in a series of 3d, 4d and 5d transition metals of groups IV−XI. IFE is a non-linear optomagnetic effect in which magnetization δO ∝ E × E ∗ is induced as a second-order response to the electric field E of a laser pulse. We calculate the light-induced spin δS and orbital δL magnetic moments by means of the Keldysh formalism, see the section Methods [\[21,](#page-7-9) [39\]](#page-7-18). We focus on light energies hω̵ of 0.25 eV and 1.55 eV which have been routinely used before in our studies within the same formalism [\[29,](#page-7-8) [39,](#page-7-18) [60\]](#page-8-7). Additionally, we choose a lifetime broadening Γ of 25 meV that corresponds to room temperature in order to account for effects of disorder on the electronic states and to provide realistic estimations of the effect. In all calculations we choose the light to be circularly polarized in the xy or yz planes with a light intensity I of 10 GW/cm<sup>2</sup> , while for magnetic materials ferromagnetic magnetization is considered to be along the z-axis. In Fig. [1](#page-1-0) we present the calculated light-induced δL<sup>z</sup> [Fig. [1\(](#page-1-0)a-c)] and δS<sup>z</sup> [Fig. [1\(](#page-1-0)b-d)] moments of the considered transition metals at the Fermi energy. Light is circularly polarized in the xy plane, with energies hω̵ of 0.25 eV [Fig. [1\(](#page-1-0)ab)] and 1.55 eV [Fig. [1\(](#page-1-0)c-d)]. Regarding the 3d magnetic elements Fe, Co, Ni, we analyze both right and left handed circular polarizations of light. The exact values of δL<sup>z</sup> and δS<sup>z</sup> are listed in Tables [II](#page-14-0) and [III](#page-15-0) of the Supplemental Material (SM). Not shown are computed transverse x and y compo- ![<span id="page-2-1"></span>FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.](path) At first sight we notice a strong dependence of the magnitude and sign of both δL<sup>z</sup> and δS<sup>z</sup> on the light frequency. We point out the case of fcc Rh where an colossal δL<sup>z</sup> = −56 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> is predicted at hω̵ = 0.25 eV and which is drastically reduced at the higher frequency, while the spin component remains suppressed in both cases. For the case of magnetic hcp Co we notice a helicity-dependent change of sign for δL<sup>z</sup> at hω̵ = 0.25 eV which is not present for δSz. In general, δL<sup>z</sup> varies stronger with the light helicity than δSz. Moreover, δL<sup>z</sup> is one to two orders of magnitude larger than δS<sup>z</sup> for the non-magnetic elements, while they are of the same magnitude for the magnetic elemental materials. The last two observations are in agreement with the findings of Ref. [\[20\]](#page-7-5) and indicate how the spontaneous magnetization strongly influences the effect in ferromagnets by the time-reversal symmetry breaking. For comparison, in Table [I](#page-2-0) we list the values of the total light-induced moments, defined as the sum of δL<sup>z</sup> and δSz, for the transitional metals studied in [\[20\]](#page-7-5) with hω̵ = 1.55 eV. Our calculated values are of the same order of magnitude, with the exceptions of Au, and of Co for the case of left-handed polarization, which can be attributed to the difference in the computational methods. From Fig. [1\(](#page-1-0)c-d) we get a clear picture of how the induced moments scale with the strength of spin-orbit coupling (SOC). The effect is overall larger for the heavier 5d elements, both in the orbital δL<sup>z</sup> and spin δS<sup>z</sup> channels. However, at the lower frequency of hω̵ = 0.25 eV it is more difficult to draw such a conclusion since the corresponding photon energy falls into the range of SOC strength, which promotes the role of electronic transitions among spin-orbit split bands occurring within a narrow range around the Fermi energy and limited regions in k-space. In contrast, the use of a much larger frequency involves transitions among manifestly orbitally-distinct states which take place over larger portions <span id="page-2-0"></span>TABLE I. Total light-induced magnetic moments for transition metals previously studied in [\[20\]](#page-7-5) in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV. of the reciprocal space with a more uniform impact of the spin-orbit strength on the magnitude of the transition probabilities. From this discussion we have to exclude the ferromagnetic elements since the spontaneous magnetization induces bands splittings on the scale of exchange strength therefore making the effect much more complex. We study the dependence on the light polarization by performing additional calculations for light circularly polarized in the yz-plane, presenting the results for the z and x components of δL and δS respectively in Figs. [6](#page-10-0) and [7](#page-11-0) of the SM for hω̵ = 0.25 eV and hω̵ = 1.55 eV. Our results for δS<sup>z</sup> and δS<sup>x</sup> for the magnetic transition metals reproduce exactly the values of Ref. [\[21\]](#page-7-9), computed with the same method. We further present the values of δL<sup>z</sup> and δLx, where we find the orbital response along the magnetization, δLz, to be by a factor of two larger than the corresponding spin response, when averaged over all magnetic elements. Generally, for the z-component the effect at the magnetic elements is sim- ![<span id="page-3-0"></span>FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level E<sup>F</sup> (dashed gray line). The horizontal dotted red lines at E<sup>F</sup> ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.](path) ilar in magnitude with the case of polarization in the xy-plane shown in Fig. [1,](#page-1-0) and even in the light helicity, while it vanishes for the non-magnetic elements. On the other hand, for the x-component the effect at the magnetic elements becomes one-two orders of magnitude smaller and is odd in the light helicity, although for the non-magnetic elements the values are comparable to the case shown in Fig. [1.](#page-1-0) It is important to note that for light circularly-polarized in the plane containing ferromagnetic magnetization, the light-induced transverse to magnetization induced moments, even though being smaller than the longitudinal ones, are related to light-induced torques that are experimentally demonstrated to lead to helicity-dependent THz emission [\[21,](#page-7-9) [33\]](#page-7-12). In order to get a better understanding of the impact of the time-reversal symmetry breaking on IFE, we consider the cases of ferromagnetic, non-relativistic (i.e. computed without SOC) ferromagnetic, and antiferromagnetic fcc Ni. In Figs. [8](#page-12-0) and [9](#page-13-0) of the SM we present the band filling dependence of the Cartesian components of the light-induced orbital moments δL, for hω̵ = 0.25 eV and hω̵ = 1.55 eV, respectively. In the non-relativistic case only a δL<sup>i</sup> parallel to the light propagation axis is induced which is odd in the helicity, and no δS<sup>i</sup> is induced, exemplifying that the orbital response is the primary non-relativistic one, whereas the spin response is generated through SOC, as was also shown in [\[20\]](#page-7-5). Similarly, for the antiferromagnetic case, only a component parallel to the light propagation axis is induced which is odd in the helicity and remains unchanged under different polarization flavors. While this is the case also for the induced components δL<sup>x</sup> and δL<sup>y</sup> which are transverse to the magnetization in the ferromagnetic case, the situation drastically changes for the induced component δL<sup>z</sup> parallel to the magnetization, as discussed earlier. We note that a tiny, odd in the helicity, δL<sup>x</sup> or δL<sup>y</sup> is additionally induced when light is rotating in a plane containing ferromagnetic magnetization. On the other hand, the induced δL<sup>x</sup> and δLy, developing normal to the polarization plane, serve as a non-relativistic "background" which is independent of the magnetization, with features due to the crystal structure driving the effect over larger regions in energy. The additional band-splittings induced by SOC result in the IFE exhibiting more features with band filling in the relativistic scenario. Remarkably, while the relativistic antiferromagnetic and non-relativistic ferromagnetic cases in principle have similar to each other behavior in energy, a larger signal arises in the antiferromagnetic case by the virtue of flatter bands (see also the discussion for Hf and Pt below). Lastly, we note that a similar behavior has been observed for the in-plane spin IFE in PT-symmetric Mn2Au, however, in the latter case additional out-of-plane moments arise due to linearly polarized light as a result of broken by the magnetization inversion symmetry [\[29\]](#page-7-8). Next, we focus on the case of 5d transition metals where in Fig. [2\(](#page-2-1)a-b) we explore the relation of δL<sup>z</sup> and δSz, respectively, to the band filling, for light circularly polarized in the xy plane and the frequency of hω̵ = 1.55 eV. When going from group IV (hcp Hf) to XI (fcc Au) we observe a smooth ![<span id="page-4-0"></span>FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.](path) variation of δL<sup>z</sup> from positive to negative values, as well as nicely shaped plateaus, where δL<sup>z</sup> remains relatively robust in a wide energy region, for hcp Hf, bcc Ta, bcc W, fcc Ir, and fcc Pt. We note that such plateaus are often characteristic of orbital effects, as witnessed for example in orbital Hall insulators [\[61](#page-8-8)[–63\]](#page-8-9) and orbital Rashba systems [\[60\]](#page-8-7). On the contrary, δS<sup>z</sup> exhibits a very erratic behavior with strong variations for each material, at the same time being much smaller in magnitude than δLz. The above observation is a clear manifestation of how differently the orbital and spin degrees of freedom behave under light excitation, with the orbital angular momentum having its origin in intrinsic structural parameters as manifested in the crystal field splitting, whereas the spin angular momentum is more sensitive to finer details of the electronic structure mediated by SOC. Indeed, the light-induced spin and orbital moments exhibit similarly erratic behavior with band filling once the frequency of the light is drastically reduced to reach the range of spin-orbit interaction, see Fig. [10](#page-13-1) of the SM for hω̵ = 0.25 eV. # Anatomy of IFE in k-space Among the 5d transition metals, hcp Hf and fcc Pt exhibit the largest computed moments at the Fermi energy for hω̵ = 1.55 eV, with the corresponding values of δL<sup>z</sup> = 9.1 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −0.8 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Hf, and δL<sup>z</sup> = −7.4 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −1.1 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Pt. Therefore, we select these two materials and explore the behavior of their light-induced moments in reciprocal space. We present the band-resolved δL<sup>z</sup> and δS<sup>z</sup> for hcp Hf in Fig. [3\(](#page-3-0)a-b), as well as band-resolved δL<sup>z</sup> and δS<sup>z</sup> for fcc Pt in Fig. [3\(](#page-3-0)c-d). For the case of Hf, transitions along the bands near the A-point are the main source of δL<sup>z</sup> and δSz. Light-induced δL<sup>z</sup> consists of hotspot-like negative contributions and secondary, but extended over energy and k-space consistently positive contributions, which are hardly visible. Overall, however, the latter lead to a large positive orbital integrated response. On the other hand, δS<sup>z</sup> For the case of Pt, δL<sup>z</sup> and δS<sup>z</sup> arise from transitions close to X and L high-symmetry points, with both originating in roughly the same regions of (E, k)-space, but often having an opposite sign to each other. Note that the bands in Pt are much more dispersive in the considered energy window, which results in an effective reduction of the regions in (E, k)-space which contribute to the spin and orbital response alike. This is in contrast to Hf, where much flatter bands reside above and below the Fermi energy within the energy window of the laser pulse, providing significant integrated, albeit very small locally, contributions. Moreover, in fcc Pt the Fermi energy cuts through the band edges of the d-states, where the effect of spin-orbit interaction is the strongest, which explains the emergence of strong hotspot-like contributions with a clear correlation in the magnitude of spin and orbital response and the resultant similar behavior of δL<sup>z</sup> and δS<sup>z</sup> with band filling around the Fermi energy. We further scrutinize the reciprocal space distribution of δL<sup>z</sup> and δSz, shown for hcp Hf and fcc Pt in Fig. [3\(](#page-3-0)e-h). For both materials δL<sup>z</sup> distributions consist of large uniform areas of either positive or negative sign. On the contrary, δS<sup>z</sup> distributions are much finer and richer in details with more areas of opposite sign, consistent with the picture we drew above from the band-resolved analysis. A similar behavior of the light-induced magnetism in reciprocal space has been recently reported for rutile altermagnets [\[39\]](#page-7-18). We also observe that, as discussed above, while for Hf the contributions are well spread throughout the Brillouin zone, for Pt the spin and orbital contributions are located at the edges of the considered k<sup>z</sup> − k<sup>y</sup> plane. Overall, this fact indicates that the microscopic behavior of light-induced magnetism varies strongly among transition metals and crucially depends on the crystal structure and position of the Fermi level with respect to the states ![<span id="page-5-0"></span>FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ### Anisotropy of light-induced magnetism We first address the anisotropy of light-induced magnetism with respect to the magnetization by examining the response of magnetic elements under different flavors of circular polarization. In Fig. [4\(](#page-4-0)a-b) we present the computed δL<sup>z</sup> and δS<sup>z</sup> in relation to scattering lifetime Γ, for the cases of magnetic bcc Fe and hcp Co under excitation by light which is circularly polarized in the xy or yz planes at hω̵ = 1.55 eV. As we have already seen in Fig. [1](#page-1-0) for light polarized in the xyplane, the responses behave differently for right (solid lines) or left (dashed lines) polarization. This difference is more pronounced for δLz, for which we even observe a change of sign for Co. Surprisingly, the situation is drastically different for the case of yz-polarization. In this case, for bcc Fe the response is even in the helicity, while for hcp Co δL<sup>z</sup> is almost even and δS<sup>z</sup> is perfectly even in the helicity. Such different behavior with respect to the light helicity between the two magnetic elements can be traced back to the additional anisotropy originating in the crystal structure itself − an effect which we discuss below. Notably, we witness a highly nonlinear behavior with respect to Γ, with the response reaching colossal values in the clean limit. For example, δL<sup>z</sup> and δS<sup>z</sup> can reach as much as 40 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> and 70 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> in Co, respectively, given the scattering lifetime of 1 meV. Besides the potential tunability of the inverse Faraday effect by the degree of the disorder of the samples, one key message that we extract from our observations is that when comparing the values for laser-induced magnetic moments obtained with different methods, special care has to be taken since the implementations of disorder effects, even within a simple constant broadening model, may differ among various approaches. Finally, we analyze the anisotropy that the crystal structure induces in the light-induced magnetism. We select the case of hcp Re and present in Fig. [5\(](#page-5-0)a-b) the x and z components of δL and δS, respectively, in relation to Γ. Due to the inequivalence of x and z axes in the hcp structure, the δL<sup>z</sup> which originates in light circularly polarized in the xy-plane (light blue line) differs from the δL<sup>x</sup> for the case of the yz polarization (brown line). The situation is similar for the case of δS<sup>z</sup> and δSx. On the contrary, as presented in Fig. [11](#page-14-1) of the SM, in the case of fcc Pt we have a perfect match between δL<sup>z</sup> and δL<sup>x</sup> (or δS<sup>z</sup> and δSx) when changing the plane of circular polarization because the x and z axes are equivalent in the fcc structure. The situation is similar for the y component when light is circularly polarized in the xz-plane. As also expected from the symmetry of the crystal structure, we confirm that the light-induced moments are perfectly odd in the helicity for the nonmagnetic elements, which is not the case for the magnetic elements, see Figs. [5](#page-5-0) and [11](#page-14-1) of the SM [\[20\]](#page-7-5). ### Discussion The main goal of our work is to showcase the importance of the orbital component of IFE and its distinct behavior from the spin counterpart. While so far it was mainly the interaction of light with the spin magnetization that was taken into consideration for the interpretation of IFE-related effects, we speculate that the orbital IFE may provide a novel way to coherently induce magnetization and manipulate the magnetic order. For example, in a recent study, different types of optical torques that may arise in ferromagnetic layers were interpreted in terms of the light-induced orbital moment and its interaction with the magnetization through the spin-orbit interaction [\[64\]](#page-8-10). Since it is known that current-induced orbital accumulation and orbital torques exhibit a long-range behavior [\[49,](#page-8-0) [65](#page-8-11)[–67\]](#page-8-12) due to a characteristic small orbital decay, a question arises whether similar behavior can be exhibited by optical torques caused by the orbital IFE. The emergence of long-ranged orbital IFE would come as no surprise given the fact that several recent experiments reported that laser excitation can drive long-range ballistic orbital currents resulting in THz emission [\[68](#page-8-13)[–71\]](#page-8-14). a perturbation by the electric field of the original ground state Hamiltonian, it is common to utilize the angular momentum of light in order to understand the interaction with the magnetic order by the means of transfer of angular momentum. The spin of light through the helicity of circularly polarized pulses plays a crucial role in helicity-dependent all-optical switching scenarios. On the other hand, there is a strong recent interest in utilizing the orbital angular momentum of light via irradiation of matter with e.g. vortex beams or twisted light, in order to probe the magnetization [\[72\]](#page-8-15), generate photocurrents [\[73\]](#page-8-16), drive ultrafast demagnetization [\[74\]](#page-8-17), and induce IFE [\[40,](#page-7-24) [75\]](#page-8-18). Therefore, it is imperative to treat both spin and orbital degrees of freedom on equal footing when exploring the lightmatter interaction. This will not only trigger further advances in the field of ultrafast magnetism and THz spintronics, but also enable a transition to the novel field of attosecond spintronics [\[76,](#page-8-19) [77\]](#page-8-20). Method. In this work we calculate the first-principles electronic structures of 3d, 4d and 5d transition metals by using the full-potential linearized augmented plane wave FLEUR code [\[78\]](#page-8-21). We describe exchange and correlation effects by using the non-relativistic PBE [\[79\]](#page-8-22) functional, while relativistic effects are described by the second-variation scheme [\[80\]](#page-9-0). The parameters of our first-principles calculations, i.e. lattice constants, muffin-tin radii, plane-wave cutoffs, etc. are taken from Table I of Ref. [\[81\]](#page-9-1). Next, we construct maximally-localized Wannier functions (MLWFs) by employing the Wannier90 code [\[82\]](#page-9-2) and its interface with the FLEUR code [\[83\]](#page-9-3). Similarly to [\[81\]](#page-9-1), we choose s, p and d orbitals for the initial projections and disentagle 18 MLWFs out of 36 Bloch states within a frozen window of 5.0 eV above the Fermi energy for each atom in the unit cell. where O<sup>i</sup> is the i-th component of either the orbital angular momentum operator L<sup>i</sup> or of the spin operator S<sup>i</sup> . Moreover, a<sup>0</sup> = 4πϵ0h̵<sup>2</sup> /(mee 2 ) is the Bohr's radius, I = ϵ0cE<sup>2</sup> 0 /2 is the intensity of the pulse, ϵ<sup>0</sup> is the vacuum permittivity, m<sup>e</sup> is the electron mass, e is the elementary charge, h̵ is the reduced Planck constant, c is the light velocity, E<sup>H</sup> = e 2 /(4πϵ0a0) is the Hartree energy, and ϵ<sup>j</sup> is the j-th component of the polarization vector of the pulse. For example, we describe right/left-handedly polarized light in the xy-plane as ϵ = (1,±i, <sup>0</sup>)/<sup>√</sup> 2, respectively, and define its propagation vector to lie along the normal to the polarization plane. A detailed form of the tensor φijk can be seen in Eq.(14) of Ref. [\[21\]](#page-7-9). For the orbital response, the prefactor in Eq. [\(1\)](#page-6-2) must be multiplied by an additional factor of 2. A 128×128×128 interpolation k-mesh is sufficient to obtain well-converged results. In all calculations the lifetime broadening Γ was set at 25 meV, the light frequency hω̵ at 0.25 eV and 1.55 eV, the intensity of light at 10 GW/cm<sup>2</sup> , and we covered an energy region of [−2.5, 2.5] eV around the Fermi energy level E<sup>F</sup> . Acknowledgements. We thank Frank Freimuth and Maximilian Merte for discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR 173/3 − 268565370 (project A11) and by the K. and A. Wallenberg Foundation (Grants No. 2022.0079 and 2023.0336). We acknowledge support from the EIC Pathfinder OPEN grant 101129641 "OBELIX". We also gratefully acknowledge the Julich Supercomputing Cen- ¨ tre and RWTH Aachen University for providing computational resources under projects jiff40 and jara0062. Author Contributions. T. A. performed numerical calculations and analysed the results. T. A. and Y. M. wrote the manuscript. 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(a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-11-0"></span>FIG. 7. (a-d) Light-induced orbital δL<sup>x</sup> (a, c) and spin δS<sup>x</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-12-0"></span>FIG. 8. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-13-0"></span>FIG. 9. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ![<span id="page-13-1"></span>FIG. 10. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-14-1"></span>FIG. 11. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic fcc Pt. The induced magnetic moments along the x-axis and along the z-axis are presented. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) <span id="page-14-0"></span>TABLE II. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 0.25 eV. <span id="page-15-0"></span>TABLE III. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV.
<span id="page-1-0"></span>FIG. 1. Map of light-induced magnetism in transition metals. (a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).	# Light-induced Orbital and Spin Magnetism in 3d, 4d, and 5d Transition Metals Understanding the coherent interplay of light with the magnetization in metals has been a long-standing problem in ultrafast magnetism. While it is known that when laser light acts on a metal it can induce magnetization via the process known as the inverse Faraday effect (IFE), the most basic ingredients of this phenomenon are still largely unexplored. In particular, given a strong recent interest in orbital non-equilibrium dynamics and its role in mediating THz emission in transition metals, the exploration of distinct features in spin and orbital IFE is pertinent. Here, we present a first complete study of the spin and orbital IFE in 3d, 4d and 5d transition metals of groups IV−XI from first-principles. By examining the dependence on the light polarization and frequency, we show that the laser-induced spin and orbital moments may vary significantly both in magnitude and sign. We underpin the interplay between the crystal field splitting and spin-orbit interaction as the key factor which determines the magnitude and key differences between the spin and orbital response. Additionally, we highlight the anisotropy of the effect with respect to the ferromagnetic magnetization and to the crystal structure. The provided complete map of IFE in transition metals is a key reference point in the field of optical magnetism. ### Introduction The demonstration of ultrafast demagnetization in ferromagnets by the application of femtosecond laser pulses [\[1\]](#page-6-0) gave rise to the field of ultrafast spintronics and set the stage for efficient manipulation of magnetism by light. By now it has been rigorously demonstrated that all-optical helicitydependent magnetization switching can be achieved in wide classes of magnetic materials [\[2](#page-6-1)[–10\]](#page-7-0), thus paving the way to contactless ultrafast magnetic recording and information processing. In interpretation of the switching experiments, the inverse Faraday effect (IFE) − i.e. the phenomenon of magnetization induced by a coherent interaction with light acting on a material − has been considered as one of the major underlying mechanisms for the magnetization reversal. Although theoretically predicted and experimentally observed decades ago [\[11–](#page-7-1)[13\]](#page-7-2), a consensus in the theoretical understanding of IFE is still lacking [\[14](#page-7-3)[–19\]](#page-7-4) – however, several microscopic methods have been recently developed for the calculation of this phenomenon in diverse setups [\[20–](#page-7-5)[27\]](#page-7-6). Apart from its established role in the magnetization switching of ferromagnets and ferrimagnents, the impact of IFE in the THz regime has also been studied in antiferromagnets like CrPt [\[28\]](#page-7-7) and Mn2Au [\[29\]](#page-7-8). Moreover, it has been shown that the component of IFE which is perpendicular to the magnetization direction behaves differently with respect to the light helicity than the parallel component [\[21\]](#page-7-9), giving rise to helicity-dependent optical torques [\[30–](#page-7-10)[32\]](#page-7-11) and THz emission [\[33\]](#page-7-12). The impact of the optical torques has been demonstrated to be significant in antiferromagnetic Mn2Au [\[34\]](#page-7-13) where transverse IFE may be responsible for the THz emission [\[35,](#page-7-14) [36\]](#page-7-15), provide an alternative way to switching the magnetization [\[37\]](#page-7-16) as well as to drive domain wall motion [\[38\]](#page-7-17). Recently it has also been shown that light can induce colossal magnetic moments in altermagnets [\[39\]](#page-7-18). Although IFE is conventionally associated with an excitation by circularly polarized light, it can also be activated by linearly-polarized laser pulses [\[25,](#page-7-19) [29,](#page-7-8) [39–](#page-7-18)[42\]](#page-7-20). In solids, two contributions to the magnetization exist: due to spin and orbital moment of electrons. And while the discussion of IFE is normally restricted to the response of spin, since recently, non-equilibrium dynamics of orbital angular momentum started to attract significant attention [\[43](#page-7-21)[–45\]](#page-7-22). Namely, the emergence of orbital currents in the context of the orbital Hall effect [\[46–](#page-7-23)[49\]](#page-8-0), orbital nature of current-induced torques on the magnetization [\[50](#page-8-1)[–52\]](#page-8-2), and current-induced orbital accumulation sizeable even in light materials [\[53–](#page-8-3)[56\]](#page-8-4) have been addressed theoretically and demonstrated experimentally. Notably, in the context of light-induced magnetism, Berritta and co-workers have predicted that the IFE in selected transition metals can exhibit a sizeable orbital component [\[20\]](#page-7-5), with the generality of this observation reaching even into the realm of altermagnets [\[39\]](#page-7-18). At the same time it is also known that within the context of plasmonic IFE the orbital magnetic moment due to electrons excited by laser pulses in small nanoparticles of noble and simple metals can reach atomic values [\[57–](#page-8-5)[59\]](#page-8-6). Despite the fact that very little is known about the interplay of spin and orbital IFE in real materials, it is believed that IFE is a very promising effect in the context of orbitronics − a field which deals with manipulation of the orbital degree of freedom by external perturbations. Although the importance of IFE in mediating the lightinduced magnetism is beyond doubt, a comprehensive indepth material-specific knowledge acquired from microscopic theoretical analysis of this effect is still missing. While acquiring this knowledge is imperative for the field of ultrafast spintronics, since the initial seminal work by Berritta and coworkers [\[20\]](#page-7-5), very little effort has been dedicated to the explorations of this phenomenon from first principles. Here, we fill this gap by providing a detailed first-principles study of the light-induced spin and orbital magnetism for the 3d, 4d and 5d transition metals of groups IV−XI. By exploring the dependence on the frequency and the polarization of light, we acquire insights into the origin and differences between spin and orbital flavors of IFE, as well as key features of their behavior. Our work provide a solid foundation for further advances in the field of optical magnetism and interaction of light and matter. # Results # Light-induced magnetism in transition metals We begin our discussion by exploring the IFE in a series of 3d, 4d and 5d transition metals of groups IV−XI. IFE is a non-linear optomagnetic effect in which magnetization δO ∝ E × E ∗ is induced as a second-order response to the electric field E of a laser pulse. We calculate the light-induced spin δS and orbital δL magnetic moments by means of the Keldysh formalism, see the section Methods [\[21,](#page-7-9) [39\]](#page-7-18). We focus on light energies hω̵ of 0.25 eV and 1.55 eV which have been routinely used before in our studies within the same formalism [\[29,](#page-7-8) [39,](#page-7-18) [60\]](#page-8-7). Additionally, we choose a lifetime broadening Γ of 25 meV that corresponds to room temperature in order to account for effects of disorder on the electronic states and to provide realistic estimations of the effect. In all calculations we choose the light to be circularly polarized in the xy or yz planes with a light intensity I of 10 GW/cm<sup>2</sup> , while for magnetic materials ferromagnetic magnetization is considered to be along the z-axis. In Fig. [1](#page-1-0) we present the calculated light-induced δL<sup>z</sup> [Fig. [1\(](#page-1-0)a-c)] and δS<sup>z</sup> [Fig. [1\(](#page-1-0)b-d)] moments of the considered transition metals at the Fermi energy. Light is circularly polarized in the xy plane, with energies hω̵ of 0.25 eV [Fig. [1\(](#page-1-0)ab)] and 1.55 eV [Fig. [1\(](#page-1-0)c-d)]. Regarding the 3d magnetic elements Fe, Co, Ni, we analyze both right and left handed circular polarizations of light. The exact values of δL<sup>z</sup> and δS<sup>z</sup> are listed in Tables [II](#page-14-0) and [III](#page-15-0) of the Supplemental Material (SM). Not shown are computed transverse x and y compo- ![<span id="page-2-1"></span>FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.](path) At first sight we notice a strong dependence of the magnitude and sign of both δL<sup>z</sup> and δS<sup>z</sup> on the light frequency. We point out the case of fcc Rh where an colossal δL<sup>z</sup> = −56 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> is predicted at hω̵ = 0.25 eV and which is drastically reduced at the higher frequency, while the spin component remains suppressed in both cases. For the case of magnetic hcp Co we notice a helicity-dependent change of sign for δL<sup>z</sup> at hω̵ = 0.25 eV which is not present for δSz. In general, δL<sup>z</sup> varies stronger with the light helicity than δSz. Moreover, δL<sup>z</sup> is one to two orders of magnitude larger than δS<sup>z</sup> for the non-magnetic elements, while they are of the same magnitude for the magnetic elemental materials. The last two observations are in agreement with the findings of Ref. [\[20\]](#page-7-5) and indicate how the spontaneous magnetization strongly influences the effect in ferromagnets by the time-reversal symmetry breaking. For comparison, in Table [I](#page-2-0) we list the values of the total light-induced moments, defined as the sum of δL<sup>z</sup> and δSz, for the transitional metals studied in [\[20\]](#page-7-5) with hω̵ = 1.55 eV. Our calculated values are of the same order of magnitude, with the exceptions of Au, and of Co for the case of left-handed polarization, which can be attributed to the difference in the computational methods. From Fig. [1\(](#page-1-0)c-d) we get a clear picture of how the induced moments scale with the strength of spin-orbit coupling (SOC). The effect is overall larger for the heavier 5d elements, both in the orbital δL<sup>z</sup> and spin δS<sup>z</sup> channels. However, at the lower frequency of hω̵ = 0.25 eV it is more difficult to draw such a conclusion since the corresponding photon energy falls into the range of SOC strength, which promotes the role of electronic transitions among spin-orbit split bands occurring within a narrow range around the Fermi energy and limited regions in k-space. In contrast, the use of a much larger frequency involves transitions among manifestly orbitally-distinct states which take place over larger portions <span id="page-2-0"></span>TABLE I. Total light-induced magnetic moments for transition metals previously studied in [\[20\]](#page-7-5) in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV. of the reciprocal space with a more uniform impact of the spin-orbit strength on the magnitude of the transition probabilities. From this discussion we have to exclude the ferromagnetic elements since the spontaneous magnetization induces bands splittings on the scale of exchange strength therefore making the effect much more complex. We study the dependence on the light polarization by performing additional calculations for light circularly polarized in the yz-plane, presenting the results for the z and x components of δL and δS respectively in Figs. [6](#page-10-0) and [7](#page-11-0) of the SM for hω̵ = 0.25 eV and hω̵ = 1.55 eV. Our results for δS<sup>z</sup> and δS<sup>x</sup> for the magnetic transition metals reproduce exactly the values of Ref. [\[21\]](#page-7-9), computed with the same method. We further present the values of δL<sup>z</sup> and δLx, where we find the orbital response along the magnetization, δLz, to be by a factor of two larger than the corresponding spin response, when averaged over all magnetic elements. Generally, for the z-component the effect at the magnetic elements is sim- ![<span id="page-3-0"></span>FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level E<sup>F</sup> (dashed gray line). The horizontal dotted red lines at E<sup>F</sup> ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.](path) ilar in magnitude with the case of polarization in the xy-plane shown in Fig. [1,](#page-1-0) and even in the light helicity, while it vanishes for the non-magnetic elements. On the other hand, for the x-component the effect at the magnetic elements becomes one-two orders of magnitude smaller and is odd in the light helicity, although for the non-magnetic elements the values are comparable to the case shown in Fig. [1.](#page-1-0) It is important to note that for light circularly-polarized in the plane containing ferromagnetic magnetization, the light-induced transverse to magnetization induced moments, even though being smaller than the longitudinal ones, are related to light-induced torques that are experimentally demonstrated to lead to helicity-dependent THz emission [\[21,](#page-7-9) [33\]](#page-7-12). In order to get a better understanding of the impact of the time-reversal symmetry breaking on IFE, we consider the cases of ferromagnetic, non-relativistic (i.e. computed without SOC) ferromagnetic, and antiferromagnetic fcc Ni. In Figs. [8](#page-12-0) and [9](#page-13-0) of the SM we present the band filling dependence of the Cartesian components of the light-induced orbital moments δL, for hω̵ = 0.25 eV and hω̵ = 1.55 eV, respectively. In the non-relativistic case only a δL<sup>i</sup> parallel to the light propagation axis is induced which is odd in the helicity, and no δS<sup>i</sup> is induced, exemplifying that the orbital response is the primary non-relativistic one, whereas the spin response is generated through SOC, as was also shown in [\[20\]](#page-7-5). Similarly, for the antiferromagnetic case, only a component parallel to the light propagation axis is induced which is odd in the helicity and remains unchanged under different polarization flavors. While this is the case also for the induced components δL<sup>x</sup> and δL<sup>y</sup> which are transverse to the magnetization in the ferromagnetic case, the situation drastically changes for the induced component δL<sup>z</sup> parallel to the magnetization, as discussed earlier. We note that a tiny, odd in the helicity, δL<sup>x</sup> or δL<sup>y</sup> is additionally induced when light is rotating in a plane containing ferromagnetic magnetization. On the other hand, the induced δL<sup>x</sup> and δLy, developing normal to the polarization plane, serve as a non-relativistic "background" which is independent of the magnetization, with features due to the crystal structure driving the effect over larger regions in energy. The additional band-splittings induced by SOC result in the IFE exhibiting more features with band filling in the relativistic scenario. Remarkably, while the relativistic antiferromagnetic and non-relativistic ferromagnetic cases in principle have similar to each other behavior in energy, a larger signal arises in the antiferromagnetic case by the virtue of flatter bands (see also the discussion for Hf and Pt below). Lastly, we note that a similar behavior has been observed for the in-plane spin IFE in PT-symmetric Mn2Au, however, in the latter case additional out-of-plane moments arise due to linearly polarized light as a result of broken by the magnetization inversion symmetry [\[29\]](#page-7-8). Next, we focus on the case of 5d transition metals where in Fig. [2\(](#page-2-1)a-b) we explore the relation of δL<sup>z</sup> and δSz, respectively, to the band filling, for light circularly polarized in the xy plane and the frequency of hω̵ = 1.55 eV. When going from group IV (hcp Hf) to XI (fcc Au) we observe a smooth ![<span id="page-4-0"></span>FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.](path) variation of δL<sup>z</sup> from positive to negative values, as well as nicely shaped plateaus, where δL<sup>z</sup> remains relatively robust in a wide energy region, for hcp Hf, bcc Ta, bcc W, fcc Ir, and fcc Pt. We note that such plateaus are often characteristic of orbital effects, as witnessed for example in orbital Hall insulators [\[61](#page-8-8)[–63\]](#page-8-9) and orbital Rashba systems [\[60\]](#page-8-7). On the contrary, δS<sup>z</sup> exhibits a very erratic behavior with strong variations for each material, at the same time being much smaller in magnitude than δLz. The above observation is a clear manifestation of how differently the orbital and spin degrees of freedom behave under light excitation, with the orbital angular momentum having its origin in intrinsic structural parameters as manifested in the crystal field splitting, whereas the spin angular momentum is more sensitive to finer details of the electronic structure mediated by SOC. Indeed, the light-induced spin and orbital moments exhibit similarly erratic behavior with band filling once the frequency of the light is drastically reduced to reach the range of spin-orbit interaction, see Fig. [10](#page-13-1) of the SM for hω̵ = 0.25 eV. # Anatomy of IFE in k-space Among the 5d transition metals, hcp Hf and fcc Pt exhibit the largest computed moments at the Fermi energy for hω̵ = 1.55 eV, with the corresponding values of δL<sup>z</sup> = 9.1 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −0.8 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Hf, and δL<sup>z</sup> = −7.4 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −1.1 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Pt. Therefore, we select these two materials and explore the behavior of their light-induced moments in reciprocal space. We present the band-resolved δL<sup>z</sup> and δS<sup>z</sup> for hcp Hf in Fig. [3\(](#page-3-0)a-b), as well as band-resolved δL<sup>z</sup> and δS<sup>z</sup> for fcc Pt in Fig. [3\(](#page-3-0)c-d). For the case of Hf, transitions along the bands near the A-point are the main source of δL<sup>z</sup> and δSz. Light-induced δL<sup>z</sup> consists of hotspot-like negative contributions and secondary, but extended over energy and k-space consistently positive contributions, which are hardly visible. Overall, however, the latter lead to a large positive orbital integrated response. On the other hand, δS<sup>z</sup> For the case of Pt, δL<sup>z</sup> and δS<sup>z</sup> arise from transitions close to X and L high-symmetry points, with both originating in roughly the same regions of (E, k)-space, but often having an opposite sign to each other. Note that the bands in Pt are much more dispersive in the considered energy window, which results in an effective reduction of the regions in (E, k)-space which contribute to the spin and orbital response alike. This is in contrast to Hf, where much flatter bands reside above and below the Fermi energy within the energy window of the laser pulse, providing significant integrated, albeit very small locally, contributions. Moreover, in fcc Pt the Fermi energy cuts through the band edges of the d-states, where the effect of spin-orbit interaction is the strongest, which explains the emergence of strong hotspot-like contributions with a clear correlation in the magnitude of spin and orbital response and the resultant similar behavior of δL<sup>z</sup> and δS<sup>z</sup> with band filling around the Fermi energy. We further scrutinize the reciprocal space distribution of δL<sup>z</sup> and δSz, shown for hcp Hf and fcc Pt in Fig. [3\(](#page-3-0)e-h). For both materials δL<sup>z</sup> distributions consist of large uniform areas of either positive or negative sign. On the contrary, δS<sup>z</sup> distributions are much finer and richer in details with more areas of opposite sign, consistent with the picture we drew above from the band-resolved analysis. A similar behavior of the light-induced magnetism in reciprocal space has been recently reported for rutile altermagnets [\[39\]](#page-7-18). We also observe that, as discussed above, while for Hf the contributions are well spread throughout the Brillouin zone, for Pt the spin and orbital contributions are located at the edges of the considered k<sup>z</sup> − k<sup>y</sup> plane. Overall, this fact indicates that the microscopic behavior of light-induced magnetism varies strongly among transition metals and crucially depends on the crystal structure and position of the Fermi level with respect to the states ![<span id="page-5-0"></span>FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ### Anisotropy of light-induced magnetism We first address the anisotropy of light-induced magnetism with respect to the magnetization by examining the response of magnetic elements under different flavors of circular polarization. In Fig. [4\(](#page-4-0)a-b) we present the computed δL<sup>z</sup> and δS<sup>z</sup> in relation to scattering lifetime Γ, for the cases of magnetic bcc Fe and hcp Co under excitation by light which is circularly polarized in the xy or yz planes at hω̵ = 1.55 eV. As we have already seen in Fig. [1](#page-1-0) for light polarized in the xyplane, the responses behave differently for right (solid lines) or left (dashed lines) polarization. This difference is more pronounced for δLz, for which we even observe a change of sign for Co. Surprisingly, the situation is drastically different for the case of yz-polarization. In this case, for bcc Fe the response is even in the helicity, while for hcp Co δL<sup>z</sup> is almost even and δS<sup>z</sup> is perfectly even in the helicity. Such different behavior with respect to the light helicity between the two magnetic elements can be traced back to the additional anisotropy originating in the crystal structure itself − an effect which we discuss below. Notably, we witness a highly nonlinear behavior with respect to Γ, with the response reaching colossal values in the clean limit. For example, δL<sup>z</sup> and δS<sup>z</sup> can reach as much as 40 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> and 70 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> in Co, respectively, given the scattering lifetime of 1 meV. Besides the potential tunability of the inverse Faraday effect by the degree of the disorder of the samples, one key message that we extract from our observations is that when comparing the values for laser-induced magnetic moments obtained with different methods, special care has to be taken since the implementations of disorder effects, even within a simple constant broadening model, may differ among various approaches. Finally, we analyze the anisotropy that the crystal structure induces in the light-induced magnetism. We select the case of hcp Re and present in Fig. [5\(](#page-5-0)a-b) the x and z components of δL and δS, respectively, in relation to Γ. Due to the inequivalence of x and z axes in the hcp structure, the δL<sup>z</sup> which originates in light circularly polarized in the xy-plane (light blue line) differs from the δL<sup>x</sup> for the case of the yz polarization (brown line). The situation is similar for the case of δS<sup>z</sup> and δSx. On the contrary, as presented in Fig. [11](#page-14-1) of the SM, in the case of fcc Pt we have a perfect match between δL<sup>z</sup> and δL<sup>x</sup> (or δS<sup>z</sup> and δSx) when changing the plane of circular polarization because the x and z axes are equivalent in the fcc structure. The situation is similar for the y component when light is circularly polarized in the xz-plane. As also expected from the symmetry of the crystal structure, we confirm that the light-induced moments are perfectly odd in the helicity for the nonmagnetic elements, which is not the case for the magnetic elements, see Figs. [5](#page-5-0) and [11](#page-14-1) of the SM [\[20\]](#page-7-5). ### Discussion The main goal of our work is to showcase the importance of the orbital component of IFE and its distinct behavior from the spin counterpart. While so far it was mainly the interaction of light with the spin magnetization that was taken into consideration for the interpretation of IFE-related effects, we speculate that the orbital IFE may provide a novel way to coherently induce magnetization and manipulate the magnetic order. For example, in a recent study, different types of optical torques that may arise in ferromagnetic layers were interpreted in terms of the light-induced orbital moment and its interaction with the magnetization through the spin-orbit interaction [\[64\]](#page-8-10). Since it is known that current-induced orbital accumulation and orbital torques exhibit a long-range behavior [\[49,](#page-8-0) [65](#page-8-11)[–67\]](#page-8-12) due to a characteristic small orbital decay, a question arises whether similar behavior can be exhibited by optical torques caused by the orbital IFE. The emergence of long-ranged orbital IFE would come as no surprise given the fact that several recent experiments reported that laser excitation can drive long-range ballistic orbital currents resulting in THz emission [\[68](#page-8-13)[–71\]](#page-8-14). a perturbation by the electric field of the original ground state Hamiltonian, it is common to utilize the angular momentum of light in order to understand the interaction with the magnetic order by the means of transfer of angular momentum. The spin of light through the helicity of circularly polarized pulses plays a crucial role in helicity-dependent all-optical switching scenarios. On the other hand, there is a strong recent interest in utilizing the orbital angular momentum of light via irradiation of matter with e.g. vortex beams or twisted light, in order to probe the magnetization [\[72\]](#page-8-15), generate photocurrents [\[73\]](#page-8-16), drive ultrafast demagnetization [\[74\]](#page-8-17), and induce IFE [\[40,](#page-7-24) [75\]](#page-8-18). Therefore, it is imperative to treat both spin and orbital degrees of freedom on equal footing when exploring the lightmatter interaction. This will not only trigger further advances in the field of ultrafast magnetism and THz spintronics, but also enable a transition to the novel field of attosecond spintronics [\[76,](#page-8-19) [77\]](#page-8-20). Method. In this work we calculate the first-principles electronic structures of 3d, 4d and 5d transition metals by using the full-potential linearized augmented plane wave FLEUR code [\[78\]](#page-8-21). We describe exchange and correlation effects by using the non-relativistic PBE [\[79\]](#page-8-22) functional, while relativistic effects are described by the second-variation scheme [\[80\]](#page-9-0). The parameters of our first-principles calculations, i.e. lattice constants, muffin-tin radii, plane-wave cutoffs, etc. are taken from Table I of Ref. [\[81\]](#page-9-1). Next, we construct maximally-localized Wannier functions (MLWFs) by employing the Wannier90 code [\[82\]](#page-9-2) and its interface with the FLEUR code [\[83\]](#page-9-3). Similarly to [\[81\]](#page-9-1), we choose s, p and d orbitals for the initial projections and disentagle 18 MLWFs out of 36 Bloch states within a frozen window of 5.0 eV above the Fermi energy for each atom in the unit cell. where O<sup>i</sup> is the i-th component of either the orbital angular momentum operator L<sup>i</sup> or of the spin operator S<sup>i</sup> . Moreover, a<sup>0</sup> = 4πϵ0h̵<sup>2</sup> /(mee 2 ) is the Bohr's radius, I = ϵ0cE<sup>2</sup> 0 /2 is the intensity of the pulse, ϵ<sup>0</sup> is the vacuum permittivity, m<sup>e</sup> is the electron mass, e is the elementary charge, h̵ is the reduced Planck constant, c is the light velocity, E<sup>H</sup> = e 2 /(4πϵ0a0) is the Hartree energy, and ϵ<sup>j</sup> is the j-th component of the polarization vector of the pulse. For example, we describe right/left-handedly polarized light in the xy-plane as ϵ = (1,±i, <sup>0</sup>)/<sup>√</sup> 2, respectively, and define its propagation vector to lie along the normal to the polarization plane. A detailed form of the tensor φijk can be seen in Eq.(14) of Ref. [\[21\]](#page-7-9). For the orbital response, the prefactor in Eq. [\(1\)](#page-6-2) must be multiplied by an additional factor of 2. A 128×128×128 interpolation k-mesh is sufficient to obtain well-converged results. In all calculations the lifetime broadening Γ was set at 25 meV, the light frequency hω̵ at 0.25 eV and 1.55 eV, the intensity of light at 10 GW/cm<sup>2</sup> , and we covered an energy region of [−2.5, 2.5] eV around the Fermi energy level E<sup>F</sup> . Acknowledgements. We thank Frank Freimuth and Maximilian Merte for discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR 173/3 − 268565370 (project A11) and by the K. and A. Wallenberg Foundation (Grants No. 2022.0079 and 2023.0336). We acknowledge support from the EIC Pathfinder OPEN grant 101129641 "OBELIX". We also gratefully acknowledge the Julich Supercomputing Cen- ¨ tre and RWTH Aachen University for providing computational resources under projects jiff40 and jara0062. Author Contributions. T. A. performed numerical calculations and analysed the results. T. A. and Y. M. wrote the manuscript. 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(a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-11-0"></span>FIG. 7. (a-d) Light-induced orbital δL<sup>x</sup> (a, c) and spin δS<sup>x</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-12-0"></span>FIG. 8. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-13-0"></span>FIG. 9. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ![<span id="page-13-1"></span>FIG. 10. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-14-1"></span>FIG. 11. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic fcc Pt. The induced magnetic moments along the x-axis and along the z-axis are presented. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) <span id="page-14-0"></span>TABLE II. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 0.25 eV. <span id="page-15-0"></span>TABLE III. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV.
FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL⁽ᶻ⁾ (a, c) and spin δS⁽ᶻ⁾ (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level Eᴮ (dashed gray line). The horizontal dotted red lines at Eᴮ ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.	# Light-induced Orbital and Spin Magnetism in 3d, 4d, and 5d Transition Metals Understanding the coherent interplay of light with the magnetization in metals has been a long-standing problem in ultrafast magnetism. While it is known that when laser light acts on a metal it can induce magnetization via the process known as the inverse Faraday effect (IFE), the most basic ingredients of this phenomenon are still largely unexplored. In particular, given a strong recent interest in orbital non-equilibrium dynamics and its role in mediating THz emission in transition metals, the exploration of distinct features in spin and orbital IFE is pertinent. Here, we present a first complete study of the spin and orbital IFE in 3d, 4d and 5d transition metals of groups IV−XI from first-principles. By examining the dependence on the light polarization and frequency, we show that the laser-induced spin and orbital moments may vary significantly both in magnitude and sign. We underpin the interplay between the crystal field splitting and spin-orbit interaction as the key factor which determines the magnitude and key differences between the spin and orbital response. Additionally, we highlight the anisotropy of the effect with respect to the ferromagnetic magnetization and to the crystal structure. The provided complete map of IFE in transition metals is a key reference point in the field of optical magnetism. ### Introduction The demonstration of ultrafast demagnetization in ferromagnets by the application of femtosecond laser pulses [\[1\]](#page-6-0) gave rise to the field of ultrafast spintronics and set the stage for efficient manipulation of magnetism by light. By now it has been rigorously demonstrated that all-optical helicitydependent magnetization switching can be achieved in wide classes of magnetic materials [\[2](#page-6-1)[–10\]](#page-7-0), thus paving the way to contactless ultrafast magnetic recording and information processing. In interpretation of the switching experiments, the inverse Faraday effect (IFE) − i.e. the phenomenon of magnetization induced by a coherent interaction with light acting on a material − has been considered as one of the major underlying mechanisms for the magnetization reversal. Although theoretically predicted and experimentally observed decades ago [\[11–](#page-7-1)[13\]](#page-7-2), a consensus in the theoretical understanding of IFE is still lacking [\[14](#page-7-3)[–19\]](#page-7-4) – however, several microscopic methods have been recently developed for the calculation of this phenomenon in diverse setups [\[20–](#page-7-5)[27\]](#page-7-6). Apart from its established role in the magnetization switching of ferromagnets and ferrimagnents, the impact of IFE in the THz regime has also been studied in antiferromagnets like CrPt [\[28\]](#page-7-7) and Mn2Au [\[29\]](#page-7-8). Moreover, it has been shown that the component of IFE which is perpendicular to the magnetization direction behaves differently with respect to the light helicity than the parallel component [\[21\]](#page-7-9), giving rise to helicity-dependent optical torques [\[30–](#page-7-10)[32\]](#page-7-11) and THz emission [\[33\]](#page-7-12). The impact of the optical torques has been demonstrated to be significant in antiferromagnetic Mn2Au [\[34\]](#page-7-13) where transverse IFE may be responsible for the THz emission [\[35,](#page-7-14) [36\]](#page-7-15), provide an alternative way to switching the magnetization [\[37\]](#page-7-16) as well as to drive domain wall motion [\[38\]](#page-7-17). Recently it has also been shown that light can induce colossal magnetic moments in altermagnets [\[39\]](#page-7-18). Although IFE is conventionally associated with an excitation by circularly polarized light, it can also be activated by linearly-polarized laser pulses [\[25,](#page-7-19) [29,](#page-7-8) [39–](#page-7-18)[42\]](#page-7-20). In solids, two contributions to the magnetization exist: due to spin and orbital moment of electrons. And while the discussion of IFE is normally restricted to the response of spin, since recently, non-equilibrium dynamics of orbital angular momentum started to attract significant attention [\[43](#page-7-21)[–45\]](#page-7-22). Namely, the emergence of orbital currents in the context of the orbital Hall effect [\[46–](#page-7-23)[49\]](#page-8-0), orbital nature of current-induced torques on the magnetization [\[50](#page-8-1)[–52\]](#page-8-2), and current-induced orbital accumulation sizeable even in light materials [\[53–](#page-8-3)[56\]](#page-8-4) have been addressed theoretically and demonstrated experimentally. Notably, in the context of light-induced magnetism, Berritta and co-workers have predicted that the IFE in selected transition metals can exhibit a sizeable orbital component [\[20\]](#page-7-5), with the generality of this observation reaching even into the realm of altermagnets [\[39\]](#page-7-18). At the same time it is also known that within the context of plasmonic IFE the orbital magnetic moment due to electrons excited by laser pulses in small nanoparticles of noble and simple metals can reach atomic values [\[57–](#page-8-5)[59\]](#page-8-6). Despite the fact that very little is known about the interplay of spin and orbital IFE in real materials, it is believed that IFE is a very promising effect in the context of orbitronics − a field which deals with manipulation of the orbital degree of freedom by external perturbations. Although the importance of IFE in mediating the lightinduced magnetism is beyond doubt, a comprehensive indepth material-specific knowledge acquired from microscopic theoretical analysis of this effect is still missing. While acquiring this knowledge is imperative for the field of ultrafast spintronics, since the initial seminal work by Berritta and coworkers [\[20\]](#page-7-5), very little effort has been dedicated to the explorations of this phenomenon from first principles. Here, we ![<span id="page-1-0"></span>FIG. 1. Map of light-induced magnetism in transition metals. (a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) fill this gap by providing a detailed first-principles study of the light-induced spin and orbital magnetism for the 3d, 4d and 5d transition metals of groups IV−XI. By exploring the dependence on the frequency and the polarization of light, we acquire insights into the origin and differences between spin and orbital flavors of IFE, as well as key features of their behavior. Our work provide a solid foundation for further advances in the field of optical magnetism and interaction of light and matter. # Results # Light-induced magnetism in transition metals We begin our discussion by exploring the IFE in a series of 3d, 4d and 5d transition metals of groups IV−XI. IFE is a non-linear optomagnetic effect in which magnetization δO ∝ E × E ∗ is induced as a second-order response to the electric field E of a laser pulse. We calculate the light-induced spin δS and orbital δL magnetic moments by means of the Keldysh formalism, see the section Methods [\[21,](#page-7-9) [39\]](#page-7-18). We focus on light energies hω̵ of 0.25 eV and 1.55 eV which have been routinely used before in our studies within the same formalism [\[29,](#page-7-8) [39,](#page-7-18) [60\]](#page-8-7). Additionally, we choose a lifetime broadening Γ of 25 meV that corresponds to room temperature in order to account for effects of disorder on the electronic states and to provide realistic estimations of the effect. In all calculations we choose the light to be circularly polarized in the xy or yz planes with a light intensity I of 10 GW/cm<sup>2</sup> , while for magnetic materials ferromagnetic magnetization is considered to be along the z-axis. In Fig. [1](#page-1-0) we present the calculated light-induced δL<sup>z</sup> [Fig. [1\(](#page-1-0)a-c)] and δS<sup>z</sup> [Fig. [1\(](#page-1-0)b-d)] moments of the considered transition metals at the Fermi energy. Light is circularly polarized in the xy plane, with energies hω̵ of 0.25 eV [Fig. [1\(](#page-1-0)ab)] and 1.55 eV [Fig. [1\(](#page-1-0)c-d)]. Regarding the 3d magnetic elements Fe, Co, Ni, we analyze both right and left handed circular polarizations of light. The exact values of δL<sup>z</sup> and δS<sup>z</sup> are listed in Tables [II](#page-14-0) and [III](#page-15-0) of the Supplemental Material (SM). Not shown are computed transverse x and y compo- ![<span id="page-2-1"></span>FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.](path) At first sight we notice a strong dependence of the magnitude and sign of both δL<sup>z</sup> and δS<sup>z</sup> on the light frequency. We point out the case of fcc Rh where an colossal δL<sup>z</sup> = −56 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> is predicted at hω̵ = 0.25 eV and which is drastically reduced at the higher frequency, while the spin component remains suppressed in both cases. For the case of magnetic hcp Co we notice a helicity-dependent change of sign for δL<sup>z</sup> at hω̵ = 0.25 eV which is not present for δSz. In general, δL<sup>z</sup> varies stronger with the light helicity than δSz. Moreover, δL<sup>z</sup> is one to two orders of magnitude larger than δS<sup>z</sup> for the non-magnetic elements, while they are of the same magnitude for the magnetic elemental materials. The last two observations are in agreement with the findings of Ref. [\[20\]](#page-7-5) and indicate how the spontaneous magnetization strongly influences the effect in ferromagnets by the time-reversal symmetry breaking. For comparison, in Table [I](#page-2-0) we list the values of the total light-induced moments, defined as the sum of δL<sup>z</sup> and δSz, for the transitional metals studied in [\[20\]](#page-7-5) with hω̵ = 1.55 eV. Our calculated values are of the same order of magnitude, with the exceptions of Au, and of Co for the case of left-handed polarization, which can be attributed to the difference in the computational methods. From Fig. [1\(](#page-1-0)c-d) we get a clear picture of how the induced moments scale with the strength of spin-orbit coupling (SOC). The effect is overall larger for the heavier 5d elements, both in the orbital δL<sup>z</sup> and spin δS<sup>z</sup> channels. However, at the lower frequency of hω̵ = 0.25 eV it is more difficult to draw such a conclusion since the corresponding photon energy falls into the range of SOC strength, which promotes the role of electronic transitions among spin-orbit split bands occurring within a narrow range around the Fermi energy and limited regions in k-space. In contrast, the use of a much larger frequency involves transitions among manifestly orbitally-distinct states which take place over larger portions <span id="page-2-0"></span>TABLE I. Total light-induced magnetic moments for transition metals previously studied in [\[20\]](#page-7-5) in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV. of the reciprocal space with a more uniform impact of the spin-orbit strength on the magnitude of the transition probabilities. From this discussion we have to exclude the ferromagnetic elements since the spontaneous magnetization induces bands splittings on the scale of exchange strength therefore making the effect much more complex. We study the dependence on the light polarization by performing additional calculations for light circularly polarized in the yz-plane, presenting the results for the z and x components of δL and δS respectively in Figs. [6](#page-10-0) and [7](#page-11-0) of the SM for hω̵ = 0.25 eV and hω̵ = 1.55 eV. Our results for δS<sup>z</sup> and δS<sup>x</sup> for the magnetic transition metals reproduce exactly the values of Ref. [\[21\]](#page-7-9), computed with the same method. We further present the values of δL<sup>z</sup> and δLx, where we find the orbital response along the magnetization, δLz, to be by a factor of two larger than the corresponding spin response, when averaged over all magnetic elements. Generally, for the z-component the effect at the magnetic elements is sim- ![<span id="page-3-0"></span>FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level E<sup>F</sup> (dashed gray line). The horizontal dotted red lines at E<sup>F</sup> ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.](path) ilar in magnitude with the case of polarization in the xy-plane shown in Fig. [1,](#page-1-0) and even in the light helicity, while it vanishes for the non-magnetic elements. On the other hand, for the x-component the effect at the magnetic elements becomes one-two orders of magnitude smaller and is odd in the light helicity, although for the non-magnetic elements the values are comparable to the case shown in Fig. [1.](#page-1-0) It is important to note that for light circularly-polarized in the plane containing ferromagnetic magnetization, the light-induced transverse to magnetization induced moments, even though being smaller than the longitudinal ones, are related to light-induced torques that are experimentally demonstrated to lead to helicity-dependent THz emission [\[21,](#page-7-9) [33\]](#page-7-12). In order to get a better understanding of the impact of the time-reversal symmetry breaking on IFE, we consider the cases of ferromagnetic, non-relativistic (i.e. computed without SOC) ferromagnetic, and antiferromagnetic fcc Ni. In Figs. [8](#page-12-0) and [9](#page-13-0) of the SM we present the band filling dependence of the Cartesian components of the light-induced orbital moments δL, for hω̵ = 0.25 eV and hω̵ = 1.55 eV, respectively. In the non-relativistic case only a δL<sup>i</sup> parallel to the light propagation axis is induced which is odd in the helicity, and no δS<sup>i</sup> is induced, exemplifying that the orbital response is the primary non-relativistic one, whereas the spin response is generated through SOC, as was also shown in [\[20\]](#page-7-5). Similarly, for the antiferromagnetic case, only a component parallel to the light propagation axis is induced which is odd in the helicity and remains unchanged under different polarization flavors. While this is the case also for the induced components δL<sup>x</sup> and δL<sup>y</sup> which are transverse to the magnetization in the ferromagnetic case, the situation drastically changes for the induced component δL<sup>z</sup> parallel to the magnetization, as discussed earlier. We note that a tiny, odd in the helicity, δL<sup>x</sup> or δL<sup>y</sup> is additionally induced when light is rotating in a plane containing ferromagnetic magnetization. On the other hand, the induced δL<sup>x</sup> and δLy, developing normal to the polarization plane, serve as a non-relativistic "background" which is independent of the magnetization, with features due to the crystal structure driving the effect over larger regions in energy. The additional band-splittings induced by SOC result in the IFE exhibiting more features with band filling in the relativistic scenario. Remarkably, while the relativistic antiferromagnetic and non-relativistic ferromagnetic cases in principle have similar to each other behavior in energy, a larger signal arises in the antiferromagnetic case by the virtue of flatter bands (see also the discussion for Hf and Pt below). Lastly, we note that a similar behavior has been observed for the in-plane spin IFE in PT-symmetric Mn2Au, however, in the latter case additional out-of-plane moments arise due to linearly polarized light as a result of broken by the magnetization inversion symmetry [\[29\]](#page-7-8). Next, we focus on the case of 5d transition metals where in Fig. [2\(](#page-2-1)a-b) we explore the relation of δL<sup>z</sup> and δSz, respectively, to the band filling, for light circularly polarized in the xy plane and the frequency of hω̵ = 1.55 eV. When going from group IV (hcp Hf) to XI (fcc Au) we observe a smooth ![<span id="page-4-0"></span>FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.](path) variation of δL<sup>z</sup> from positive to negative values, as well as nicely shaped plateaus, where δL<sup>z</sup> remains relatively robust in a wide energy region, for hcp Hf, bcc Ta, bcc W, fcc Ir, and fcc Pt. We note that such plateaus are often characteristic of orbital effects, as witnessed for example in orbital Hall insulators [\[61](#page-8-8)[–63\]](#page-8-9) and orbital Rashba systems [\[60\]](#page-8-7). On the contrary, δS<sup>z</sup> exhibits a very erratic behavior with strong variations for each material, at the same time being much smaller in magnitude than δLz. The above observation is a clear manifestation of how differently the orbital and spin degrees of freedom behave under light excitation, with the orbital angular momentum having its origin in intrinsic structural parameters as manifested in the crystal field splitting, whereas the spin angular momentum is more sensitive to finer details of the electronic structure mediated by SOC. Indeed, the light-induced spin and orbital moments exhibit similarly erratic behavior with band filling once the frequency of the light is drastically reduced to reach the range of spin-orbit interaction, see Fig. [10](#page-13-1) of the SM for hω̵ = 0.25 eV. # Anatomy of IFE in k-space Among the 5d transition metals, hcp Hf and fcc Pt exhibit the largest computed moments at the Fermi energy for hω̵ = 1.55 eV, with the corresponding values of δL<sup>z</sup> = 9.1 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −0.8 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Hf, and δL<sup>z</sup> = −7.4 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −1.1 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Pt. Therefore, we select these two materials and explore the behavior of their light-induced moments in reciprocal space. We present the band-resolved δL<sup>z</sup> and δS<sup>z</sup> for hcp Hf in Fig. [3\(](#page-3-0)a-b), as well as band-resolved δL<sup>z</sup> and δS<sup>z</sup> for fcc Pt in Fig. [3\(](#page-3-0)c-d). For the case of Hf, transitions along the bands near the A-point are the main source of δL<sup>z</sup> and δSz. Light-induced δL<sup>z</sup> consists of hotspot-like negative contributions and secondary, but extended over energy and k-space consistently positive contributions, which are hardly visible. Overall, however, the latter lead to a large positive orbital integrated response. On the other hand, δS<sup>z</sup> For the case of Pt, δL<sup>z</sup> and δS<sup>z</sup> arise from transitions close to X and L high-symmetry points, with both originating in roughly the same regions of (E, k)-space, but often having an opposite sign to each other. Note that the bands in Pt are much more dispersive in the considered energy window, which results in an effective reduction of the regions in (E, k)-space which contribute to the spin and orbital response alike. This is in contrast to Hf, where much flatter bands reside above and below the Fermi energy within the energy window of the laser pulse, providing significant integrated, albeit very small locally, contributions. Moreover, in fcc Pt the Fermi energy cuts through the band edges of the d-states, where the effect of spin-orbit interaction is the strongest, which explains the emergence of strong hotspot-like contributions with a clear correlation in the magnitude of spin and orbital response and the resultant similar behavior of δL<sup>z</sup> and δS<sup>z</sup> with band filling around the Fermi energy. We further scrutinize the reciprocal space distribution of δL<sup>z</sup> and δSz, shown for hcp Hf and fcc Pt in Fig. [3\(](#page-3-0)e-h). For both materials δL<sup>z</sup> distributions consist of large uniform areas of either positive or negative sign. On the contrary, δS<sup>z</sup> distributions are much finer and richer in details with more areas of opposite sign, consistent with the picture we drew above from the band-resolved analysis. A similar behavior of the light-induced magnetism in reciprocal space has been recently reported for rutile altermagnets [\[39\]](#page-7-18). We also observe that, as discussed above, while for Hf the contributions are well spread throughout the Brillouin zone, for Pt the spin and orbital contributions are located at the edges of the considered k<sup>z</sup> − k<sup>y</sup> plane. Overall, this fact indicates that the microscopic behavior of light-induced magnetism varies strongly among transition metals and crucially depends on the crystal structure and position of the Fermi level with respect to the states ![<span id="page-5-0"></span>FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ### Anisotropy of light-induced magnetism We first address the anisotropy of light-induced magnetism with respect to the magnetization by examining the response of magnetic elements under different flavors of circular polarization. In Fig. [4\(](#page-4-0)a-b) we present the computed δL<sup>z</sup> and δS<sup>z</sup> in relation to scattering lifetime Γ, for the cases of magnetic bcc Fe and hcp Co under excitation by light which is circularly polarized in the xy or yz planes at hω̵ = 1.55 eV. As we have already seen in Fig. [1](#page-1-0) for light polarized in the xyplane, the responses behave differently for right (solid lines) or left (dashed lines) polarization. This difference is more pronounced for δLz, for which we even observe a change of sign for Co. Surprisingly, the situation is drastically different for the case of yz-polarization. In this case, for bcc Fe the response is even in the helicity, while for hcp Co δL<sup>z</sup> is almost even and δS<sup>z</sup> is perfectly even in the helicity. Such different behavior with respect to the light helicity between the two magnetic elements can be traced back to the additional anisotropy originating in the crystal structure itself − an effect which we discuss below. Notably, we witness a highly nonlinear behavior with respect to Γ, with the response reaching colossal values in the clean limit. For example, δL<sup>z</sup> and δS<sup>z</sup> can reach as much as 40 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> and 70 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> in Co, respectively, given the scattering lifetime of 1 meV. Besides the potential tunability of the inverse Faraday effect by the degree of the disorder of the samples, one key message that we extract from our observations is that when comparing the values for laser-induced magnetic moments obtained with different methods, special care has to be taken since the implementations of disorder effects, even within a simple constant broadening model, may differ among various approaches. Finally, we analyze the anisotropy that the crystal structure induces in the light-induced magnetism. We select the case of hcp Re and present in Fig. [5\(](#page-5-0)a-b) the x and z components of δL and δS, respectively, in relation to Γ. Due to the inequivalence of x and z axes in the hcp structure, the δL<sup>z</sup> which originates in light circularly polarized in the xy-plane (light blue line) differs from the δL<sup>x</sup> for the case of the yz polarization (brown line). The situation is similar for the case of δS<sup>z</sup> and δSx. On the contrary, as presented in Fig. [11](#page-14-1) of the SM, in the case of fcc Pt we have a perfect match between δL<sup>z</sup> and δL<sup>x</sup> (or δS<sup>z</sup> and δSx) when changing the plane of circular polarization because the x and z axes are equivalent in the fcc structure. The situation is similar for the y component when light is circularly polarized in the xz-plane. As also expected from the symmetry of the crystal structure, we confirm that the light-induced moments are perfectly odd in the helicity for the nonmagnetic elements, which is not the case for the magnetic elements, see Figs. [5](#page-5-0) and [11](#page-14-1) of the SM [\[20\]](#page-7-5). ### Discussion The main goal of our work is to showcase the importance of the orbital component of IFE and its distinct behavior from the spin counterpart. While so far it was mainly the interaction of light with the spin magnetization that was taken into consideration for the interpretation of IFE-related effects, we speculate that the orbital IFE may provide a novel way to coherently induce magnetization and manipulate the magnetic order. For example, in a recent study, different types of optical torques that may arise in ferromagnetic layers were interpreted in terms of the light-induced orbital moment and its interaction with the magnetization through the spin-orbit interaction [\[64\]](#page-8-10). Since it is known that current-induced orbital accumulation and orbital torques exhibit a long-range behavior [\[49,](#page-8-0) [65](#page-8-11)[–67\]](#page-8-12) due to a characteristic small orbital decay, a question arises whether similar behavior can be exhibited by optical torques caused by the orbital IFE. The emergence of long-ranged orbital IFE would come as no surprise given the fact that several recent experiments reported that laser excitation can drive long-range ballistic orbital currents resulting in THz emission [\[68](#page-8-13)[–71\]](#page-8-14). a perturbation by the electric field of the original ground state Hamiltonian, it is common to utilize the angular momentum of light in order to understand the interaction with the magnetic order by the means of transfer of angular momentum. The spin of light through the helicity of circularly polarized pulses plays a crucial role in helicity-dependent all-optical switching scenarios. On the other hand, there is a strong recent interest in utilizing the orbital angular momentum of light via irradiation of matter with e.g. vortex beams or twisted light, in order to probe the magnetization [\[72\]](#page-8-15), generate photocurrents [\[73\]](#page-8-16), drive ultrafast demagnetization [\[74\]](#page-8-17), and induce IFE [\[40,](#page-7-24) [75\]](#page-8-18). Therefore, it is imperative to treat both spin and orbital degrees of freedom on equal footing when exploring the lightmatter interaction. This will not only trigger further advances in the field of ultrafast magnetism and THz spintronics, but also enable a transition to the novel field of attosecond spintronics [\[76,](#page-8-19) [77\]](#page-8-20). Method. In this work we calculate the first-principles electronic structures of 3d, 4d and 5d transition metals by using the full-potential linearized augmented plane wave FLEUR code [\[78\]](#page-8-21). We describe exchange and correlation effects by using the non-relativistic PBE [\[79\]](#page-8-22) functional, while relativistic effects are described by the second-variation scheme [\[80\]](#page-9-0). The parameters of our first-principles calculations, i.e. lattice constants, muffin-tin radii, plane-wave cutoffs, etc. are taken from Table I of Ref. [\[81\]](#page-9-1). Next, we construct maximally-localized Wannier functions (MLWFs) by employing the Wannier90 code [\[82\]](#page-9-2) and its interface with the FLEUR code [\[83\]](#page-9-3). Similarly to [\[81\]](#page-9-1), we choose s, p and d orbitals for the initial projections and disentagle 18 MLWFs out of 36 Bloch states within a frozen window of 5.0 eV above the Fermi energy for each atom in the unit cell. where O<sup>i</sup> is the i-th component of either the orbital angular momentum operator L<sup>i</sup> or of the spin operator S<sup>i</sup> . Moreover, a<sup>0</sup> = 4πϵ0h̵<sup>2</sup> /(mee 2 ) is the Bohr's radius, I = ϵ0cE<sup>2</sup> 0 /2 is the intensity of the pulse, ϵ<sup>0</sup> is the vacuum permittivity, m<sup>e</sup> is the electron mass, e is the elementary charge, h̵ is the reduced Planck constant, c is the light velocity, E<sup>H</sup> = e 2 /(4πϵ0a0) is the Hartree energy, and ϵ<sup>j</sup> is the j-th component of the polarization vector of the pulse. For example, we describe right/left-handedly polarized light in the xy-plane as ϵ = (1,±i, <sup>0</sup>)/<sup>√</sup> 2, respectively, and define its propagation vector to lie along the normal to the polarization plane. A detailed form of the tensor φijk can be seen in Eq.(14) of Ref. [\[21\]](#page-7-9). For the orbital response, the prefactor in Eq. [\(1\)](#page-6-2) must be multiplied by an additional factor of 2. A 128×128×128 interpolation k-mesh is sufficient to obtain well-converged results. In all calculations the lifetime broadening Γ was set at 25 meV, the light frequency hω̵ at 0.25 eV and 1.55 eV, the intensity of light at 10 GW/cm<sup>2</sup> , and we covered an energy region of [−2.5, 2.5] eV around the Fermi energy level E<sup>F</sup> . Acknowledgements. We thank Frank Freimuth and Maximilian Merte for discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR 173/3 − 268565370 (project A11) and by the K. and A. Wallenberg Foundation (Grants No. 2022.0079 and 2023.0336). We acknowledge support from the EIC Pathfinder OPEN grant 101129641 "OBELIX". We also gratefully acknowledge the Julich Supercomputing Cen- ¨ tre and RWTH Aachen University for providing computational resources under projects jiff40 and jara0062. Author Contributions. T. A. performed numerical calculations and analysed the results. T. A. and Y. M. wrote the manuscript. 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(a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-11-0"></span>FIG. 7. (a-d) Light-induced orbital δL<sup>x</sup> (a, c) and spin δS<sup>x</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-12-0"></span>FIG. 8. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-13-0"></span>FIG. 9. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ![<span id="page-13-1"></span>FIG. 10. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-14-1"></span>FIG. 11. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic fcc Pt. The induced magnetic moments along the x-axis and along the z-axis are presented. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) <span id="page-14-0"></span>TABLE II. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 0.25 eV. <span id="page-15-0"></span>TABLE III. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV.
FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.	# Light-induced Orbital and Spin Magnetism in 3d, 4d, and 5d Transition Metals Understanding the coherent interplay of light with the magnetization in metals has been a long-standing problem in ultrafast magnetism. While it is known that when laser light acts on a metal it can induce magnetization via the process known as the inverse Faraday effect (IFE), the most basic ingredients of this phenomenon are still largely unexplored. In particular, given a strong recent interest in orbital non-equilibrium dynamics and its role in mediating THz emission in transition metals, the exploration of distinct features in spin and orbital IFE is pertinent. Here, we present a first complete study of the spin and orbital IFE in 3d, 4d and 5d transition metals of groups IV−XI from first-principles. By examining the dependence on the light polarization and frequency, we show that the laser-induced spin and orbital moments may vary significantly both in magnitude and sign. We underpin the interplay between the crystal field splitting and spin-orbit interaction as the key factor which determines the magnitude and key differences between the spin and orbital response. Additionally, we highlight the anisotropy of the effect with respect to the ferromagnetic magnetization and to the crystal structure. The provided complete map of IFE in transition metals is a key reference point in the field of optical magnetism. ### Introduction The demonstration of ultrafast demagnetization in ferromagnets by the application of femtosecond laser pulses [\[1\]](#page-6-0) gave rise to the field of ultrafast spintronics and set the stage for efficient manipulation of magnetism by light. By now it has been rigorously demonstrated that all-optical helicitydependent magnetization switching can be achieved in wide classes of magnetic materials [\[2](#page-6-1)[–10\]](#page-7-0), thus paving the way to contactless ultrafast magnetic recording and information processing. In interpretation of the switching experiments, the inverse Faraday effect (IFE) − i.e. the phenomenon of magnetization induced by a coherent interaction with light acting on a material − has been considered as one of the major underlying mechanisms for the magnetization reversal. Although theoretically predicted and experimentally observed decades ago [\[11–](#page-7-1)[13\]](#page-7-2), a consensus in the theoretical understanding of IFE is still lacking [\[14](#page-7-3)[–19\]](#page-7-4) – however, several microscopic methods have been recently developed for the calculation of this phenomenon in diverse setups [\[20–](#page-7-5)[27\]](#page-7-6). Apart from its established role in the magnetization switching of ferromagnets and ferrimagnents, the impact of IFE in the THz regime has also been studied in antiferromagnets like CrPt [\[28\]](#page-7-7) and Mn2Au [\[29\]](#page-7-8). Moreover, it has been shown that the component of IFE which is perpendicular to the magnetization direction behaves differently with respect to the light helicity than the parallel component [\[21\]](#page-7-9), giving rise to helicity-dependent optical torques [\[30–](#page-7-10)[32\]](#page-7-11) and THz emission [\[33\]](#page-7-12). The impact of the optical torques has been demonstrated to be significant in antiferromagnetic Mn2Au [\[34\]](#page-7-13) where transverse IFE may be responsible for the THz emission [\[35,](#page-7-14) [36\]](#page-7-15), provide an alternative way to switching the magnetization [\[37\]](#page-7-16) as well as to drive domain wall motion [\[38\]](#page-7-17). Recently it has also been shown that light can induce colossal magnetic moments in altermagnets [\[39\]](#page-7-18). Although IFE is conventionally associated with an excitation by circularly polarized light, it can also be activated by linearly-polarized laser pulses [\[25,](#page-7-19) [29,](#page-7-8) [39–](#page-7-18)[42\]](#page-7-20). In solids, two contributions to the magnetization exist: due to spin and orbital moment of electrons. And while the discussion of IFE is normally restricted to the response of spin, since recently, non-equilibrium dynamics of orbital angular momentum started to attract significant attention [\[43](#page-7-21)[–45\]](#page-7-22). Namely, the emergence of orbital currents in the context of the orbital Hall effect [\[46–](#page-7-23)[49\]](#page-8-0), orbital nature of current-induced torques on the magnetization [\[50](#page-8-1)[–52\]](#page-8-2), and current-induced orbital accumulation sizeable even in light materials [\[53–](#page-8-3)[56\]](#page-8-4) have been addressed theoretically and demonstrated experimentally. Notably, in the context of light-induced magnetism, Berritta and co-workers have predicted that the IFE in selected transition metals can exhibit a sizeable orbital component [\[20\]](#page-7-5), with the generality of this observation reaching even into the realm of altermagnets [\[39\]](#page-7-18). At the same time it is also known that within the context of plasmonic IFE the orbital magnetic moment due to electrons excited by laser pulses in small nanoparticles of noble and simple metals can reach atomic values [\[57–](#page-8-5)[59\]](#page-8-6). Despite the fact that very little is known about the interplay of spin and orbital IFE in real materials, it is believed that IFE is a very promising effect in the context of orbitronics − a field which deals with manipulation of the orbital degree of freedom by external perturbations. Although the importance of IFE in mediating the lightinduced magnetism is beyond doubt, a comprehensive indepth material-specific knowledge acquired from microscopic theoretical analysis of this effect is still missing. While acquiring this knowledge is imperative for the field of ultrafast spintronics, since the initial seminal work by Berritta and coworkers [\[20\]](#page-7-5), very little effort has been dedicated to the explorations of this phenomenon from first principles. Here, we ![<span id="page-1-0"></span>FIG. 1. Map of light-induced magnetism in transition metals. (a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) fill this gap by providing a detailed first-principles study of the light-induced spin and orbital magnetism for the 3d, 4d and 5d transition metals of groups IV−XI. By exploring the dependence on the frequency and the polarization of light, we acquire insights into the origin and differences between spin and orbital flavors of IFE, as well as key features of their behavior. Our work provide a solid foundation for further advances in the field of optical magnetism and interaction of light and matter. # Results # Light-induced magnetism in transition metals We begin our discussion by exploring the IFE in a series of 3d, 4d and 5d transition metals of groups IV−XI. IFE is a non-linear optomagnetic effect in which magnetization δO ∝ E × E ∗ is induced as a second-order response to the electric field E of a laser pulse. We calculate the light-induced spin δS and orbital δL magnetic moments by means of the Keldysh formalism, see the section Methods [\[21,](#page-7-9) [39\]](#page-7-18). We focus on light energies hω̵ of 0.25 eV and 1.55 eV which have been routinely used before in our studies within the same formalism [\[29,](#page-7-8) [39,](#page-7-18) [60\]](#page-8-7). Additionally, we choose a lifetime broadening Γ of 25 meV that corresponds to room temperature in order to account for effects of disorder on the electronic states and to provide realistic estimations of the effect. In all calculations we choose the light to be circularly polarized in the xy or yz planes with a light intensity I of 10 GW/cm<sup>2</sup> , while for magnetic materials ferromagnetic magnetization is considered to be along the z-axis. In Fig. [1](#page-1-0) we present the calculated light-induced δL<sup>z</sup> [Fig. [1\(](#page-1-0)a-c)] and δS<sup>z</sup> [Fig. [1\(](#page-1-0)b-d)] moments of the considered transition metals at the Fermi energy. Light is circularly polarized in the xy plane, with energies hω̵ of 0.25 eV [Fig. [1\(](#page-1-0)ab)] and 1.55 eV [Fig. [1\(](#page-1-0)c-d)]. Regarding the 3d magnetic elements Fe, Co, Ni, we analyze both right and left handed circular polarizations of light. The exact values of δL<sup>z</sup> and δS<sup>z</sup> are listed in Tables [II](#page-14-0) and [III](#page-15-0) of the Supplemental Material (SM). Not shown are computed transverse x and y compo- ![<span id="page-2-1"></span>FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.](path) At first sight we notice a strong dependence of the magnitude and sign of both δL<sup>z</sup> and δS<sup>z</sup> on the light frequency. We point out the case of fcc Rh where an colossal δL<sup>z</sup> = −56 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> is predicted at hω̵ = 0.25 eV and which is drastically reduced at the higher frequency, while the spin component remains suppressed in both cases. For the case of magnetic hcp Co we notice a helicity-dependent change of sign for δL<sup>z</sup> at hω̵ = 0.25 eV which is not present for δSz. In general, δL<sup>z</sup> varies stronger with the light helicity than δSz. Moreover, δL<sup>z</sup> is one to two orders of magnitude larger than δS<sup>z</sup> for the non-magnetic elements, while they are of the same magnitude for the magnetic elemental materials. The last two observations are in agreement with the findings of Ref. [\[20\]](#page-7-5) and indicate how the spontaneous magnetization strongly influences the effect in ferromagnets by the time-reversal symmetry breaking. For comparison, in Table [I](#page-2-0) we list the values of the total light-induced moments, defined as the sum of δL<sup>z</sup> and δSz, for the transitional metals studied in [\[20\]](#page-7-5) with hω̵ = 1.55 eV. Our calculated values are of the same order of magnitude, with the exceptions of Au, and of Co for the case of left-handed polarization, which can be attributed to the difference in the computational methods. From Fig. [1\(](#page-1-0)c-d) we get a clear picture of how the induced moments scale with the strength of spin-orbit coupling (SOC). The effect is overall larger for the heavier 5d elements, both in the orbital δL<sup>z</sup> and spin δS<sup>z</sup> channels. However, at the lower frequency of hω̵ = 0.25 eV it is more difficult to draw such a conclusion since the corresponding photon energy falls into the range of SOC strength, which promotes the role of electronic transitions among spin-orbit split bands occurring within a narrow range around the Fermi energy and limited regions in k-space. In contrast, the use of a much larger frequency involves transitions among manifestly orbitally-distinct states which take place over larger portions <span id="page-2-0"></span>TABLE I. Total light-induced magnetic moments for transition metals previously studied in [\[20\]](#page-7-5) in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV. of the reciprocal space with a more uniform impact of the spin-orbit strength on the magnitude of the transition probabilities. From this discussion we have to exclude the ferromagnetic elements since the spontaneous magnetization induces bands splittings on the scale of exchange strength therefore making the effect much more complex. We study the dependence on the light polarization by performing additional calculations for light circularly polarized in the yz-plane, presenting the results for the z and x components of δL and δS respectively in Figs. [6](#page-10-0) and [7](#page-11-0) of the SM for hω̵ = 0.25 eV and hω̵ = 1.55 eV. Our results for δS<sup>z</sup> and δS<sup>x</sup> for the magnetic transition metals reproduce exactly the values of Ref. [\[21\]](#page-7-9), computed with the same method. We further present the values of δL<sup>z</sup> and δLx, where we find the orbital response along the magnetization, δLz, to be by a factor of two larger than the corresponding spin response, when averaged over all magnetic elements. Generally, for the z-component the effect at the magnetic elements is sim- ![<span id="page-3-0"></span>FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level E<sup>F</sup> (dashed gray line). The horizontal dotted red lines at E<sup>F</sup> ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.](path) ilar in magnitude with the case of polarization in the xy-plane shown in Fig. [1,](#page-1-0) and even in the light helicity, while it vanishes for the non-magnetic elements. On the other hand, for the x-component the effect at the magnetic elements becomes one-two orders of magnitude smaller and is odd in the light helicity, although for the non-magnetic elements the values are comparable to the case shown in Fig. [1.](#page-1-0) It is important to note that for light circularly-polarized in the plane containing ferromagnetic magnetization, the light-induced transverse to magnetization induced moments, even though being smaller than the longitudinal ones, are related to light-induced torques that are experimentally demonstrated to lead to helicity-dependent THz emission [\[21,](#page-7-9) [33\]](#page-7-12). In order to get a better understanding of the impact of the time-reversal symmetry breaking on IFE, we consider the cases of ferromagnetic, non-relativistic (i.e. computed without SOC) ferromagnetic, and antiferromagnetic fcc Ni. In Figs. [8](#page-12-0) and [9](#page-13-0) of the SM we present the band filling dependence of the Cartesian components of the light-induced orbital moments δL, for hω̵ = 0.25 eV and hω̵ = 1.55 eV, respectively. In the non-relativistic case only a δL<sup>i</sup> parallel to the light propagation axis is induced which is odd in the helicity, and no δS<sup>i</sup> is induced, exemplifying that the orbital response is the primary non-relativistic one, whereas the spin response is generated through SOC, as was also shown in [\[20\]](#page-7-5). Similarly, for the antiferromagnetic case, only a component parallel to the light propagation axis is induced which is odd in the helicity and remains unchanged under different polarization flavors. While this is the case also for the induced components δL<sup>x</sup> and δL<sup>y</sup> which are transverse to the magnetization in the ferromagnetic case, the situation drastically changes for the induced component δL<sup>z</sup> parallel to the magnetization, as discussed earlier. We note that a tiny, odd in the helicity, δL<sup>x</sup> or δL<sup>y</sup> is additionally induced when light is rotating in a plane containing ferromagnetic magnetization. On the other hand, the induced δL<sup>x</sup> and δLy, developing normal to the polarization plane, serve as a non-relativistic "background" which is independent of the magnetization, with features due to the crystal structure driving the effect over larger regions in energy. The additional band-splittings induced by SOC result in the IFE exhibiting more features with band filling in the relativistic scenario. Remarkably, while the relativistic antiferromagnetic and non-relativistic ferromagnetic cases in principle have similar to each other behavior in energy, a larger signal arises in the antiferromagnetic case by the virtue of flatter bands (see also the discussion for Hf and Pt below). Lastly, we note that a similar behavior has been observed for the in-plane spin IFE in PT-symmetric Mn2Au, however, in the latter case additional out-of-plane moments arise due to linearly polarized light as a result of broken by the magnetization inversion symmetry [\[29\]](#page-7-8). Next, we focus on the case of 5d transition metals where in Fig. [2\(](#page-2-1)a-b) we explore the relation of δL<sup>z</sup> and δSz, respectively, to the band filling, for light circularly polarized in the xy plane and the frequency of hω̵ = 1.55 eV. When going from group IV (hcp Hf) to XI (fcc Au) we observe a smooth ![<span id="page-4-0"></span>FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.](path) variation of δL<sup>z</sup> from positive to negative values, as well as nicely shaped plateaus, where δL<sup>z</sup> remains relatively robust in a wide energy region, for hcp Hf, bcc Ta, bcc W, fcc Ir, and fcc Pt. We note that such plateaus are often characteristic of orbital effects, as witnessed for example in orbital Hall insulators [\[61](#page-8-8)[–63\]](#page-8-9) and orbital Rashba systems [\[60\]](#page-8-7). On the contrary, δS<sup>z</sup> exhibits a very erratic behavior with strong variations for each material, at the same time being much smaller in magnitude than δLz. The above observation is a clear manifestation of how differently the orbital and spin degrees of freedom behave under light excitation, with the orbital angular momentum having its origin in intrinsic structural parameters as manifested in the crystal field splitting, whereas the spin angular momentum is more sensitive to finer details of the electronic structure mediated by SOC. Indeed, the light-induced spin and orbital moments exhibit similarly erratic behavior with band filling once the frequency of the light is drastically reduced to reach the range of spin-orbit interaction, see Fig. [10](#page-13-1) of the SM for hω̵ = 0.25 eV. # Anatomy of IFE in k-space Among the 5d transition metals, hcp Hf and fcc Pt exhibit the largest computed moments at the Fermi energy for hω̵ = 1.55 eV, with the corresponding values of δL<sup>z</sup> = 9.1 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −0.8 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Hf, and δL<sup>z</sup> = −7.4 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −1.1 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Pt. Therefore, we select these two materials and explore the behavior of their light-induced moments in reciprocal space. We present the band-resolved δL<sup>z</sup> and δS<sup>z</sup> for hcp Hf in Fig. [3\(](#page-3-0)a-b), as well as band-resolved δL<sup>z</sup> and δS<sup>z</sup> for fcc Pt in Fig. [3\(](#page-3-0)c-d). For the case of Hf, transitions along the bands near the A-point are the main source of δL<sup>z</sup> and δSz. Light-induced δL<sup>z</sup> consists of hotspot-like negative contributions and secondary, but extended over energy and k-space consistently positive contributions, which are hardly visible. Overall, however, the latter lead to a large positive orbital integrated response. On the other hand, δS<sup>z</sup> For the case of Pt, δL<sup>z</sup> and δS<sup>z</sup> arise from transitions close to X and L high-symmetry points, with both originating in roughly the same regions of (E, k)-space, but often having an opposite sign to each other. Note that the bands in Pt are much more dispersive in the considered energy window, which results in an effective reduction of the regions in (E, k)-space which contribute to the spin and orbital response alike. This is in contrast to Hf, where much flatter bands reside above and below the Fermi energy within the energy window of the laser pulse, providing significant integrated, albeit very small locally, contributions. Moreover, in fcc Pt the Fermi energy cuts through the band edges of the d-states, where the effect of spin-orbit interaction is the strongest, which explains the emergence of strong hotspot-like contributions with a clear correlation in the magnitude of spin and orbital response and the resultant similar behavior of δL<sup>z</sup> and δS<sup>z</sup> with band filling around the Fermi energy. We further scrutinize the reciprocal space distribution of δL<sup>z</sup> and δSz, shown for hcp Hf and fcc Pt in Fig. [3\(](#page-3-0)e-h). For both materials δL<sup>z</sup> distributions consist of large uniform areas of either positive or negative sign. On the contrary, δS<sup>z</sup> distributions are much finer and richer in details with more areas of opposite sign, consistent with the picture we drew above from the band-resolved analysis. A similar behavior of the light-induced magnetism in reciprocal space has been recently reported for rutile altermagnets [\[39\]](#page-7-18). We also observe that, as discussed above, while for Hf the contributions are well spread throughout the Brillouin zone, for Pt the spin and orbital contributions are located at the edges of the considered k<sup>z</sup> − k<sup>y</sup> plane. Overall, this fact indicates that the microscopic behavior of light-induced magnetism varies strongly among transition metals and crucially depends on the crystal structure and position of the Fermi level with respect to the states ![<span id="page-5-0"></span>FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ### Anisotropy of light-induced magnetism We first address the anisotropy of light-induced magnetism with respect to the magnetization by examining the response of magnetic elements under different flavors of circular polarization. In Fig. [4\(](#page-4-0)a-b) we present the computed δL<sup>z</sup> and δS<sup>z</sup> in relation to scattering lifetime Γ, for the cases of magnetic bcc Fe and hcp Co under excitation by light which is circularly polarized in the xy or yz planes at hω̵ = 1.55 eV. As we have already seen in Fig. [1](#page-1-0) for light polarized in the xyplane, the responses behave differently for right (solid lines) or left (dashed lines) polarization. This difference is more pronounced for δLz, for which we even observe a change of sign for Co. Surprisingly, the situation is drastically different for the case of yz-polarization. In this case, for bcc Fe the response is even in the helicity, while for hcp Co δL<sup>z</sup> is almost even and δS<sup>z</sup> is perfectly even in the helicity. Such different behavior with respect to the light helicity between the two magnetic elements can be traced back to the additional anisotropy originating in the crystal structure itself − an effect which we discuss below. Notably, we witness a highly nonlinear behavior with respect to Γ, with the response reaching colossal values in the clean limit. For example, δL<sup>z</sup> and δS<sup>z</sup> can reach as much as 40 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> and 70 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> in Co, respectively, given the scattering lifetime of 1 meV. Besides the potential tunability of the inverse Faraday effect by the degree of the disorder of the samples, one key message that we extract from our observations is that when comparing the values for laser-induced magnetic moments obtained with different methods, special care has to be taken since the implementations of disorder effects, even within a simple constant broadening model, may differ among various approaches. Finally, we analyze the anisotropy that the crystal structure induces in the light-induced magnetism. We select the case of hcp Re and present in Fig. [5\(](#page-5-0)a-b) the x and z components of δL and δS, respectively, in relation to Γ. Due to the inequivalence of x and z axes in the hcp structure, the δL<sup>z</sup> which originates in light circularly polarized in the xy-plane (light blue line) differs from the δL<sup>x</sup> for the case of the yz polarization (brown line). The situation is similar for the case of δS<sup>z</sup> and δSx. On the contrary, as presented in Fig. [11](#page-14-1) of the SM, in the case of fcc Pt we have a perfect match between δL<sup>z</sup> and δL<sup>x</sup> (or δS<sup>z</sup> and δSx) when changing the plane of circular polarization because the x and z axes are equivalent in the fcc structure. The situation is similar for the y component when light is circularly polarized in the xz-plane. As also expected from the symmetry of the crystal structure, we confirm that the light-induced moments are perfectly odd in the helicity for the nonmagnetic elements, which is not the case for the magnetic elements, see Figs. [5](#page-5-0) and [11](#page-14-1) of the SM [\[20\]](#page-7-5). ### Discussion The main goal of our work is to showcase the importance of the orbital component of IFE and its distinct behavior from the spin counterpart. While so far it was mainly the interaction of light with the spin magnetization that was taken into consideration for the interpretation of IFE-related effects, we speculate that the orbital IFE may provide a novel way to coherently induce magnetization and manipulate the magnetic order. For example, in a recent study, different types of optical torques that may arise in ferromagnetic layers were interpreted in terms of the light-induced orbital moment and its interaction with the magnetization through the spin-orbit interaction [\[64\]](#page-8-10). Since it is known that current-induced orbital accumulation and orbital torques exhibit a long-range behavior [\[49,](#page-8-0) [65](#page-8-11)[–67\]](#page-8-12) due to a characteristic small orbital decay, a question arises whether similar behavior can be exhibited by optical torques caused by the orbital IFE. The emergence of long-ranged orbital IFE would come as no surprise given the fact that several recent experiments reported that laser excitation can drive long-range ballistic orbital currents resulting in THz emission [\[68](#page-8-13)[–71\]](#page-8-14). a perturbation by the electric field of the original ground state Hamiltonian, it is common to utilize the angular momentum of light in order to understand the interaction with the magnetic order by the means of transfer of angular momentum. The spin of light through the helicity of circularly polarized pulses plays a crucial role in helicity-dependent all-optical switching scenarios. On the other hand, there is a strong recent interest in utilizing the orbital angular momentum of light via irradiation of matter with e.g. vortex beams or twisted light, in order to probe the magnetization [\[72\]](#page-8-15), generate photocurrents [\[73\]](#page-8-16), drive ultrafast demagnetization [\[74\]](#page-8-17), and induce IFE [\[40,](#page-7-24) [75\]](#page-8-18). Therefore, it is imperative to treat both spin and orbital degrees of freedom on equal footing when exploring the lightmatter interaction. This will not only trigger further advances in the field of ultrafast magnetism and THz spintronics, but also enable a transition to the novel field of attosecond spintronics [\[76,](#page-8-19) [77\]](#page-8-20). Method. In this work we calculate the first-principles electronic structures of 3d, 4d and 5d transition metals by using the full-potential linearized augmented plane wave FLEUR code [\[78\]](#page-8-21). We describe exchange and correlation effects by using the non-relativistic PBE [\[79\]](#page-8-22) functional, while relativistic effects are described by the second-variation scheme [\[80\]](#page-9-0). The parameters of our first-principles calculations, i.e. lattice constants, muffin-tin radii, plane-wave cutoffs, etc. are taken from Table I of Ref. [\[81\]](#page-9-1). Next, we construct maximally-localized Wannier functions (MLWFs) by employing the Wannier90 code [\[82\]](#page-9-2) and its interface with the FLEUR code [\[83\]](#page-9-3). Similarly to [\[81\]](#page-9-1), we choose s, p and d orbitals for the initial projections and disentagle 18 MLWFs out of 36 Bloch states within a frozen window of 5.0 eV above the Fermi energy for each atom in the unit cell. where O<sup>i</sup> is the i-th component of either the orbital angular momentum operator L<sup>i</sup> or of the spin operator S<sup>i</sup> . Moreover, a<sup>0</sup> = 4πϵ0h̵<sup>2</sup> /(mee 2 ) is the Bohr's radius, I = ϵ0cE<sup>2</sup> 0 /2 is the intensity of the pulse, ϵ<sup>0</sup> is the vacuum permittivity, m<sup>e</sup> is the electron mass, e is the elementary charge, h̵ is the reduced Planck constant, c is the light velocity, E<sup>H</sup> = e 2 /(4πϵ0a0) is the Hartree energy, and ϵ<sup>j</sup> is the j-th component of the polarization vector of the pulse. For example, we describe right/left-handedly polarized light in the xy-plane as ϵ = (1,±i, <sup>0</sup>)/<sup>√</sup> 2, respectively, and define its propagation vector to lie along the normal to the polarization plane. A detailed form of the tensor φijk can be seen in Eq.(14) of Ref. [\[21\]](#page-7-9). For the orbital response, the prefactor in Eq. [\(1\)](#page-6-2) must be multiplied by an additional factor of 2. A 128×128×128 interpolation k-mesh is sufficient to obtain well-converged results. In all calculations the lifetime broadening Γ was set at 25 meV, the light frequency hω̵ at 0.25 eV and 1.55 eV, the intensity of light at 10 GW/cm<sup>2</sup> , and we covered an energy region of [−2.5, 2.5] eV around the Fermi energy level E<sup>F</sup> . Acknowledgements. We thank Frank Freimuth and Maximilian Merte for discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR 173/3 − 268565370 (project A11) and by the K. and A. Wallenberg Foundation (Grants No. 2022.0079 and 2023.0336). We acknowledge support from the EIC Pathfinder OPEN grant 101129641 "OBELIX". We also gratefully acknowledge the Julich Supercomputing Cen- ¨ tre and RWTH Aachen University for providing computational resources under projects jiff40 and jara0062. Author Contributions. T. A. performed numerical calculations and analysed the results. T. A. and Y. M. wrote the manuscript. 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(a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-11-0"></span>FIG. 7. (a-d) Light-induced orbital δL<sup>x</sup> (a, c) and spin δS<sup>x</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-12-0"></span>FIG. 8. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-13-0"></span>FIG. 9. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ![<span id="page-13-1"></span>FIG. 10. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-14-1"></span>FIG. 11. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic fcc Pt. The induced magnetic moments along the x-axis and along the z-axis are presented. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) <span id="page-14-0"></span>TABLE II. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 0.25 eV. <span id="page-15-0"></span>TABLE III. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV.
FIG. 8. (a-i) Cartesian components of the light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.	# Light-induced Orbital and Spin Magnetism in 3d, 4d, and 5d Transition Metals Understanding the coherent interplay of light with the magnetization in metals has been a long-standing problem in ultrafast magnetism. While it is known that when laser light acts on a metal it can induce magnetization via the process known as the inverse Faraday effect (IFE), the most basic ingredients of this phenomenon are still largely unexplored. In particular, given a strong recent interest in orbital non-equilibrium dynamics and its role in mediating THz emission in transition metals, the exploration of distinct features in spin and orbital IFE is pertinent. Here, we present a first complete study of the spin and orbital IFE in 3d, 4d and 5d transition metals of groups IV−XI from first-principles. By examining the dependence on the light polarization and frequency, we show that the laser-induced spin and orbital moments may vary significantly both in magnitude and sign. We underpin the interplay between the crystal field splitting and spin-orbit interaction as the key factor which determines the magnitude and key differences between the spin and orbital response. Additionally, we highlight the anisotropy of the effect with respect to the ferromagnetic magnetization and to the crystal structure. The provided complete map of IFE in transition metals is a key reference point in the field of optical magnetism. ### Introduction The demonstration of ultrafast demagnetization in ferromagnets by the application of femtosecond laser pulses [\[1\]](#page-6-0) gave rise to the field of ultrafast spintronics and set the stage for efficient manipulation of magnetism by light. By now it has been rigorously demonstrated that all-optical helicitydependent magnetization switching can be achieved in wide classes of magnetic materials [\[2](#page-6-1)[–10\]](#page-7-0), thus paving the way to contactless ultrafast magnetic recording and information processing. In interpretation of the switching experiments, the inverse Faraday effect (IFE) − i.e. the phenomenon of magnetization induced by a coherent interaction with light acting on a material − has been considered as one of the major underlying mechanisms for the magnetization reversal. Although theoretically predicted and experimentally observed decades ago [\[11–](#page-7-1)[13\]](#page-7-2), a consensus in the theoretical understanding of IFE is still lacking [\[14](#page-7-3)[–19\]](#page-7-4) – however, several microscopic methods have been recently developed for the calculation of this phenomenon in diverse setups [\[20–](#page-7-5)[27\]](#page-7-6). Apart from its established role in the magnetization switching of ferromagnets and ferrimagnents, the impact of IFE in the THz regime has also been studied in antiferromagnets like CrPt [\[28\]](#page-7-7) and Mn2Au [\[29\]](#page-7-8). Moreover, it has been shown that the component of IFE which is perpendicular to the magnetization direction behaves differently with respect to the light helicity than the parallel component [\[21\]](#page-7-9), giving rise to helicity-dependent optical torques [\[30–](#page-7-10)[32\]](#page-7-11) and THz emission [\[33\]](#page-7-12). The impact of the optical torques has been demonstrated to be significant in antiferromagnetic Mn2Au [\[34\]](#page-7-13) where transverse IFE may be responsible for the THz emission [\[35,](#page-7-14) [36\]](#page-7-15), provide an alternative way to switching the magnetization [\[37\]](#page-7-16) as well as to drive domain wall motion [\[38\]](#page-7-17). Recently it has also been shown that light can induce colossal magnetic moments in altermagnets [\[39\]](#page-7-18). Although IFE is conventionally associated with an excitation by circularly polarized light, it can also be activated by linearly-polarized laser pulses [\[25,](#page-7-19) [29,](#page-7-8) [39–](#page-7-18)[42\]](#page-7-20). In solids, two contributions to the magnetization exist: due to spin and orbital moment of electrons. And while the discussion of IFE is normally restricted to the response of spin, since recently, non-equilibrium dynamics of orbital angular momentum started to attract significant attention [\[43](#page-7-21)[–45\]](#page-7-22). Namely, the emergence of orbital currents in the context of the orbital Hall effect [\[46–](#page-7-23)[49\]](#page-8-0), orbital nature of current-induced torques on the magnetization [\[50](#page-8-1)[–52\]](#page-8-2), and current-induced orbital accumulation sizeable even in light materials [\[53–](#page-8-3)[56\]](#page-8-4) have been addressed theoretically and demonstrated experimentally. Notably, in the context of light-induced magnetism, Berritta and co-workers have predicted that the IFE in selected transition metals can exhibit a sizeable orbital component [\[20\]](#page-7-5), with the generality of this observation reaching even into the realm of altermagnets [\[39\]](#page-7-18). At the same time it is also known that within the context of plasmonic IFE the orbital magnetic moment due to electrons excited by laser pulses in small nanoparticles of noble and simple metals can reach atomic values [\[57–](#page-8-5)[59\]](#page-8-6). Despite the fact that very little is known about the interplay of spin and orbital IFE in real materials, it is believed that IFE is a very promising effect in the context of orbitronics − a field which deals with manipulation of the orbital degree of freedom by external perturbations. Although the importance of IFE in mediating the lightinduced magnetism is beyond doubt, a comprehensive indepth material-specific knowledge acquired from microscopic theoretical analysis of this effect is still missing. While acquiring this knowledge is imperative for the field of ultrafast spintronics, since the initial seminal work by Berritta and coworkers [\[20\]](#page-7-5), very little effort has been dedicated to the explorations of this phenomenon from first principles. Here, we ![<span id="page-1-0"></span>FIG. 1. Map of light-induced magnetism in transition metals. (a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) fill this gap by providing a detailed first-principles study of the light-induced spin and orbital magnetism for the 3d, 4d and 5d transition metals of groups IV−XI. By exploring the dependence on the frequency and the polarization of light, we acquire insights into the origin and differences between spin and orbital flavors of IFE, as well as key features of their behavior. Our work provide a solid foundation for further advances in the field of optical magnetism and interaction of light and matter. # Results # Light-induced magnetism in transition metals We begin our discussion by exploring the IFE in a series of 3d, 4d and 5d transition metals of groups IV−XI. IFE is a non-linear optomagnetic effect in which magnetization δO ∝ E × E ∗ is induced as a second-order response to the electric field E of a laser pulse. We calculate the light-induced spin δS and orbital δL magnetic moments by means of the Keldysh formalism, see the section Methods [\[21,](#page-7-9) [39\]](#page-7-18). We focus on light energies hω̵ of 0.25 eV and 1.55 eV which have been routinely used before in our studies within the same formalism [\[29,](#page-7-8) [39,](#page-7-18) [60\]](#page-8-7). Additionally, we choose a lifetime broadening Γ of 25 meV that corresponds to room temperature in order to account for effects of disorder on the electronic states and to provide realistic estimations of the effect. In all calculations we choose the light to be circularly polarized in the xy or yz planes with a light intensity I of 10 GW/cm<sup>2</sup> , while for magnetic materials ferromagnetic magnetization is considered to be along the z-axis. In Fig. [1](#page-1-0) we present the calculated light-induced δL<sup>z</sup> [Fig. [1\(](#page-1-0)a-c)] and δS<sup>z</sup> [Fig. [1\(](#page-1-0)b-d)] moments of the considered transition metals at the Fermi energy. Light is circularly polarized in the xy plane, with energies hω̵ of 0.25 eV [Fig. [1\(](#page-1-0)ab)] and 1.55 eV [Fig. [1\(](#page-1-0)c-d)]. Regarding the 3d magnetic elements Fe, Co, Ni, we analyze both right and left handed circular polarizations of light. The exact values of δL<sup>z</sup> and δS<sup>z</sup> are listed in Tables [II](#page-14-0) and [III](#page-15-0) of the Supplemental Material (SM). Not shown are computed transverse x and y compo- ![<span id="page-2-1"></span>FIG. 2. Band filling dependence of light-induced magnetic moments in 5d transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 1.55 eV.](path) At first sight we notice a strong dependence of the magnitude and sign of both δL<sup>z</sup> and δS<sup>z</sup> on the light frequency. We point out the case of fcc Rh where an colossal δL<sup>z</sup> = −56 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> is predicted at hω̵ = 0.25 eV and which is drastically reduced at the higher frequency, while the spin component remains suppressed in both cases. For the case of magnetic hcp Co we notice a helicity-dependent change of sign for δL<sup>z</sup> at hω̵ = 0.25 eV which is not present for δSz. In general, δL<sup>z</sup> varies stronger with the light helicity than δSz. Moreover, δL<sup>z</sup> is one to two orders of magnitude larger than δS<sup>z</sup> for the non-magnetic elements, while they are of the same magnitude for the magnetic elemental materials. The last two observations are in agreement with the findings of Ref. [\[20\]](#page-7-5) and indicate how the spontaneous magnetization strongly influences the effect in ferromagnets by the time-reversal symmetry breaking. For comparison, in Table [I](#page-2-0) we list the values of the total light-induced moments, defined as the sum of δL<sup>z</sup> and δSz, for the transitional metals studied in [\[20\]](#page-7-5) with hω̵ = 1.55 eV. Our calculated values are of the same order of magnitude, with the exceptions of Au, and of Co for the case of left-handed polarization, which can be attributed to the difference in the computational methods. From Fig. [1\(](#page-1-0)c-d) we get a clear picture of how the induced moments scale with the strength of spin-orbit coupling (SOC). The effect is overall larger for the heavier 5d elements, both in the orbital δL<sup>z</sup> and spin δS<sup>z</sup> channels. However, at the lower frequency of hω̵ = 0.25 eV it is more difficult to draw such a conclusion since the corresponding photon energy falls into the range of SOC strength, which promotes the role of electronic transitions among spin-orbit split bands occurring within a narrow range around the Fermi energy and limited regions in k-space. In contrast, the use of a much larger frequency involves transitions among manifestly orbitally-distinct states which take place over larger portions <span id="page-2-0"></span>TABLE I. Total light-induced magnetic moments for transition metals previously studied in [\[20\]](#page-7-5) in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV. of the reciprocal space with a more uniform impact of the spin-orbit strength on the magnitude of the transition probabilities. From this discussion we have to exclude the ferromagnetic elements since the spontaneous magnetization induces bands splittings on the scale of exchange strength therefore making the effect much more complex. We study the dependence on the light polarization by performing additional calculations for light circularly polarized in the yz-plane, presenting the results for the z and x components of δL and δS respectively in Figs. [6](#page-10-0) and [7](#page-11-0) of the SM for hω̵ = 0.25 eV and hω̵ = 1.55 eV. Our results for δS<sup>z</sup> and δS<sup>x</sup> for the magnetic transition metals reproduce exactly the values of Ref. [\[21\]](#page-7-9), computed with the same method. We further present the values of δL<sup>z</sup> and δLx, where we find the orbital response along the magnetization, δLz, to be by a factor of two larger than the corresponding spin response, when averaged over all magnetic elements. Generally, for the z-component the effect at the magnetic elements is sim- ![<span id="page-3-0"></span>FIG. 3. Reciprocal space anatomy of light-induced magnetism in nonmagnetic transition metals. (a-d) Band-resolved light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in hcp Hf (a-b) and fcc Pt (c-d) relatively to the Fermi energy level E<sup>F</sup> (dashed gray line). The horizontal dotted red lines at E<sup>F</sup> ± 1.55 eV correspond to the frequency of the light. (e-h) Reciprocal space distribution of light-induced orbital (e, g) and spin (f, h) magnetic moments in hcp Hf (e-f) and fcc Pt (g-h) at the Fermi energy. In all calculations light is circularly polarized in the xy-plane at the frequency of hω̵ = 1.55 eV.](path) ilar in magnitude with the case of polarization in the xy-plane shown in Fig. [1,](#page-1-0) and even in the light helicity, while it vanishes for the non-magnetic elements. On the other hand, for the x-component the effect at the magnetic elements becomes one-two orders of magnitude smaller and is odd in the light helicity, although for the non-magnetic elements the values are comparable to the case shown in Fig. [1.](#page-1-0) It is important to note that for light circularly-polarized in the plane containing ferromagnetic magnetization, the light-induced transverse to magnetization induced moments, even though being smaller than the longitudinal ones, are related to light-induced torques that are experimentally demonstrated to lead to helicity-dependent THz emission [\[21,](#page-7-9) [33\]](#page-7-12). In order to get a better understanding of the impact of the time-reversal symmetry breaking on IFE, we consider the cases of ferromagnetic, non-relativistic (i.e. computed without SOC) ferromagnetic, and antiferromagnetic fcc Ni. In Figs. [8](#page-12-0) and [9](#page-13-0) of the SM we present the band filling dependence of the Cartesian components of the light-induced orbital moments δL, for hω̵ = 0.25 eV and hω̵ = 1.55 eV, respectively. In the non-relativistic case only a δL<sup>i</sup> parallel to the light propagation axis is induced which is odd in the helicity, and no δS<sup>i</sup> is induced, exemplifying that the orbital response is the primary non-relativistic one, whereas the spin response is generated through SOC, as was also shown in [\[20\]](#page-7-5). Similarly, for the antiferromagnetic case, only a component parallel to the light propagation axis is induced which is odd in the helicity and remains unchanged under different polarization flavors. While this is the case also for the induced components δL<sup>x</sup> and δL<sup>y</sup> which are transverse to the magnetization in the ferromagnetic case, the situation drastically changes for the induced component δL<sup>z</sup> parallel to the magnetization, as discussed earlier. We note that a tiny, odd in the helicity, δL<sup>x</sup> or δL<sup>y</sup> is additionally induced when light is rotating in a plane containing ferromagnetic magnetization. On the other hand, the induced δL<sup>x</sup> and δLy, developing normal to the polarization plane, serve as a non-relativistic "background" which is independent of the magnetization, with features due to the crystal structure driving the effect over larger regions in energy. The additional band-splittings induced by SOC result in the IFE exhibiting more features with band filling in the relativistic scenario. Remarkably, while the relativistic antiferromagnetic and non-relativistic ferromagnetic cases in principle have similar to each other behavior in energy, a larger signal arises in the antiferromagnetic case by the virtue of flatter bands (see also the discussion for Hf and Pt below). Lastly, we note that a similar behavior has been observed for the in-plane spin IFE in PT-symmetric Mn2Au, however, in the latter case additional out-of-plane moments arise due to linearly polarized light as a result of broken by the magnetization inversion symmetry [\[29\]](#page-7-8). Next, we focus on the case of 5d transition metals where in Fig. [2\(](#page-2-1)a-b) we explore the relation of δL<sup>z</sup> and δSz, respectively, to the band filling, for light circularly polarized in the xy plane and the frequency of hω̵ = 1.55 eV. When going from group IV (hcp Hf) to XI (fcc Au) we observe a smooth ![<span id="page-4-0"></span>FIG. 4. Anisotropy of light-induced magnetism in ferromagnetic transition metals. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the lifetime broadening for ferromagnetic bcc Fe and hcp Co. Light is circularly polarized in the xy-plane (brown curves for bcc Fe and light green curves for hcp Co) or in the yz-plane (light blue curves for bcc Fe and light purple curves for hcp Co). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The ferromagnetic magnetization is along the z-axis. The light frequency is set at hω̵ = 1.55 eV.](path) variation of δL<sup>z</sup> from positive to negative values, as well as nicely shaped plateaus, where δL<sup>z</sup> remains relatively robust in a wide energy region, for hcp Hf, bcc Ta, bcc W, fcc Ir, and fcc Pt. We note that such plateaus are often characteristic of orbital effects, as witnessed for example in orbital Hall insulators [\[61](#page-8-8)[–63\]](#page-8-9) and orbital Rashba systems [\[60\]](#page-8-7). On the contrary, δS<sup>z</sup> exhibits a very erratic behavior with strong variations for each material, at the same time being much smaller in magnitude than δLz. The above observation is a clear manifestation of how differently the orbital and spin degrees of freedom behave under light excitation, with the orbital angular momentum having its origin in intrinsic structural parameters as manifested in the crystal field splitting, whereas the spin angular momentum is more sensitive to finer details of the electronic structure mediated by SOC. Indeed, the light-induced spin and orbital moments exhibit similarly erratic behavior with band filling once the frequency of the light is drastically reduced to reach the range of spin-orbit interaction, see Fig. [10](#page-13-1) of the SM for hω̵ = 0.25 eV. # Anatomy of IFE in k-space Among the 5d transition metals, hcp Hf and fcc Pt exhibit the largest computed moments at the Fermi energy for hω̵ = 1.55 eV, with the corresponding values of δL<sup>z</sup> = 9.1 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −0.8 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Hf, and δL<sup>z</sup> = −7.4 ⋅ 10<sup>−</sup><sup>3</sup>µB, δS<sup>z</sup> = −1.1 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> for Pt. Therefore, we select these two materials and explore the behavior of their light-induced moments in reciprocal space. We present the band-resolved δL<sup>z</sup> and δS<sup>z</sup> for hcp Hf in Fig. [3\(](#page-3-0)a-b), as well as band-resolved δL<sup>z</sup> and δS<sup>z</sup> for fcc Pt in Fig. [3\(](#page-3-0)c-d). For the case of Hf, transitions along the bands near the A-point are the main source of δL<sup>z</sup> and δSz. Light-induced δL<sup>z</sup> consists of hotspot-like negative contributions and secondary, but extended over energy and k-space consistently positive contributions, which are hardly visible. Overall, however, the latter lead to a large positive orbital integrated response. On the other hand, δS<sup>z</sup> For the case of Pt, δL<sup>z</sup> and δS<sup>z</sup> arise from transitions close to X and L high-symmetry points, with both originating in roughly the same regions of (E, k)-space, but often having an opposite sign to each other. Note that the bands in Pt are much more dispersive in the considered energy window, which results in an effective reduction of the regions in (E, k)-space which contribute to the spin and orbital response alike. This is in contrast to Hf, where much flatter bands reside above and below the Fermi energy within the energy window of the laser pulse, providing significant integrated, albeit very small locally, contributions. Moreover, in fcc Pt the Fermi energy cuts through the band edges of the d-states, where the effect of spin-orbit interaction is the strongest, which explains the emergence of strong hotspot-like contributions with a clear correlation in the magnitude of spin and orbital response and the resultant similar behavior of δL<sup>z</sup> and δS<sup>z</sup> with band filling around the Fermi energy. We further scrutinize the reciprocal space distribution of δL<sup>z</sup> and δSz, shown for hcp Hf and fcc Pt in Fig. [3\(](#page-3-0)e-h). For both materials δL<sup>z</sup> distributions consist of large uniform areas of either positive or negative sign. On the contrary, δS<sup>z</sup> distributions are much finer and richer in details with more areas of opposite sign, consistent with the picture we drew above from the band-resolved analysis. A similar behavior of the light-induced magnetism in reciprocal space has been recently reported for rutile altermagnets [\[39\]](#page-7-18). We also observe that, as discussed above, while for Hf the contributions are well spread throughout the Brillouin zone, for Pt the spin and orbital contributions are located at the edges of the considered k<sup>z</sup> − k<sup>y</sup> plane. Overall, this fact indicates that the microscopic behavior of light-induced magnetism varies strongly among transition metals and crucially depends on the crystal structure and position of the Fermi level with respect to the states ![<span id="page-5-0"></span>FIG. 5. Crystalline anisotropy of light-induced magnetism in nonmagnetic transition metals. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic hcp Re. The induced magnetic moments along the x-axis and along the z-axis are shown. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ### Anisotropy of light-induced magnetism We first address the anisotropy of light-induced magnetism with respect to the magnetization by examining the response of magnetic elements under different flavors of circular polarization. In Fig. [4\(](#page-4-0)a-b) we present the computed δL<sup>z</sup> and δS<sup>z</sup> in relation to scattering lifetime Γ, for the cases of magnetic bcc Fe and hcp Co under excitation by light which is circularly polarized in the xy or yz planes at hω̵ = 1.55 eV. As we have already seen in Fig. [1](#page-1-0) for light polarized in the xyplane, the responses behave differently for right (solid lines) or left (dashed lines) polarization. This difference is more pronounced for δLz, for which we even observe a change of sign for Co. Surprisingly, the situation is drastically different for the case of yz-polarization. In this case, for bcc Fe the response is even in the helicity, while for hcp Co δL<sup>z</sup> is almost even and δS<sup>z</sup> is perfectly even in the helicity. Such different behavior with respect to the light helicity between the two magnetic elements can be traced back to the additional anisotropy originating in the crystal structure itself − an effect which we discuss below. Notably, we witness a highly nonlinear behavior with respect to Γ, with the response reaching colossal values in the clean limit. For example, δL<sup>z</sup> and δS<sup>z</sup> can reach as much as 40 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> and 70 ⋅ 10<sup>−</sup><sup>3</sup>µ<sup>B</sup> in Co, respectively, given the scattering lifetime of 1 meV. Besides the potential tunability of the inverse Faraday effect by the degree of the disorder of the samples, one key message that we extract from our observations is that when comparing the values for laser-induced magnetic moments obtained with different methods, special care has to be taken since the implementations of disorder effects, even within a simple constant broadening model, may differ among various approaches. Finally, we analyze the anisotropy that the crystal structure induces in the light-induced magnetism. We select the case of hcp Re and present in Fig. [5\(](#page-5-0)a-b) the x and z components of δL and δS, respectively, in relation to Γ. Due to the inequivalence of x and z axes in the hcp structure, the δL<sup>z</sup> which originates in light circularly polarized in the xy-plane (light blue line) differs from the δL<sup>x</sup> for the case of the yz polarization (brown line). The situation is similar for the case of δS<sup>z</sup> and δSx. On the contrary, as presented in Fig. [11](#page-14-1) of the SM, in the case of fcc Pt we have a perfect match between δL<sup>z</sup> and δL<sup>x</sup> (or δS<sup>z</sup> and δSx) when changing the plane of circular polarization because the x and z axes are equivalent in the fcc structure. The situation is similar for the y component when light is circularly polarized in the xz-plane. As also expected from the symmetry of the crystal structure, we confirm that the light-induced moments are perfectly odd in the helicity for the nonmagnetic elements, which is not the case for the magnetic elements, see Figs. [5](#page-5-0) and [11](#page-14-1) of the SM [\[20\]](#page-7-5). ### Discussion The main goal of our work is to showcase the importance of the orbital component of IFE and its distinct behavior from the spin counterpart. While so far it was mainly the interaction of light with the spin magnetization that was taken into consideration for the interpretation of IFE-related effects, we speculate that the orbital IFE may provide a novel way to coherently induce magnetization and manipulate the magnetic order. For example, in a recent study, different types of optical torques that may arise in ferromagnetic layers were interpreted in terms of the light-induced orbital moment and its interaction with the magnetization through the spin-orbit interaction [\[64\]](#page-8-10). Since it is known that current-induced orbital accumulation and orbital torques exhibit a long-range behavior [\[49,](#page-8-0) [65](#page-8-11)[–67\]](#page-8-12) due to a characteristic small orbital decay, a question arises whether similar behavior can be exhibited by optical torques caused by the orbital IFE. The emergence of long-ranged orbital IFE would come as no surprise given the fact that several recent experiments reported that laser excitation can drive long-range ballistic orbital currents resulting in THz emission [\[68](#page-8-13)[–71\]](#page-8-14). a perturbation by the electric field of the original ground state Hamiltonian, it is common to utilize the angular momentum of light in order to understand the interaction with the magnetic order by the means of transfer of angular momentum. The spin of light through the helicity of circularly polarized pulses plays a crucial role in helicity-dependent all-optical switching scenarios. On the other hand, there is a strong recent interest in utilizing the orbital angular momentum of light via irradiation of matter with e.g. vortex beams or twisted light, in order to probe the magnetization [\[72\]](#page-8-15), generate photocurrents [\[73\]](#page-8-16), drive ultrafast demagnetization [\[74\]](#page-8-17), and induce IFE [\[40,](#page-7-24) [75\]](#page-8-18). Therefore, it is imperative to treat both spin and orbital degrees of freedom on equal footing when exploring the lightmatter interaction. This will not only trigger further advances in the field of ultrafast magnetism and THz spintronics, but also enable a transition to the novel field of attosecond spintronics [\[76,](#page-8-19) [77\]](#page-8-20). Method. In this work we calculate the first-principles electronic structures of 3d, 4d and 5d transition metals by using the full-potential linearized augmented plane wave FLEUR code [\[78\]](#page-8-21). We describe exchange and correlation effects by using the non-relativistic PBE [\[79\]](#page-8-22) functional, while relativistic effects are described by the second-variation scheme [\[80\]](#page-9-0). The parameters of our first-principles calculations, i.e. lattice constants, muffin-tin radii, plane-wave cutoffs, etc. are taken from Table I of Ref. [\[81\]](#page-9-1). Next, we construct maximally-localized Wannier functions (MLWFs) by employing the Wannier90 code [\[82\]](#page-9-2) and its interface with the FLEUR code [\[83\]](#page-9-3). Similarly to [\[81\]](#page-9-1), we choose s, p and d orbitals for the initial projections and disentagle 18 MLWFs out of 36 Bloch states within a frozen window of 5.0 eV above the Fermi energy for each atom in the unit cell. where O<sup>i</sup> is the i-th component of either the orbital angular momentum operator L<sup>i</sup> or of the spin operator S<sup>i</sup> . Moreover, a<sup>0</sup> = 4πϵ0h̵<sup>2</sup> /(mee 2 ) is the Bohr's radius, I = ϵ0cE<sup>2</sup> 0 /2 is the intensity of the pulse, ϵ<sup>0</sup> is the vacuum permittivity, m<sup>e</sup> is the electron mass, e is the elementary charge, h̵ is the reduced Planck constant, c is the light velocity, E<sup>H</sup> = e 2 /(4πϵ0a0) is the Hartree energy, and ϵ<sup>j</sup> is the j-th component of the polarization vector of the pulse. For example, we describe right/left-handedly polarized light in the xy-plane as ϵ = (1,±i, <sup>0</sup>)/<sup>√</sup> 2, respectively, and define its propagation vector to lie along the normal to the polarization plane. A detailed form of the tensor φijk can be seen in Eq.(14) of Ref. [\[21\]](#page-7-9). For the orbital response, the prefactor in Eq. [\(1\)](#page-6-2) must be multiplied by an additional factor of 2. A 128×128×128 interpolation k-mesh is sufficient to obtain well-converged results. In all calculations the lifetime broadening Γ was set at 25 meV, the light frequency hω̵ at 0.25 eV and 1.55 eV, the intensity of light at 10 GW/cm<sup>2</sup> , and we covered an energy region of [−2.5, 2.5] eV around the Fermi energy level E<sup>F</sup> . Acknowledgements. We thank Frank Freimuth and Maximilian Merte for discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) − TRR 173/3 − 268565370 (project A11) and by the K. and A. Wallenberg Foundation (Grants No. 2022.0079 and 2023.0336). We acknowledge support from the EIC Pathfinder OPEN grant 101129641 "OBELIX". We also gratefully acknowledge the Julich Supercomputing Cen- ¨ tre and RWTH Aachen University for providing computational resources under projects jiff40 and jara0062. Author Contributions. T. A. performed numerical calculations and analysed the results. T. A. and Y. M. wrote the manuscript. 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(a-d) Light-induced orbital δL<sup>z</sup> (a, c) and spin δS<sup>z</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-11-0"></span>FIG. 7. (a-d) Light-induced orbital δL<sup>x</sup> (a, c) and spin δS<sup>x</sup> (b, d) magnetic moments in 3d (red circles), 4d (green squares) and 5d (blue triangles) transition metals of groups IV−XI. In all calculations light is considered to be circularly polarized in the yz-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for left-handedly polarized light are shown with light red crosses. The light frequency is hω̵ = 0.25 eV in (a-b) and hω̵ = 1.55 eV in (c-d).](path) ![<span id="page-12-0"></span>FIG. 8. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-13-0"></span>FIG. 9. (a-i) Cartesian components of the light-induced orbital δL magnetic moments in relation to the band filling for the cases of ferromagnetic (a-c), non-relativistic (d-f), and antiferromagnetic (g-i) fcc Ni. Light is circularly polarized in the xy (pink line), yz (orange line), and zx (light blue line) planes, with both right-handed (solid lines) and left-handed (dashed lines) polarizations being displayed. The light frequency is set at hω̵ = 1.55 eV.](path) ![<span id="page-13-1"></span>FIG. 10. (a-b) Light-induced orbital δL<sup>z</sup> (a) and spin δS<sup>z</sup> (b) magnetic moments in relation to the band filling for the 5d transition metals of groups IV−XI. Light is circularly polarized in the xy-plane and the light frequency is set at hω̵ = 0.25 eV.](path) ![<span id="page-14-1"></span>FIG. 11. (a-b) Light-induced orbital δL (a) and spin δS (c) magnetic moments in relation to the lifetime broadening for nonmagnetic fcc Pt. The induced magnetic moments along the x-axis and along the z-axis are presented. Light is considered to be circularly polarized in the xy-plane (brown curves) or in the yz-plane (light blue curves). Both right-handed (solid lines) and left-handed (dashed lines) polarizations are displayed. The light frequency is set at hω̵ = 1.55 eV.](path) <span id="page-14-0"></span>TABLE II. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 0.25 eV. <span id="page-15-0"></span>TABLE III. Light-induced orbital δL<sup>z</sup> and spin δS<sup>z</sup> magnetic moments for the transition metals of groups IV−XI at the Fermi level, in units of 10<sup>−</sup><sup>3</sup> µ<sup>B</sup> per unit cell. Light is circularly polarized in the xy-plane. For the 3d magnetic elements (Fe, Co, Ni) the moments that arise for both right/left-handedly polarized light are listed. The light frequency is hω̵ = 1.55 eV.
Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (C<sup>p</sup>); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, C<sup>p</sup>=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52–54]) and two (C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53,54]) components, respectively.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd³ phase.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup>, AlII, AlIII, and AlIV.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd<sup>3</sup>. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between NdI and its surrounding Al atoms, along the bonds between various Al atoms around NdI , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e3 and f1-f3 are the same, respectively. For a better comparison, the lengths of the bonds are normalized.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 7. Local chemical environment (LCE) in α-Al11Nd³. (a1 a2) LCEs of Ndᴵ and Ndᴵᴵ. (b1-b4) LCEs of Alᴵ to Alᴵᴵ. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Ndⁱ. (a1−a3) The bonds between Ndⁱ and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Ndⁱ. In both a and b, from left to right, the bond length increases gradually.	# Deformability, inherent mechanical properties and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd target material Microstructure uniformity of the Al-Nd target materials with Al11Nd<sup>3</sup> significantly affects the performance of the fabricated film, which is widely used as wiring material in large-size thin-film transistor liquid crystal display (TFT-LCD) panels. Understanding the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is crucial for homogenizing the Al-Nd target. Here, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> are investigated comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix and the eutectic α-Al and a high stable α-Al11Nd<sup>3</sup> phases. During the plastic deformation, the eutectic microstructure transforms from a cellular to a lamellar shape, while the morphology and dimension of α-Al11Nd<sup>3</sup> are not changed significantly. By examining ideal tensile strength, elastic moduli, hardness and brittleness-ductility, the hardness-brittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for its difficulties of plastic deformation and fragmentation. Combining band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two types of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with electron transfer from Nd to Al, while the latter, dominated by both 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work is expected to lay a foundation for Al-Nd alloy and catalyze the fabrication of high-quality Al-Nd target materials. ### I. INTRODUCTION Aluminum is widely used as wiring material in large-size, high-response and high-precision thin-film transistor liquid crystal display (TFT-LCD) panels due to its advantages of low electron resistivity and low cost [1–5]. Instead of pure Al, the Al-3wt%Nd alloy has come into practical use since adding a small amount of Nd can effectively suppress the formation of harmful hemispheric hillocks during the heating process [2, 6, 7]. In industry, the Al-Nd wiring materials in TFT-LCD are fabricated with the magnetron sputtering technique by using the Al-Nd alloy as the target material. It is reported that the microstructure uniformity of the Al-Nd target significantly affects the overall performance of the fabricated Al-Nd film [4, 8]. In the Al-Nd target material, the Nd element mainly exists in the form of eutectic Al11Nd<sup>3</sup> phase [9]. The challenge of homogenizing the microstructure of Al-Nd target material lies just in this eutectic phase. Therefore, uncovering the phase stability, deformability, inherent mechanical properties and chemical bonds of Al11Nd<sup>3</sup> is inevitably necessary to design and fabricate high-quality Al-Nd target materials and further advanced large-size TFT-LCD panels. Unlike the extensive studies on other Al-Nd compounds, such as Al3Nd, Al2Nd and AlNd [10–13], the knowledge of Al11Nd<sup>3</sup> is extremely limited in the open literature. From the Al-Nd phase diagram [14], it is known that Al11Nd<sup>3</sup> has two kinds of crystal structures, i.e., the high-temperature β phase and the low-temperature α phase, with a transitional temperature around 950 C°. S. Lv and coworkers [15] studied the thermodynamic stabilities of a series of Al11RE<sup>3</sup> where RE represents the rare earth element, demonstrating that the Al11Nd<sup>3</sup> phase exhibits high thermodynamic stability against the decomposition into Al2Nd and Al at temperatures below 1000 K. H. Yamamoto and coworkers [16] determined the standard formation entropy of α-Al11Nd<sup>3</sup> by heat capacity measurement from near absolute zero Kelvin. Very recently, T. Fan and coworkers [17] studied the elastic properties of several Al11RE<sup>3</sup> (RE= La, Ce, Pr, Nd and Sm) compounds, claiming that Al11RE<sup>3</sup> can be utilized as the strengthening phase to effectively improve the mechanical properties of high-performance heat-resistant Al alloys. Nevertheless, until now, there is still no comprehensive knowledge about the stability, deformability, and inherent mechanical properties of the Al11Nd<sup>3</sup> phase in the Al-3wt%Nd alloy. Furthermore, the electron structure and chemical bonding information of the Al11Nd<sup>3</sup> phase, which is essential for understanding the inherent mechanical properties of materials, has not been elucidated. To bridge these knowledge gaps, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. In Section III A, the crystal structure, chemical composition, and morphological and crystallographic orientation features of the microstruc- ture of as-cast Al-3wt%Nd alloy are investigated. The thermodynamics and elastic stabilities of Al11Nd<sup>3</sup> are then evaluated. In Section III B, the deformability, intrinsic mechanical strength and brittleness-ductility of Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy are examined. The deformability is evaluated by tracking the microstructure evolution during the asymmetric cold rolling. The intrinsic mechanical strength is examined by ideal tensile strength (σmax), bulk modulus (B), Young's modulus (E), shear modulus (G) and hardness. The inherent brittleness-ductility is probed by different brittleness-ductility criteria, including Pugh's ratio (G/B), Cauchy pressure (Cp), and Poisson's ratio ν. In Section III C, the electronic structures and chemical bonds of Al11Nd<sup>3</sup> are investigated systematically. First, the local chemical environment, which decides electron structure and chemical bonding, is analyzed. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge population analysis, quantum theory of atoms in molecular (QTAIM) and electron localization function (ELF) topological analysis, crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are studied. ## II. METHODOLOGY ### A. Experimental details An ingot of the Al-3wt%Nd alloy was fabricated using vacuum induction melting technology. The samples with an initial thickness of 20 mm were used for asymmetrical rolling at a speed ratio of 1.3. The rolling was carried out at room temperature and was cooled in ice water between passes. After rolling, the sheets with thickness reductions of 25% and 50% were obtained. For crystal structure and microstructure characterization, the rectangular samples with dimensions of 5×5×1 mm<sup>3</sup> were cut out. To remove surface scratches, the samples were mechanically ground with sandpapers and polished with MgO suspension. The electrolytic polishing technique was then utilized to remove the strained layer near the surface. The solution of electrolytic polishing is a mixture of 20% HClO<sup>4</sup> and 80% C2H5OH. The voltage and time are 20 V and 20 s, respectively. The crystal structure was examined using X-ray diffraction (XRD, Rigaku Smartlab) with Cu-Kα radiation. To examine the microstructure with the optical microscope, the specimen was chemically etched to highlight the grain boundary network. Keller's reagent with a solution ratio of 2.5% HNO3, 1.5% HCl, 1.0% HF, 95% H2O in volume was used. The morphology and the crystallographic orientation were measured by scanning electron microscope (SEM, JSM 7001F) with an electron backscatter diffraction (EBSD) acquisition camera. ### B. Calculation methods First-principles calculations were performed using density functional theory (DFT) implemented in the Vienna ab-initio Simulation Package (VASP 6.3) [21, 22]. The Perdew-Burke-Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA) was employed to describe the exchangecorrelation function. The electron-ion interactions were described by the projector augmented wave (PAW) pseudopotential approach. The valence electron configurations of 3s <sup>2</sup>3p 1 for Al and 5s <sup>2</sup>5p <sup>6</sup>4 f <sup>4</sup>6s 2 for Nd were adopted. A kinetic energy cutoff of 500 eV was adopted for wave-function expansion. The k-point meshes with an interval of 2π×0.02 Å−<sup>1</sup> for the Brillouin zone to ensure that the total energy converges within 10−<sup>6</sup> eV/atom. During the structural relaxation, the Hellman-Feynman force on each atom was relaxed to be less than 10−<sup>3</sup> eV/Å. The independent elastic constants were determined by computing the second-order derivatives of the total energy with respect to the position of the ions using a finite difference approach [23]. The ideal tensile stress-strain curves along the a, b, and c axes of orthorhombic lattice of Al11Nd<sup>3</sup> were calculated with the method detailed in Ref. [24]. For the electronic structure calculation, the tetrahedron method with the Blochl ¨ correction [25] was used to integrate the Brillouin zone and a denser k-point mesh with an interval of 2π×0.01 Å−<sup>1</sup> was adopted. Crystalline orbital Hamiltonian population (COHP) and crystal orbital bond index (COBI) were calculated by using the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) program [26, 27]. The 3s and 3p for Al and 5s, 5p, 4 f and 6s for Nd were taken as basis sets. The atom charge analyses were performed using the Bader, Mulliken and Lowdin decomposition techniques [ ¨ 26–28]. The local chemical environment was analyzed with robocrystallographer toolkit [29]. The topological analyses of charge density were performed using the CRITIC2 software package [30]. The formation energy (Ef) of α-Al11Nd<sup>3</sup> was calculated by where Etotal is the total energy, E Al solid and E Nd solid are the averaging energies per Al and Nd atom in their ground-state structures, respectively. Here, the face-centered cubic (fcc) structure with a space group of Fm3m for Al and the α-La-type hexagonal structure with a space group of P63/mmc for pure Nd were adopted. NAl and NNd are the atom numbers of Al and Nd at the adopted structural models, respectively. ### III. RESULTS ### A. Microstructure and phase stability ### 1. Microstructure and crystal structure Figs. 1 shows the microstructure and crystal structure of the casted Al-3wt%Nd alloy. The sample is composed of the pre-eutectic Al matrix with a grain size around 100 µm and the cellular eutectic microstructure with a width of 6∼35 µm (Figs. 1a and b). The volume fraction of the Al matrix and the eutectic microstructure are measured to be around 92% and ![Figure 1. Microstructure of the as-casted Al-3wt%Nd alloy. (a) Optical image. (b) Backscattered electron (BSE) image. (c1-c2) Distributions of Al and Nd for the region indicated in the dashed box in b. (d) Rietveld refinement of XRD pattern of the bulk sample. (e1-e2) Crystal structures of α-Al and α-Al11Nd3. (f1-f2) Measured and simulated Kikuchi patterns for α-Al and α-Al11Nd3. (g1-g4) Band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation map colored using the index of inverse pole figure along the horizontal direction of the sample (X0). The boundaries between the α-Al matrix and the eutectic phases are highlighted in bold black lines. The probing region in g1-g<sup>4</sup> is exactly the same as that of c1-c2. Note that some subgrains (∼1 µm) are detected in the Al matrix, which might be associated with the prominent thermal stress generated during the rapid cooling process.](path) 8%, respectively, which is in good agreement with the theoretical value (90.6 wt.% for the pre-eutectic Al matrix) deduced from the Al-Nd equilibrium phase diagram [14]. Composition examination shows that the eutectic microstructure is composed of an enriched Nd phase with a composition of Al11Nd<sup>3</sup> (wt%, Fig. 1c2) and a pure Al phase (Fig. 1c1). The fraction of Al11Nd<sup>3</sup> is around 50% in the eutectic microstructure, in line with the deduced theoretical value (46.4% in weight ratio) [14]. The Al11Nd<sup>3</sup> phase is in a rod shape with a radius of around 0.5∼4 µm and a length of around 0.6∼19 µm (Fig. 1b and Supplementary Materials Fig. S1). Note that in both the Al matrix and the Al phase in the eutectic microstructure, almost no Nd element is detected, which is in good agreement with the limited solid solubility of Nd in Al [14]. By the Rietveld full-profile refinement, as shown in Fig. 1d, the Al matrix and the Al phase in the eutectic microstructure are identified to possess the face-centered cubic (fcc) structure with a space group of Fm3m (group number: 225, Fig. 1e1), i.e., the α-Al phase. The lattice parameter is determined to be a=4.0496(92) Å. This value is very close to that of pure Al (4.0494 Å [31]), aligning well with the limited Nd solid solubility in Al. The Al3Nd phase is identified to possess an orthorhombic structure with a space group of Immm (group number: 71, Fig. 1e2), i.e., α-Al11Nd<sup>3</sup> following the nomenclature of Refs. [9, 32]. The lattice parameters are refined to be a= 4.3648(41) Å, b= 10.0284(19) Å and c= 12.9684(24) Å (see detailed structural information of Al3Nd in Supplementary Materials Table S1). With the structure information of α-Al11Nd<sup>3</sup> and α-Al, the morphology and crystallographic features of the microstructure of the as-casted Al-3wt%Nd alloy are characterized by using electron backscatter diffraction (EBSD) technique. Figs. 1f<sup>1</sup> and f<sup>2</sup> show the overlapped measured and simulated electron backscattered diffraction (i.e., Kikuchi) patterns for α-Al and α-Al11Nd3, respectively. A good match between measured and simulated Kikuchi patterns guarantees the accuracy of EBSD measurements. Figs. 1g<sup>1</sup> to g<sup>4</sup> display, respectively, band contrast (BC), phase distribution (PD), stacked BC and PD, and crystallographic orientation distribution plotted by using the index of inverse pole figure (IPF) along the horizontal direction of the sample (X0). From phase distribution, it is confirmed that both the Al matrix and the Al phase possess the fcc structure, and the enriched-Nd Al11Nd<sup>3</sup> phase is of orthorhombic structure (α-Al11Nd3). Misorientation analyses show that no specific orientation relationship between the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. Fig. 2 illustrates the formation mechanism of the microstructure of the Al-3wt%Nd alloy. With the cooling of temperature, when the solidification temperature is reached, the preeutectic α-Al phase begins to crystallize and gradually grows from the liquid phase. When the eutectic point is reached, the remaining liquid simultaneously transforms into the eutectic α-Al11Nd<sup>3</sup> and α-Al phases. ![Figure 2. Illustration of formation mechanism of the microstructure of the Al-3wt%Nd alloy.](path) ### 2. Thermodynamic and elastic stability To clarify the stability of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy, ab-initio calculations in the frame of density functional theory (DFT) are carried out. First, with the determined crystal structure information from the XRD refinement as input, a full structural relaxation is carried out. The optimized lattice constants of α-Al11Nd<sup>3</sup> are a=4.3801(10) Å, b=10.0260(24) Å and c=13.0315(66) Å (see detailed information in Supplementary Materials Section S2). This result aligns well with the experimental results and previous reports [15, 33] (Table I), suggesting the reliability of the adopted theoretical methods and computational parameters. With the optimized structural model, the thermomechanical stability of α-Al11Nd<sup>3</sup> is evaluated. E<sup>f</sup> of α-Al11Nd<sup>3</sup> is determined to be −0.389 eV/atom. The negative value of E<sup>f</sup> suggests a thermodynamic stability of α-Al11Nd<sup>3</sup> against the decomposition into constituent elements. Fig. 3 compares E<sup>f</sup> of α-Al11Nd<sup>3</sup> with the known convexhull phases of the binary Al-Nd system [19, 34–36], including the Ni3Sn-type Al3Nd with D019 structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 structure (Fd3m), the orthorhombic AlNd structure (Pbcm). In the upper and lower panels, the referenced data is extracted from the Materials Project [37] and OQMD [19, 35, 36] database, respectively. Clearly, E<sup>f</sup> of α-Al11Nd<sup>3</sup> lies at the outside of the connection line between α-Al and Al3Nd, as highlighted in the insets of Fig. 3. In the upper and lower panels, the energy distances of α-Al11Nd<sup>3</sup> to the connection line between Al and Al3Nd are 10.3 and 4.6 meV/atom, respectively. Thus, α-Al11Nd<sup>3</sup> would be a convex-hull phase and has a high thermomechanical stability against the decomposition into Al and Al3Nd (and other binary convex-hull phases). This result, aligning well with the previous report [15], accounts for the experimental observation of α-Al11Nd<sup>3</sup> in the Al-3wt%Nd alloy. ![Figure 3. Comparison of formation energy (Ef) of α-Al11Nd<sup>3</sup> with the other known convex-hull phases of binary Al-Nd system. They include the fcc α-Al (Fm3m), the Ni3Sn-type Al3Nd with D0<sup>19</sup> structure (P63/mmc), the Cu2Mg-type Al2Nd with C15 laves structure (Fd3m), the orthorhombic AlNd structure (Pbcm), and the α-La-type Nd with hexagonal structure (P63/mmc). Reference data in the upper and lower panels are obtained from the Materials Project [37] and OQMD [19, 35, 36] databases, respectively. The inset in each panel is the zoomed figure around Al11Nd3.](path) Based on the Born-Huang's lattice dynamics theory [38], the elastic stability of α-Al11Nd<sup>3</sup> is evaluated. First, using the strain-stress method [23], the independent lattice constants (Ci j) of α-Al11Nd<sup>3</sup> are determined. Different from cubic crystals possessing three independent Ci j, the orthorhombic crystal has 9 independent Ci j, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C<sup>66</sup> [39]. Table II lists the determined independent Ci j of α-Al11Nd<sup>3</sup> along with the one reported in the literature [17]. We see that the determined Ci j is consistent with the previous report [17]. According to the Born-Huang's lattice dynamics theory [38], the elastic stable orthorhombic system should satisfy the following three conditions simultaneously [39]: For α-Al11Nd3, all Cii (i = 1, 4, 5, and 6) are larger than 0, in which C<sup>55</sup> is the minimum one with a value of 42.3 GPa. Thus, the first condition is satisfied. For the second and third conditions, ∆<sup>1</sup> and ∆<sup>2</sup> are determined to be 0.37×10<sup>4</sup> GPa<sup>2</sup> and 0.95×10<sup>6</sup> GPa<sup>3</sup> , respectively. Clearly, both values are larger than 0 significantly, suggesting the satisfaction of these two conditions simultaneously. The above results evidence a high elastic stability of α-Al11Nd3. ### B. Deformability and intrinsic material properties ### 1. Deformability during cold rolling Combining experimental and ab-initio theoretical study, the deformability of the Al-3wt%Nd alloy and the intrinsic mechanical properties of α-Al11Nd<sup>3</sup> are studied. First, the microstructure evolution of the Al-3wt%Nd alloy during the asymmetrical cold rolling with a speed ratio of 1.3 is investigated. The selection of asymmetrical rolling aims to increase the deformability of metallic materials [40]. Figs. 4a1, b<sup>1</sup> and c<sup>1</sup> show the backscattered electron (BSE) images of the Al-3wt%Nd alloy at the deformations of 0%, 25% and 50%, respectively. Figs. 4a2, b<sup>2</sup> and c<sup>2</sup> are the zoomed images outlined by the dashed boxes in Figs. 4a1, b<sup>1</sup> and c1, respectively. Clearly, with the plastic deformation, the α-Al matrix is obviously elongated along the rolling direction (RD, vertical direction in Fig. 4), aligning well with the excellent inherent ductility of Al and the sufficient dislocation slip systems of fcc metal. Accompanied by the elongation of α-Al matrix, the eutectic microstructure is reshaped significantly. With the increased deformation, the cellular eutectic microstructure gradually turns to a lamellar shape with the lamella normal along the normal direction of the sample (ND, horizontal direction in Fig. 4). At the deformation of 50%, the thickness of the eutectic microstructure is measured to be 1∼9 µm (Fig. 4c2), which is much smaller than that of the undeformed one (6∼35 µm, Fig. 4a2). This result evidences the feasibility of homogenizing the microstructure of the Al-3wt%Nd alloy by severe plastic deformation. Nevertheless, despite a significant redistribution of the eutectic microstructure, the morphology and Figure 4. Microstructure evaluation during asymmetric cold rolling. (a1) 0%. (b1) 25%. (c1) 50%. (a2-c2) are the zoomed figures highlighted in a1-c1, respectively. RD and ND represent the rolling and normal direction, respectively. the dimensions of the eutectic α-Al11Nd<sup>3</sup> phase do not change significantly. After the deformation of 50%, α-Al11Nd<sup>3</sup> is still in a rod shape. The radius and lengths of α-Al11Nd<sup>3</sup> are measured to be 0.4∼4 µm and 0.6∼15 µm, respectively. These values are very close to those of the as-casted ones (0.5∼4 µm for the radius and 0.6∼19 µm for the length). This result implies a difficulty of plastic deformation and fragmentation of α-Al11Nd<sup>3</sup> during cold rolling. ### 2. Intrinsic strength and brittleness-ductility a. Elastic moduli, hardness and ideal tensile strength To understand the deformability of α-Al11Nd3, its intrinsic mechanical properties, including elastic moduli, hardess, ideal tensile strength and brittleness-ductility, are calculated. Figs. 5a, b and c show, respectively, the estimated polycrystalline bulk modulus (B), Young's modulus (E) and shear modulus (G) of α-Al11Nd<sup>3</sup> along with those of pure α-Al as a reference. As a comparison, the elastic moduli of Al3Nd, Al2Nd and AlNd are also plotted [11, 12, 17]. Here, the Voigt-Reuss-Hill (VRH) averaging approach [39] (see calculation details in Appendix) is adopted. We see that B of α-Al11Nd<sup>3</sup> (72.9 GPa) is comparable to that of pure α-Al (76 GPa). On the contrary, E (107 GPa) and G (42.6 GPa) of α-Al11Nd<sup>3</sup> are much higher than those of pure α-Al, i.e., E=70 GPa and G=26 GPa [41]. Furthermore, with the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44], the hardess of α-Al11Nd<sup>3</sup> is estimated, as shown in Fig. 5d. For all models, the hardness of α-Al11Nd<sup>3</sup> is estimated to be around 6.0 GPa (Fig. 5d), which is significantly larger than those of pure α-Al, i.e., 160-350 MPa in Vickers hardness and 160-550 MPa in Brinell hardness. Compared with other Al-Nd binary com- Table II. Independent elastic constants (Ci j) of α-Al11Nd3. The orthorhombic crystal possesses nine independent elastic constants, namely, C11, C12, C13, C22, C23, C33, C44, C<sup>55</sup> and C66. For comparison, Ci j of α-Al11Nd3, cubic Ni3Al-type (Pm3m) and hexagonal Ni3Sn-type (P63/mmc) Al3Nd, cubic Cu2Mg-type Al2Nd (Fd3m) and cubic CsCl-type AlNd (Pm3m) reported in the literature [11, 12, 17] are also listed. pounds, it is found that the intrinsic mechanical strength of α-Al11Nd<sup>3</sup> could be weaker than those of Al3Nd and Al2Nd, but is stronger than that of AlNd, even though their differences are not prominent. The chemical bond-dominated ideal tensile strengths, which represent the maximum stress at elastic instability (yield or break) when applying an increased stress to an infinite defect-free crystal [45, 46], are evaluated. Fig. 5e displays the ideal tensile stress-strain curves of α-Al11Nd<sup>3</sup> along the a-, b- and c-axes of the orthorhombic cell. During the calculation, the lattice is first deformed gradually along the tensile direction. Then, at each strain, the structure is relaxed until all stresses perpendicular to the tensile direction disappear with a threshold of Hellman-Feynman stress less than 0.1 GPa [47– 49]. The ideal tensile strains (εmax) along the a-, b- and c-axes of α-Al11Nd<sup>3</sup> are measured to be 14%, 19%, and 14%, respectively. These values are much smaller than εmax of pure α-Al along [100] (36% [46]). The ideal tensile stress (σmax) along the a-, b- and c-axes are determined to be 8.8 GPa, 8.3 GPa, and 9.2 GPa, respectively, which are also lower than that of pure α-Al along [100] (12.5 GPa [46]). Both smaller εmax and lower σmax of α-Al11Nd<sup>3</sup> against pure α-Al imply a brittle nature of this compound. b. Inherent brittleness-ductility By using Pugh's, Pettifor's, and Poisson's criteria, the intrinsic brittleness-ductility of α-Al11Nd<sup>3</sup> are quantitatively examined. In these three criteria, the brittleness-ductility are represented by B/G, Cauchy pressure Cp, and Poisson's ratio ν (defined by E/2G − 1 [51]), respectively. As comparisons, the brittleness-ductilities of other pure α-Al and other binary Al-Nd compounds are also evaluated. Note that unlike the cubic crystal having one component of C<sup>p</sup> (defined by C12−C44), for the orthorhombic (α-Al11Nd3) and the hexagonal (Al3Nd) crystals, their C<sup>p</sup> possess three (i.e., C a <sup>p</sup>=C<sup>23</sup> − C44, C b <sup>p</sup>=C<sup>13</sup> − C<sup>55</sup> and C c <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [52– 54]) and two (i.e., C a <sup>p</sup>=C<sup>13</sup> − C<sup>44</sup> and C b <sup>p</sup>=C<sup>12</sup> − C<sup>66</sup> [53, 54]) components, respectively. Figs. 6a, b and c show, respectively, the determined B/G, Cp, and ν of α-Al11Nd<sup>3</sup> along with those of pure α-Al (dashdot lines), Al3Nd, Al2Nd and AlNd. For clarity, the critical values of ductility-brittleness in different criteria, namely, B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50], are indicated by dashed ![Figure 5. Intrinsic mechanical strength of α-Al11Nd3. (a) Bulk modulus (B). (b) Young's modulus (E). (c) Shear modulus (G). (d) Hardness. (e) Ideal stress-strain curves along a-, b- and c-axes. The dashed lines represent the pure α-Al. Data of cubic (cub.) Ni3Al-type [12] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [12] and cubic CsCl-type AlNd [12] are also included as comparison. For hardness, the models proposed by D.M. Teter [42], X.-Q. Chen [43] and Y. Tian [44] are adopted.](path) ![Figure 6. Intrinsic ductility-brittleness of α-Al11Nd3. (a) Pugh's ratio (B/G); (b) Cauchy pressure (Cp); (c) Poisson's ratio (v). As a reference, the data of cubic (cub.) Ni3Al-type [32] and hexagonal (hex.) Ni3Sn-type [32] Al3Nd, cubic Cu2Mg-type Al2Nd [32] and cubic CsCl-type AlNd [12] are also plotted. The dash-doted line represents the pure α-Al. The dashed line highlights the ductility-brittleness critical values, i.e., B/G=1.75 in Pugh's theory, Cp=0 in Pettifor's model and ν= 0.26 in Poisson's picture [49, 50]. Note that for C<sup>p</sup> of orthorhombic and hexagonal crystals, there are three (C a p , C b p and C c p ) and two (C a p and C b p ) components, respectively.](path) lines. Clearly, the pure α-Al with B/G of 2.9, C<sup>p</sup> of 33.9 GPa and ν of 0.35 is inherent ductility. For α-Al11Nd3, B/G and ν are calculated to be 1.7 and 0.26, respectively. Clearly, both of them are located at the brittle regions in Pugh's (Fig. 6a) and Poisson's (Fig. 6c) criteria. The Cauchy pressure C a p , C b p and C c p are determined to be −6.7 GPa, 10.3 GPa and −4.1 GPa, respectively (Fig. 6b). The large deviation between various Cauchy pressure parameters implies a strong anisotropy of chemical bonding. Owing to the presence of two negative C<sup>p</sup> parameters, α-Al11Nd<sup>3</sup> should be considered inherently brittle from Pettifor's criteria. Despite the brittle nature, from all Pugh's, Pettifor's, and Poisson's criteria, the ductility of α-Al11Nd<sup>3</sup> should be superior to those of Al3Nd and Al2Nd. On the contrary, it is inferior to the B2-type AlNd, which should be classified as ductile since B/G, C<sup>p</sup> and ν are larger than 1.75, 0 and 0.26, respectively. The ductility nature of B2 type AlNd aligns well with the result of B2-type AlSc [55]. The above quantitative investigation evidences the hard and brittle nature of α-Al11Nd3, explaining well the difficulty in deformation and fragmentation of this phase in Al-3wt%Nd alloy during plastic deformation. ### C. Electronic structures To understand the inherent mechanical properties of α-Al11Nd3, the electronic structures and chemical bonds of this compound are investigated systematically. First, the local chemical environments (LCEs), which decides electron structure and chemical bonding, are studied. Second, the electron band structure (BS) and density of states (DOS) are calculated. Third, the atom charge, quantum theory of atoms in molecular (QTAIM), crystal orbital Hamilton population (COHP) [18, 19] and crystal orbital bond index (COBI) [20] are investigated. ### 1. Local chemical environment From the crystal structure of α-Al11Nd3, it is found that there are two inequivalent Nd atoms, occupying 2a (NdI) and 4i (NdII) sites, respectively, and four inequivalent Al atoms, located at 2d (AlI), 4h (AlII) and two 8l (AlIII and AlIV) sites, respectively (see details in Supplementary Materials Table S1). Notably, in the unit cell of α-Al11Nd3, the numbers of two kinds of Nd atoms and four kinds of Al atoms are not the same, which can be seen from the multiplicity factor in the Wyckoff symbol. Specifically, in the unit cell, the number of NdII (4i) is two times larger than that of Nd<sup>I</sup> (2a), and the numbers of AlIII and AlIV (8l) are two times and four times larger than AlII (4h) and Al<sup>I</sup> (2d), respectively. Figs. 7a and b illustrate the local chemical environments of the two distinct Nd and four independent Al atoms, respectively. The distributions of bond lengths around different atoms are summarized in Fig. 7c. Nd<sup>I</sup> is bonded with 16 Al atoms to form distorted NdIAl<sup>16</sup> cuboctahedra (Fig. 7a1), including four AlII with a length of 3.60 Å, four AlIV with a length of 3.28 Å, and eight AlIII atoms with a length of 3.35 Å (Fig. 7c). NdII is bonded in a 12-coordinate geometry (Fig. 7a2) to form distorted NdIIAl<sup>14</sup> cuboctahedra, including two Al<sup>I</sup> with a length of 3.23 Å, two AlII with a length of 3.21 Å, four AlIII with a length of 3.23 Å, four AlIV with a length of 3.20 Å and two AlIV with a length of 3.63 Å (Fig. 7c). Notably, all Nd atoms are bonded with Al atoms with bond lengths larger than 3.1 Å, and no Nd-Nd bond exists. Like the Nd atoms, four kinds of Al atoms, i.e., Al<sup>I</sup> (Fig. 7b1), AlII (Fig. 7b2), AlIII (Fig. 7b3) and AlIV (Fig. 7b4), possess different local chemical environments. Apart from the Al-Nd bond, some Al-Al bonds around the Al atom are observed. Notably, as shown in Fig. 7c, all the lengths of the Al-Al bonds (<3.1 Å) are shorter than those of the Al-Nd bonds (>3.1Å). Remarkably, the crystal of α-Al11Nd<sup>3</sup> can be constructed by stacking the polyhedrons of Nd<sup>I</sup> (NdIAl16, Fig. 7a1) and NdII (NdIIAl14, Fig. 7a2) alternately. For easy illustration, the polyhedrons of NdIAl<sup>16</sup> and NdIIAl<sup>14</sup> are dubbed A and B, respectively. First, by stacking the polyhedron A and B along ![Figure 7. Local chemical environment (LCE) in α-Al11Nd3. (a1 a2) LCEs of Nd<sup>I</sup> and NdII. (b1-b4) LCEs of Al<sup>I</sup> to AlIV. (c) Distribution of bond length around distinct Nd and Al atoms. All Nd-Al bonds are longer than 3.1 Å, whereas all Al-Al bonds are shorter than 3.1 Å. The number in parentheses in the annotation represents the bond number. For instance, NdI-AlII(4) indicate that there exists four NdI-AlII bonds. (d1-d2) Column A' and B' polyhedron units constructed by stacking A and B polyhedrons along the a-axis. (e) Planar-shaped A'B'B'A' substructure constructed by connecting one A' and two B' columns alternately along the c-axis; (f) Structural model by stacking A'B'B'A' substructures along the b-axis. Note that the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously.](path) the a-axis, the column A' (Fig. 7d1) and B' (Fig. 7d2) polyhedron units can be obtained, respectively. Second, by connecting two A' and two B' column units alternately along the c-axis, the planar-shaped A'B'B'A' substructure along the ac plane is formed (Fig. 7e). Third, the whole crystal can be constructed (Fig. 7f) by stacking A'B'B'A' substructures one by one along the b-axis. Notably, the two adjacent A'B'B'A' planar-shaped substructures are mutually shifted 1/2 lattice length along the a and c axes simultaneously. ### 2. Band structure and density of states Fig. 8a<sup>1</sup> displays the electron band structure (BS) of α-Al11Nd3. In terms of dispersion degree, the BS of α-Al11Nd<sup>3</sup> could be classified into three distinct regions. For the energies being lower and higher Fermi energy (EF), the bands show obvious dispersion along various high-symmetry k-point paths. In contrast, the nearly dispersionless flat bands are observed around EF. From the zoomed BS around EF, it is seen that the flat bands span from −0.2 eV to 0.5 eV (Fig. 8a2). Among them, two bands are located below EF, one is crossed with EF, and the others are above EF. Fig. 8b<sup>1</sup> shows the total density of states (TDOS) and the atom-resolved partial density of states (PDOS) of α-Al11Nd<sup>3</sup> with the low-density region highlighted in Fig. 8b2. Remarkably, the states around E<sup>F</sup> are much higher than those of lower and higher EF, aligning well with the dispersion feature of BS. Comparing TDOS and PDOS of different atoms, it is evident that the high-density states around E<sup>F</sup> mostly come from Nd<sup>I</sup> and NdII, while the low-density states being away from E<sup>F</sup> are majorly associated with the Al<sup>I</sup> to AlIV atoms. Figs. 8c and d display the orbital-resolved DOS for the two independent Nd and the four distinct Al atoms, respectively. Unexpectedly, the atom and orbital-resolved DOS structures of Nd<sup>I</sup> (Fig. 8c1) and NdII (Fig. 8c2) are very similar, even though they possess totally different local chemical environments. Clearly, the high-density states of Nd around E<sup>F</sup> originate from 4f electrons, in good line with the observation in other materials containing rare earth element [56]. Besides, some low-density states (<0.2 states/eV) stemming from 6s electrons of Nd are also observed. For the Al atoms (Figs. 8d1 d4), there exist four obvious state peaks centered at −6 eV, −4 eV, −2 eV and EF, respectively. The peak centered at −6 eV majorly comes from 3s electrons, the one centered at −4 eV is contributed by both 3s and 3p electrons, and the peak centered at −2 eV and E<sup>F</sup> are mainly originated from 3p electrons. Unlike the similar DOSs of Nd<sup>I</sup> and NdII, there exist obvious discrepancies for the four kinds of Al atoms, suggesting a high sensitivity of the electron structure of Al with respect to the local chemical environment. From Figs. 8c and d, there could exist the interaction between 6s electron of Nd and 3s electron of Al at energies around −6 eV and −4 eV and the interaction of 6s/4f electron of Nd and 3p electron of Al around EF. However, owing to the weak density, these interactions between Al and Nd would be not strong. Comparing the PDOSs of four kinds of Al atoms, it is deduced that there could exist 3s-3p hybridization at energies centered −4 eV and 3p-3p interaction at energies centered −2 eV between two neighboring Al atoms. ### 3. Chemical bonding a. Population analysis To reveal the features of chemical bonds in α-Al11Nd3, the atom charges are calculated. Table III lists the transferred charges of different Nd and Al atoms during the formation of chemical bonding calculated following the charge division algorithms proposed by Bader, Mulliken and Lowdin [ ¨ 26–28]. Despite the distinct charge values, the conclusion about charge transfer is consistent. Specifically, during the bonding, both Nd<sup>I</sup> and NdII atoms lose electrons and all Al atoms get electrons, which is in accordance with the fact of more positive electronegativities of Nd (1.14 in the Pauling scale) compared with Al (1.61 in the Pauling scale). ![9](path) ![Figure 8. Electronic band structure (BS) and density of states (DOSs) of α-Al11Nd3. (a1) BS along the high-symmetry points of the first Brillouin zone. (a2) Zoomed BS around the Fermi energy (EF). (b1) Total DOS (TDOS) and atom-resolved partial DOS (PDOS). (b2) Zoomed DOS at the low-density region. (c1-c2) Atom and orbital-resolved DOSs of Nd<sup>I</sup> and NdII. (d1-d4) Atom and orbital-resolved DOSs of Al<sup>I</sup> , AlII, AlIII, and AlIV.](path) This result implies that the ionic bond could form between Nd and Al atoms in α-Al11Nd3. As shown in Table III, despite different local chemical environments, the transferred charges for Nd<sup>I</sup> and NdII are similar, and so are they for all kinds of Al atoms (Table III). Thus, the strength of Coulomb electrostatic interaction between different Nd-Al pairs could be compared. b. QTAIM and ELF topological analysis By using the quantum theory of atoms in molecular (QTAIM) [57], the topological analyses on the charge density of α-Al11Nd<sup>3</sup> are performed. In this work, the charge density (CHG, ρ(r)), charge density difference (CDD, ∆ρ(r)=ρ(r)−ρ(r)unbonding) reflecting electron transfer during the bonding, electron localization function (ELF, η(r)) characterizing electron localization with a range of [0, 1] [58] and Laplacian of charge density (LAP, ∇ <sup>2</sup>ρ(r)) determining local concentration or depletion of electron [59], are calculated. Figs. 9a and b show the 2-dimensional (2D) distributions of ρ(r), ∆ρ(r), η(r) and ∇ <sup>2</sup>ρ(r) on (1 0 0) and (0.5 0 0) of α-Al11Nd3, respectively. The grey line represents the bond path, a single line of maximum electron density linking the nuclei of two chemically bonded atoms [60, 61]. The open cycle indicates the bond critical points (BCP), a single point with minimum electron density along the bond path [57]. The values of ρ, ∆ρ, η and ∇ <sup>2</sup>ρ at the BCPs, which can reflect the bonding features [60, 61], are listed in Table IV. Notably, there is no obvious charge accumulation between Nd and Al (Figs. 9a2, b2, a<sup>3</sup> and b3). At the BCPs of the Nd-Al bonds, the localization of electrons is very weak (Figs. 9a<sup>4</sup> and b4). The Laplacian of charge density exhibits positive values, suggesting a local depletion of electron, i.e., the electron density is less than the average density in the immediate neighborhood [62]. At the BCPs of the Nd-Al bonds, ρ, ∆ρ, η and ∇ <sup>2</sup>ρ are around 0.02 e/bohr<sup>3</sup> , 0.001 e/bohr<sup>3</sup> , 0.2, and 0.02 e/bohr<sup>5</sup> , respectively (Table IV). All these features confirm the ionic-type nature of the Nd-Al bonds in α-Al11Nd3. ![Figure 9. Topological analysis of charge density. (a1-b1) Atom arrangement on (1 0 0) and (0.5 0 0). (a2-b2) Charge density (CHG). (a3-b3) Charge density difference (CDD). (a4-b4) Electron localization function (ELF). (a5-b5) Laplacian of charge density (LAP). The grey lines and the open cycles represent the bond paths (BPs) and the bond critical points (BCPs), respectively. (c1-c3), (d1-d3), (e1-e3) and (f1-f3) are lineprofiles of ELF, CDD and LAP along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, along the bonds between various Al atoms around Nd<sup>I</sup> , along the bonds between NdII and its surrounding Al atoms and along the bonds between various Al atoms around NdII, respectively. The legend in c1-c3, d1-d3, e1-e<sup>3</sup> and f1-f<sup>3</sup> are the same, respectively. For a better comparison, the lengths of the bonds are normalized.](path) around 0.03-0.04 e/bohr<sup>3</sup> , which is two times higher than that of the Nd-Al bond. From the distribution of ELF, the electrons between two neighboring Al atoms are found to possess obvious localization (Fig. 9a<sup>4</sup> and b4). The η values at the BCPs of the Al-Al bonds are around 0.8. Furthermore, unlike the positive value of the Nd-Al bond, ∇ <sup>2</sup>ρ at the BCPs of the Al-Al bonds are negative, implying a local concentration of electrons. All these features suggest a covalent-type nature of the Al-Al bond in α-Al11Nd3. To highlight the differences between various Nd-Al bonds and between various Al-Al bonds, the 1-dimensional (1D) line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) are plotted. Figs. 9c, d, e and f show the line profiles of η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) along the bonds between Nd<sup>I</sup> and its surrounding Al atoms, the bonds between various Al atoms around the Nd<sup>I</sup> atom, the bonds between the NdII and its surrounding Al atoms, the bonds Table IV. Charge density (ρ), charge density difference (∆ρ), electron localization function (η), and Laplacian charge density (∇ <sup>2</sup>ρ) at bond critical points (BCPs) of different Al-Nd and Al-Al bonds. between various Al atoms around the NdII atom, respectively. For an easy comparison, the lengths of all bonds are normalized. From all η(r) (Figs. 9c<sup>1</sup> and e1), ∆ρ(r) (Figs. 9c<sup>2</sup> and e2) and ∇ <sup>2</sup>ρ(r) (Figs. 9c<sup>3</sup> and e3), all ionic-type Nd-Al bonds exhibit similar features, which aligns well with the similar atom charges of the Nd and Al atoms (Table III). This result shows that the ionic-type Nd-Al bond in α-Al11Nd<sup>3</sup> is insensitive to bond length and local chemical environment. Unlike the Nd-Al bond, η(r), ∆ρ(r) and ∇ <sup>2</sup>ρ(r) of different Al-Al bonds exhibit obvious discrepancies, which is in agreement with the distinct DOS structures of the Al atoms (Fig. 8d). It shows that compared with the Al-Nd bond, the Al-Al bond is relatively sensitive to bond length and local environment. c. COHP, ICOHP and ICOBI Figs. 10 and 11 display the crystal orbital Hamilton population (COHP) and the orbitalresolved COHP curves for the chemical bonds around Nd<sup>I</sup> and NdII, respectively. The integration of total and orbitalresolved COHPs up to E<sup>F</sup> (ICOHPs), an indicator of bonding strength [18, 63], are listed in Table V. Remarkably, the absolute values of ICOHP (\|ICOHP\|) of the Nd-Al bonds are generally smaller than 0.5 eV, while \|ICOHP\| of the Al-Al bonds are around 1.5-2.5 eV. Higher bonding strength of the Al-Al bond may link with their shorter bond length (<3.1 Å) with respect to the Nd-Al bond (>3.1 Å). For the Al-Al bond, as shown in Table V, the bonding strength decreases significantly with the increase in bond length. For instance, \|ICOHP\| of AlIII-AlIII with a length of 2.63 Å is 2.68 eV, while \|ICOHP\| drops to 1.51 eV when the length increases to 2.82 Å (AlII-AlIV). This result further evidences the high sensitivity of the Al-Al bonds on local environment. Comparing the total and orbital-resolved \|ICOHP\|, it is evident that the bonding strength of the Al-Al bond is governed by both 3s-3p and 3p-3p hybridization, aligning well with the DOS results (Section III C 2). Therein, the former, i.e., 3s-3p interaction, plays a slightly larger contribution. From Figs. 10b and 11b, it is seen that the bonding states of 3s-3p interaction are located at the low-energy regions with the state peaks centered around −6 and −4 eV. For the Al-Al bonds with shorter distances, such as AlIII-AlIII bonds (Fig. 10b<sup>1</sup> and Fig. 11b1), some anti-bonding states of 3s-3p interaction are observed around EF. Nevertheless, these antibonding states disappear gradually with the increase in bond length. Compared with the 3s-3p interaction, the 3p-3p hybridization occurs at the energies being closer to EF. For all Al-Al bonds, an obvious bonding state peak centered around −2 eV is observed. Notably, with the increase in bond length, the intensity of the bonding state of both 3s-3p and 3p-3p interaction decreases significantly, which accounts for the high sensitivity of the bonding strength of Al-Al bonds. Lastly, the crystal orbital bond index (COBI) for different bonds of α-Al11Nd<sup>3</sup> are calculated. Table V lists the integration of COBI up to E<sup>F</sup> (ICOBI), which is an index of bond order in solid [20]. It is seen that the ICOBIs of all Nd-Al bonds are around 0.1, which is comparable to the Na-Cl bond (0.09 [20]) in NaCl. This result evidences the strong ionic bonding nature of the Nd-Al bond. For the Al-Al bonds, ICOBIs are determined to be around 0.5, which is around half of C-C bond in diamond (0.95 [20]). Thus, despite the pres- ![Figure 10. COHP and orbital-resolved COHP curves of chemical bonds around Nd<sup>I</sup> . (a1−a3) The bonds between Nd<sup>I</sup> and its surrounding Al atoms. (b1−b5) The bonds between different Al atoms around Nd<sup>I</sup> . In both a and b, from left to right, the bond length increases gradually.](path) ![Figure 11. COHP and orbital-resolved COHP curves of chemical bonds around NdII. (a1−a5) The bonds between NdII and its surrounding Al atoms. (b1−b6) The bonds between different Al atoms around NdII. In both a and b, from left to right, the bond length increases gradually.](path) ence of electron localization between Al-Al, the covalencies of the Al-Al bonds in α-Al11Nd<sup>3</sup> are much weaker than those of typical covalent compounds, such as diamond or silicon. This result is consistent with the metallic bonding of Al-Al bonding in pure Al. As is known, it is not easy to distinguish weak covalent interaction and metallic bonding in intermetallics [64]. Therefore, it can be concluded that the Nd-Al is of typical ionic-type chemical bonding, while the Al-Al bonds around Nd possess weak-covalent or metallic bonds. The mixed ionic and weak-covalent bond is responsible for the hardness-brittleness of α-Al11Nd3. ### IV. CONCLUSIONS In summary, by a combined experimental and ab-initio theoretical study, the microstructure and deformability of the Al-3wt%Nd alloy and the inherent mechanical properties, electron structures and chemical bonds of α-Al11Nd<sup>3</sup> are studied comprehensively. The Al-3wt%Nd alloy is composed of the pre-eutectic α-Al matrix with a volume fraction of ∼92% and the eutectic microstructure constituted by the alternately arranged α-Al and α-Al11Nd<sup>3</sup> phases. Under the asymmetric cold rolling, the Al-3wt%Nd alloy possesses excellent plastic deformability. With the plastic deformation, the α-Al matrix is elongated significantly along the rolling direction, and the eutectic microstructure transforms from a cellular to a lamellar shape. Nevertheless, the morphology of α-Al11Nd<sup>3</sup> is not changed obviously. Theoretical calculations show that α-Al11Nd<sup>3</sup> exhibits excellent thermodynamic and elastic stabilities. By examining ideal tensile strength, elastic moduli, hardness and inherent brittleness-ductility, the hardnessbrittleness of α-Al11Nd<sup>3</sup> is quantitatively evaluated, accounting for the difficulty in plastic deformation and fragmentation during cold rolling. By a combined investigation of band structure, population analysis, topological analysis and crystal orbital Hamilton population, it is revealed that α-Al11Nd<sup>3</sup> possesses two kinds of chemical bonds: the Nd-Al and Al-Al bonds. The former is a typical ionic bond with the electron transfer from Nd to Al, while the latter, originated from 3s-3p and 3p-3p interactions, is a weak covalent bond. The mixed chemical bond is responsible for the high hardness-brittleness of α-Al11Nd3. This work shows that the microstructure uniformity of the Al-3wt%Nd alloy can be significantly improved by plastic deformation. Meanwhile, it is confirmed that the plastic deformation or fragmentation of α-Al11Nd<sup>3</sup> is relatively difficult. As is known, the size of the eutectic α-Al11Nd<sup>3</sup> phase can be tailored by tuning the processing parameters during the casting, e.g., supercooling degree. It, thus, should be essential for a multiple-process collaborative control, e.g., casting and plastic deformation, to fabricate the high-quality Al-Nd target. The study is expected to lay a foundation for the Al-3wt%Nd alloy and thus catalyze the fabrication of highquality Al-Nd targets and further the advanced large-size TFT-LCD panels. ### V. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of China (2022YFB3504401). ### Appendix A: Calculations of isotropic elastic moduli Isotropic bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (ν) are calculated by Voigt–Reuss–Hill (V-R-H) approximation. For the orthorhombic system, the Voigt bulk modulus (BV) and shear modulus (GV) can be calculated by [51]: $$\begin{aligned} G\_{\rm V} &= \frac{1}{15} \left[ C\_{11} + C\_{22} + C\_{33} - C\_{12} - C\_{13} - C\_{23} \right] \\ &+ \frac{1}{5} \left[ (C\_{44} + C\_{55} + C\_{66}) \right] \end{aligned} \qquad (\text{A2})$$ $$\begin{aligned} B\_R &= \Delta \left[ C\_{11} (C\_{22} + C\_{33} - 2C\_{23}) + C\_{22} (C\_{33} - 2C\_{13}) \right. \\ &\left. - 2C\_{33} C\_{12} + C\_{12} (2C\_{23} - C\_{12}) + C\_{13} (2C\_{12} - C\_{13}) \right. \\ &\left. + C\_{23} (2C\_{13} - C\_{23}) \right]^{-1} \end{aligned} \tag{A3}$$ $$\begin{aligned} G\_R &= 15\{4[C\_{11}(C\_{22} + C\_{33} + C\_{23}) + C\_{22}(C\_{33} + C\_{13}) + C\_{33}C\_{12} \\ &- C\_{12}(C\_{23} + C\_{12}) - C\_{13}(C\_{12} + C\_{13}) - C\_{23}(C\_{13} + C\_{23})\}/\Delta t \\ &+ 3[(1/C\_{44}) + (1/C\_{55}) + (1/C\_{66})]^{-1} \end{aligned} \tag{A4}$$ # arXiv:2404.18050v1 [cond-mat.mtrl-sci] 28 Apr 2024 # Supplementary Materials to "Deformability, inherent brittleness-ductility and chemical bonding of Al11Nd<sup>3</sup> in Al-Nd sputtering target material" Xue-Qian Wang,<sup>1</sup> Run-Xin Song,<sup>1</sup> Xu Guan,<sup>1</sup> Shuan Li,<sup>2</sup> Shuchen Sun,<sup>3</sup> Hongbo Yang,<sup>2</sup> Daogao Wu,2, <sup>∗</sup> Ganfeng Tu,<sup>3</sup> Song Li,1, † Hai-Le Yan,1, ‡ and Liang Zuo<sup>1</sup> School of Material Science and Engineering, Northeastern University, Shenyang 110819, China. # Supplementary Caption: The Supplementary Materials contain the detailed crystal structure information of α-Al11Nd3, the relaxed structure file in the format of POSCAR by ab-initio calculation and the zoomed eutectic microstructure of the casted Al-3wt% alloy. # Content: ### I. CRYSTAL STRUCTURE INFORMATION OF α-AL11ND<sup>3</sup> ### II. RELAXED STRUCTURAL MODEL OF α-AL11ND<sup>3</sup> Al11Nd3 in POSCAR format 1.000000 4.3801096498628382 0.0000000000000000 0.0000000000000000 0.0000000000000000 10.0260238680764271 0.0000000000000000 0.0000000000000000 0.0000000000000000 13.0315663977200114 Nd Al 6 22 Direct 0.0000000000000000 0.0000000000000000 0.0000000000000000 Nd1 0.5000000000000000 0.5000000000000000 0.8176408510047521 Nd2 0.5000000000000000 0.5000000000000000 0.1823591489952479 Nd3 0.5000000000000000 0.5000000000000000 0.5000000000000000 Nd4 0.0000000000000000 0.0000000000000000 0.3176408510047521 Nd5 0.0000000000000000 0.0000000000000000 0.6823591489952479 Nd6 0.0000000000000000 0.5000000000000000 0.0000000000000000 Al1 0.5000000000000000 0.2852782166261220 0.0000000000000000 Al2 0.5000000000000000 0.7147217833738780 0.0000000000000000 Al3 0.0000000000000000 0.6312487585391187 0.3338148796572754 Al4 0.0000000000000000 0.3687512414608813 0.6661851203427246 Al5 0.0000000000000000 0.3687512414608813 0.3338148796572754 Al6 0.0000000000000000 0.6312487585391187 0.6661851203427246 Al7 0.0000000000000000 0.7254842833147103 0.1365571627913562 Al8 0.0000000000000000 0.2745157166852897 0.8634428372086438 Al9 0.0000000000000000 0.2745157166852897 0.1365571627913562 Al10 0.0000000000000000 0.7254842833147103 0.8634428372086438 Al11 0.5000000000000000 0.0000000000000000 0.5000000000000000 Al12 0.0000000000000000 0.7852782166261220 0.5000000000000000 Al13 0.0000000000000000 0.2147217833738782 0.5000000000000000 Al14 0.5000000000000000 0.1312487585391187 0.8338148796572754 Al15 0.5000000000000000 0.8687512414608813 0.1661851203427247 Al16 0.5000000000000000 0.8687512414608813 0.8338148796572754 Al17 0.5000000000000000 0.1312487585391187 0.1661851203427247 Al18 0.5000000000000000 0.2254842833147102 0.6365571627913562 Al19 0.5000000000000000 0.7745157166852897 0.3634428372086438 Al20 0.5000000000000000 0.7745157166852897 0.6365571627913562 Al21 0.5000000000000000 0.2254842833147102 0.3634428372086438 Al22 ![FIG. 1. Backscattered electron (BSE) image of the casted Al-3wt%Nd alloy. The compounds with the white color is the eutectic α-Al11Nd<sup>3</sup> phase.](path)
Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage
Figure 3: 1D diffraction scans showing that the difference in monolayer-MoS<sup>2</sup> structure when placed directly onto SiO<sup>2</sup> (gray) versus when a few-layer hBN buffer is used between the monolayer and SiO<sup>2</sup> (red). Bulk MoS<sup>2</sup> and direct measurement of SiO<sup>2</sup> are included as references for known ordered/disordered scattering, respectively. Figure reproduced with permission from Radi´c et al. <sup>6</sup>	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage can leverage the spatial resolution and exclusive surface sensitivity of the technique to investigate inter-layer coupling strength in otherwise difficult to measure systems such as van Der Waals heterostructures where comparable techniques like LEED/M, PL or Raman struggle due to the inherent transmission of the probe through the sample. #### Thermal expansion coefficient Structural information, such as the lattice constant, can be accurately measured using 2D diffraction scans, found to be within 1%. <sup>1</sup> 2D diffraction scans offer increased accuracy over 1D scans, like those in figure 2, because more diffraction provide a statistical advantage. Initially, we measure the 2D diffraction scan on the surface of bulk MoS2, shown in figure S1, and determine the lattice constant to be 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A, matching literature values within experimental uncertainty. We extend the method to mechanically exfoliated monolayer-MoS2/hBN/SiO2, with diffraction pattern shown in figure 4, to showcase the exclusive surface sensitivity of the technique and determine that the lattice constant is <sup>3</sup>.14±0.07˚A, which is within experimental error ![Figure 4: A 2D diffraction pattern taken on monolayer-MoS2/hBN/SiO2, with the identified centres of the diffraction peaks shown as red dots. All of the lattice spacings between adjacent peaks were measured and averaged give a lattice constant 3.<sup>14</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) The thermal expansion coefficient (TEC) of 2D materials is crucial to understanding their behaviour for successful integration into optoelectronic devices, these materials also provide direct measurement of the anharmonic vibrational modes of low-dimensional systems, 7,8 however, difficult to measure due to their optical transparency. 9,10 Using helium atom microdiffraction one can measure the lattice constant, and therefore TEC, of microscopic optically transparent materials, whether supported by a substrate or free-standing. The diffraction pattern was measured along the principle ⟨10⟩ azimuth of MoS<sup>2</sup> at temperatures from 60 −450 ◦C (the data is presented in figure 4), as with work on HAS no change in the lattice constant was observed, allowing us to put an upper bound on the expansion coefficient of 16 × 10−<sup>6</sup> K−<sup>1</sup> . Our measurement is consistent with the upper bound determined as α < 14 × 10−<sup>6</sup> K−<sup>1</sup> reported by Anemone et al.,<sup>11</sup> and α = 7.6×10−<sup>6</sup> K−<sup>1</sup> measured by Zhang et al. using micro-Raman spectroscopy.<sup>10</sup> Our method is currently limited in angular resolution (radial dimension of figure 4) by the size of the limiting aperture that lies between the sample and detector. The current configuration uses a nominal 0.5 mm diameter hole for an in-plane angular resolution ∼ 7.9 ◦ , representing a compromise between measurement time and helium signal. By changing the limiting aperture to a rectangle with dimensions 0.075 × 0.5 mm we can integrate the signal in the axis perpendicular to any change in lattice parameter, the radial direction in figure 4, while improving in-plane angular resolution. We estimate that moving from the current circular aperture to the proposed rectangular aperture decreases helium signal by a factor of 5, but improves in-plane angular resolution by a factor of 6.7 from 7.9 ◦ → 1.2 ◦ which would be sufficient to resolve the TEC over the current temperature range. Details of the current instrument's spatial and angular resolutions, alongside their definitions, can be found in section 'Experimental Methods - Helium atom micro-diffraction instrument details'. ### Electron-phonon coupling Electron-phonon coupling is a major decoherence mechanism in semiconductors, often causing electron-phonon scattering, and eventually energy dissipation, in turn harming optoelectronic device performance. Optimisation of electron-phonon coupling is therefore critical in device design, both in cases where it must be minimised, or maximised. Electron-phonon coupling is typically measured using inelastic optical techniques, such as Raman or Brillouin scattering, and are primarily limited to optically transparent materials and access to electronic coupling to optical phonons only. It has been shown that helium atom scattering is an effective tool for determining the coupling strength between 2D materials and their substrates via Debye-Waller attenuation of helium scattering.12–15 However, previous atom scattering techniques have been limited to millimetre-scale spot sizes which made devicescale samples inaccessible. Here we introduce helium atom micro-diffraction as a technique well suited to the measurement of electronphonon coupling in a range of materials where its non-damaging, surface sensitive and microscopic neutral probe can be used to access electronic couplings to both optical and acoustic phonon modes in 2D materials. The temperature dependence of reflected and diffracted helium intensities can be modelled by Debye-Waller attenuation, 12,13 which describes the increasing motion of the surface atoms with temperature and causes an increase in inelastic scattering. Thus, the ordered scattering intensity is reduced. The attenuation is described by where 2W is the D-W factor. As the helium scattering occurs from the valence electron density rather than from the ionic cores themselves, one can link the D-W factor to electron phonon coupling, as described by Al Taleb et al. <sup>13</sup> D-W attenuation has been measured previously for macro-scale samples for both LiF and bulk MoS2, however without the spatial resolution enabled by SHeM, monolayer MoS<sup>2</sup> flakes could not be measured.<sup>13</sup> ![Figure 5: Log diffracted intensity as a function of temperature for monolayer MoS<sup>2</sup> on few-layer hBN on glass. In theory all diffraction orders should yield the same D-W, and therefore electron-phonon coupling, constants but the signal-to-noise ratios of each peak are a function of relative peak heights and instrument geometry, resulting in small differences.](path) From figure 5 we can extract the D-W factor as the straight-line gradient, and from that the electron-phonon coupling constant λ, contained in table 1 for monolayer and bulk MoS2. The exponents are compared relative to each other, and to literature values for the bulk case, and found to be in agreement with the expected behaviour with parallel momentum transfer (∆K). Table 1: Debye-Waller factors and electronphonon coupling constants extracted from the data presented in figure 5 using the model in equation 1 and theoretical relations outlined by Anemone et al.,<sup>13</sup> respectively. We report the electron-phonon coupling constant in bulk MoS<sup>2</sup> as λbulk ≈ 0.51 × 10−<sup>3</sup> K−<sup>1</sup> , showing good agreement to the literature values λ ≈ 0.41 × 10−<sup>3</sup> K−<sup>1</sup> , 0.49 × 10−<sup>3</sup> K−<sup>1</sup> measured using a typical helium atom scattering instrument with few-millimetre spot size, reported by Anemone et al. <sup>13</sup> We also find λML ≈ 0.40 × 10−<sup>3</sup> K−<sup>1</sup> for ML-MoS<sup>2</sup> on fewlayer hBN. We find the electron-phonon coupling constant is ∼ 20 % smaller in the monolayer compared to bulk MoS2. By leveraging the current best reported ≈ 350 nm spot size, the method can be extended to perform spatial mapping of the Debye-Waller factor, and therefore electron-phonon coupling constant, without sample preparation or damage in thin films and delicate materials, a class of materials typically difficult to characterise using standard optical and electron beam techniques. #### Vacancy-type defect density Precise control of defect density in semiconductors is instrumental for both current and future semiconductor device development. In particular, the optoelectronic properties of two-dimensional TMD semiconductors such as MoS2, can be tuned using single-atom defects. <sup>16</sup> Applications of these materials includes catalysis<sup>17</sup> and a plethora of devices, 18–21 all of which can affected by the material's defect density. In many applications a balance must be made between a sufficiently high number of defects for the material to acquire the desired properties while the material remaining sufficiently ordered to not degrade electronic performance, <sup>22</sup> or even entirely degrade the lattice structure. However, quantification of defect densities in 2D materials remains a significant experimental challenge, where typically used methods are XPS<sup>22</sup> and STEM, with conductive AFM (CAFM) being explored recently. <sup>23</sup> As all of these methods commonly require complicated sample preparation processes, there is a characterisation shortcoming that is only going to grow more acute as devices using 2D materials start being produced on an industrial scale, and therefore enter the commercial sphere. In this work we demonstrate how helium atom micro-diffraction can be used to characterise the vacancy-type defect density on the surface of few-layer materials using data reproduced from Radi´c et al. with permission. <sup>6</sup> Three mechanically exfoliated monolayer flakes of MoS<sup>2</sup> with increasing defect densities, ranging from the intrinsic 0.1 × 10<sup>14</sup> cm−<sup>2</sup> to 1.8 × 10<sup>14</sup> cm−<sup>2</sup> , and ≈ 15 µm lateral size were produced using high-temperature annealing under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere as outlined by Zhu et al.<sup>22</sup> Exact annealing parameters, and the resulting defect densities inferred using photoluminescence spectroscopy, which has been calibrated against stoichiometric beam-line XPS, are contained in Table 2. Figure 6 shows that diffracted helium intensity decreases approximately linearly as a function of defect density within the single-vacancy limit. Detailed analysis and discussion of the results in figure 6 can be found in the full manuscript.<sup>6</sup> ![Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS<sup>2</sup> flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>](path) tion can be employed as a lab-based method to quantify point-defect density without specific sample preparation or damage. The method is agnostic to sample chemistry, and thickness, because the mechanism through which it measures defect density is purely geometric, relying upon the degree of order of the sample surface, meaning that the method can be trivially extended to any system whose macroscopic properties are mediated by surface defects or dopants. Examples of current systems include hBN,<sup>24</sup> graphene,<sup>25</sup> doped systems such as diamond,<sup>26</sup> alongside other TMDs. In conjunction with its electrostatically neutral and low energy beam, microscopic helium atom diffraction presents itself as an ideal technique for the investigation of these few-layer materials. Further improvements in the instruments' lateral resolution resolution while performing diffraction measurements, down to the previously reported ∼ 300 nm spot sizes,<sup>2</sup> would allow for sub-micron scale mapping of defect density across samples. The technique is not currently capable of acting as a standalone, stoichimetric measure of defect density, like XPS, once an accurate helium-defect scattering crosssection is calculated this will also be possible. ## Conclusion The findings reported here have demonstrated the effectiveness of helium atom microdiffraction as a powerful, and completely noninvasive characterisation tool for 2D materials. The technique's ability to probe only the outermost atomic layers with micron spatial resolution makes it uniquely suited to addressing the challenges of analysing 2D materials, where established techniques often struggle. The results and discussion have highlighted several key applications of helium atom microdiffraction, including the measurement of surface cleanliness and contamination without altering the sample, a critical feature for ensuring the integrity of materials used in device fabrication. We also investigated the impact of substrate on monolayer properties, confirming that the use of a few-layer hBN buffer can preserve the structural integrity of monolayer MoS2, whereas direct contact with SiO2 results in significant disorder.? We have also explored the potential of helium atom micro-diffraction for determining the thermal expansion coefficient and electronphonon coupling in monolayer MoS2, and have demonstrated its capability for precise structural and thermal analysis. The technique's ability to quantify defect densities without sample preparation or damage also highlights it as an obvious tool for optimizing 2D materials in ever expanding areas of application, from optoelectronics to catalysis and more. Future enhancements to helium atom microdiffraction, such as achieving sub-50 nm spatial resolution, and incorporating out-of-plane scattering capabilities, will further expand its applicability, particularly for complex heterostructures and device-grade materials. In summary, helium atom micro-diffraction promises to provide a robust and adaptable platform for characterisation and development of 2D materials, paving the way for advancements in nanotechnology and material science. ## Experimental Methods ## Helium atom micro-diffraction instrument details In the presented work we used a Scanning Helium Microscope (SHeM) with 5 µm spatial and 7.9 ◦ in-plane angular resolution at the specular condition, representing a factor of 2 improvement in spatial resolution over the first iteration published by von Jeinsen et al. <sup>1</sup> Spatial resolution is defined as the full-width at halfmaximum of the beam spot on the sample at the designed working distance. In-plane angular resolution is given by the angle subtended by the limiting detector aperture and beam spot on the sample, with the correcting factor <sup>√</sup> 1 2 applied to account for the 45◦ instrument geometry. ## Sample details Monolayers of MoS<sup>2</sup> were produced by mechanical exfoliation of a bulk MoS<sup>2</sup> crystal, purchased from 2D Semiconductors Ltd., and were deposited onto few-layer thick hexgonal boron nitride (hBN) which in turn lies on a SiO<sup>2</sup> substrate (high-precision glass microscope slide). Each monolayer measured ≈ 15 µm laterally. The increased defect density monolayer MoS<sup>2</sup> samples, presented in section , were produced the same as previously described in this section with the addition of a thermal annealing process under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere for 0.5 h, following the method as described by Zhu et al.<sup>22</sup> Varying annealing temperature and time allows for control of defect density. Exact annealing parameters, with final defect densities characterised by stoichiometric XPS, are shown in Table 2. # Acknowledgements The work was supported by EPSRC grant EP/R008272/1, Innovate UK/Ionoptika Ltd. through Knowledge Transfer Partnership 10000925. The work was performed in part at CORDE, the Collaborative R&D Environment established to provide access to physics related facilities at the Cavendish Laboratory, University of Cambridge and EPSRC award EP/T00634X/1. SML acknowledges support from EPSRC grant EP/X525686/1. # References - 1. von Jeinsen, N. A.; Lambrick, S. M.; Bergin, M.; Radi´c, A.; Liu, B.; Seremet, D.; Jardine, A. P.; Ward, D. J. 2D Helium Atom Diffraction from a Microscopic Spot. Physical Review Letters 2023, 131, 236202, Publisher: American Physical Society. - 5. Man, M. K. L.; Deckoff-Jones, S.; Winchester, A.; Shi, G.; Gupta, G.; Mohite, A. D.; Kar, S.; Kioupakis, E.; Talapatra, S.; Dani, K. M. Protecting the properties of monolayer MoS2 on silicon based substrates with an atomically thin buffer. Scientific Reports 2016, 6, 20890, Number: 1 Publisher: Nature Publishing Group. - 6. Radi´c, A.; von Jeinsen, N.; Wang, K.; Zhu, Y.; Sami, I.; Perez, V.; Ward, D.; Jardine, A.; Chhowwalla, M.; Lambrick, S. Defect density quantification in monolayer MoS<sup>2</sup> using helium atom micro-diffraction. arXiv 2024, - 7. Late, D. J.; Shirodkar, S. N.; Waghmare, U. V.; Dravid, V. P.; Rao, C. N. R. Thermal Expansion, Anharmonicity and Temperature-Dependent Raman Spectra of Single- and Few-Layer MoSe2 and WSe2. ChemPhysChem 2014, 15, 1592–1598. - 10. Zhang, L.; Lu, Z.; Song, Y.; Zhao, L.; Bhatia, B.; Bagnall, K. R.; Wang, E. N. Thermal Expansion Coefficient of Monolayer Molybdenum Disulfide Using Micro-Raman Spectroscopy. Nano Letters 2019, 19, 4745–4751. - 11. Anemone, G.; Taleb, A. A.; Politano, A.; Kuo, C.-N.; Lue, C. S.; Miranda, R.; Far´ıas, D. Setting the limit for the lateral thermal expansion of layered crystals via helium atom scattering. Physical Chemistry Chemical Physics 2022, 24, 13229–13233, Publisher: The Royal Society of Chemistry. - 14. Anemone, G.; Taleb, A. A.; Benedek, G.; Castellanos-Gomez, A.; Far´ıas, D. Electron–Phonon Coupling Constant of 2H-MoS2(0001) from Helium-Atom Scattering. The Journal of Physical Chemistry C 2019, 123, 3682–3686, Publisher: American Chemical Society. - 15. Anemone, G.; Casado Aguilar, P.; Garnica, M.; Calleja, F.; Al Taleb, A.; Kuo, C.- N.; Lue, C. S.; Politano, A.; V´azquez de Parga, A. L.; Benedek, G.; Far´ıas, D.; Miranda, R. Electron–phonon coupling in superconducting 1T-PdTe2. npj 2D Materials and Applications 2021, 5, 1–7, Number: 1 Publisher: Nature Publishing Group. - 16. Regan, E. C.; Wang, D.; Paik, E. Y.; Zeng, Y.; Zhang, L.; Zhu, J.; MacDonald, A. H.; Deng, H.; Wang, F. Emerging exciton physics in transition metal dichalcogenide heterobilayers. 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Nature Materials 2024, # Helium atom micro-diffraction as a characterisation tool for 2D materials - supporting information # Bulk MoS<sup>2</sup> diffraction pattern ![Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure 5: Log diffracted intensity as a function of temperature for monolayer MoS<sup>2</sup> on few-layer hBN on glass. In theory all diffraction orders should yield the same D-W, and therefore electron-phonon coupling, constants but the signal-to-noise ratios of each peak are a function of relative peak heights and instrument geometry, resulting in small differences.	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage From figure 5 we can extract the D-W factor as the straight-line gradient, and from that the electron-phonon coupling constant λ, contained in table 1 for monolayer and bulk MoS2. The exponents are compared relative to each other, and to literature values for the bulk case, and found to be in agreement with the expected behaviour with parallel momentum transfer (∆K). Table 1: Debye-Waller factors and electronphonon coupling constants extracted from the data presented in figure 5 using the model in equation 1 and theoretical relations outlined by Anemone et al.,<sup>13</sup> respectively. We report the electron-phonon coupling constant in bulk MoS<sup>2</sup> as λbulk ≈ 0.51 × 10−<sup>3</sup> K−<sup>1</sup> , showing good agreement to the literature values λ ≈ 0.41 × 10−<sup>3</sup> K−<sup>1</sup> , 0.49 × 10−<sup>3</sup> K−<sup>1</sup> measured using a typical helium atom scattering instrument with few-millimetre spot size, reported by Anemone et al. <sup>13</sup> We also find λML ≈ 0.40 × 10−<sup>3</sup> K−<sup>1</sup> for ML-MoS<sup>2</sup> on fewlayer hBN. We find the electron-phonon coupling constant is ∼ 20 % smaller in the monolayer compared to bulk MoS2. By leveraging the current best reported ≈ 350 nm spot size, the method can be extended to perform spatial mapping of the Debye-Waller factor, and therefore electron-phonon coupling constant, without sample preparation or damage in thin films and delicate materials, a class of materials typically difficult to characterise using standard optical and electron beam techniques. #### Vacancy-type defect density Precise control of defect density in semiconductors is instrumental for both current and future semiconductor device development. In particular, the optoelectronic properties of two-dimensional TMD semiconductors such as MoS2, can be tuned using single-atom defects. <sup>16</sup> Applications of these materials includes catalysis<sup>17</sup> and a plethora of devices, 18–21 all of which can affected by the material's defect density. In many applications a balance must be made between a sufficiently high number of defects for the material to acquire the desired properties while the material remaining sufficiently ordered to not degrade electronic performance, <sup>22</sup> or even entirely degrade the lattice structure. However, quantification of defect densities in 2D materials remains a significant experimental challenge, where typically used methods are XPS<sup>22</sup> and STEM, with conductive AFM (CAFM) being explored recently. <sup>23</sup> As all of these methods commonly require complicated sample preparation processes, there is a characterisation shortcoming that is only going to grow more acute as devices using 2D materials start being produced on an industrial scale, and therefore enter the commercial sphere. In this work we demonstrate how helium atom micro-diffraction can be used to characterise the vacancy-type defect density on the surface of few-layer materials using data reproduced from Radi´c et al. with permission. <sup>6</sup> Three mechanically exfoliated monolayer flakes of MoS<sup>2</sup> with increasing defect densities, ranging from the intrinsic 0.1 × 10<sup>14</sup> cm−<sup>2</sup> to 1.8 × 10<sup>14</sup> cm−<sup>2</sup> , and ≈ 15 µm lateral size were produced using high-temperature annealing under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere as outlined by Zhu et al.<sup>22</sup> Exact annealing parameters, and the resulting defect densities inferred using photoluminescence spectroscopy, which has been calibrated against stoichiometric beam-line XPS, are contained in Table 2. Figure 6 shows that diffracted helium intensity decreases approximately linearly as a function of defect density within the single-vacancy limit. Detailed analysis and discussion of the results in figure 6 can be found in the full manuscript.<sup>6</sup> ![Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS<sup>2</sup> flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>](path) tion can be employed as a lab-based method to quantify point-defect density without specific sample preparation or damage. The method is agnostic to sample chemistry, and thickness, because the mechanism through which it measures defect density is purely geometric, relying upon the degree of order of the sample surface, meaning that the method can be trivially extended to any system whose macroscopic properties are mediated by surface defects or dopants. Examples of current systems include hBN,<sup>24</sup> graphene,<sup>25</sup> doped systems such as diamond,<sup>26</sup> alongside other TMDs. In conjunction with its electrostatically neutral and low energy beam, microscopic helium atom diffraction presents itself as an ideal technique for the investigation of these few-layer materials. Further improvements in the instruments' lateral resolution resolution while performing diffraction measurements, down to the previously reported ∼ 300 nm spot sizes,<sup>2</sup> would allow for sub-micron scale mapping of defect density across samples. The technique is not currently capable of acting as a standalone, stoichimetric measure of defect density, like XPS, once an accurate helium-defect scattering crosssection is calculated this will also be possible. ## Conclusion The findings reported here have demonstrated the effectiveness of helium atom microdiffraction as a powerful, and completely noninvasive characterisation tool for 2D materials. The technique's ability to probe only the outermost atomic layers with micron spatial resolution makes it uniquely suited to addressing the challenges of analysing 2D materials, where established techniques often struggle. The results and discussion have highlighted several key applications of helium atom microdiffraction, including the measurement of surface cleanliness and contamination without altering the sample, a critical feature for ensuring the integrity of materials used in device fabrication. We also investigated the impact of substrate on monolayer properties, confirming that the use of a few-layer hBN buffer can preserve the structural integrity of monolayer MoS2, whereas direct contact with SiO2 results in significant disorder.? We have also explored the potential of helium atom micro-diffraction for determining the thermal expansion coefficient and electronphonon coupling in monolayer MoS2, and have demonstrated its capability for precise structural and thermal analysis. The technique's ability to quantify defect densities without sample preparation or damage also highlights it as an obvious tool for optimizing 2D materials in ever expanding areas of application, from optoelectronics to catalysis and more. Future enhancements to helium atom microdiffraction, such as achieving sub-50 nm spatial resolution, and incorporating out-of-plane scattering capabilities, will further expand its applicability, particularly for complex heterostructures and device-grade materials. In summary, helium atom micro-diffraction promises to provide a robust and adaptable platform for characterisation and development of 2D materials, paving the way for advancements in nanotechnology and material science. ## Experimental Methods ## Helium atom micro-diffraction instrument details In the presented work we used a Scanning Helium Microscope (SHeM) with 5 µm spatial and 7.9 ◦ in-plane angular resolution at the specular condition, representing a factor of 2 improvement in spatial resolution over the first iteration published by von Jeinsen et al. <sup>1</sup> Spatial resolution is defined as the full-width at halfmaximum of the beam spot on the sample at the designed working distance. In-plane angular resolution is given by the angle subtended by the limiting detector aperture and beam spot on the sample, with the correcting factor <sup>√</sup> 1 2 applied to account for the 45◦ instrument geometry. ## Sample details Monolayers of MoS<sup>2</sup> were produced by mechanical exfoliation of a bulk MoS<sup>2</sup> crystal, purchased from 2D Semiconductors Ltd., and were deposited onto few-layer thick hexgonal boron nitride (hBN) which in turn lies on a SiO<sup>2</sup> substrate (high-precision glass microscope slide). Each monolayer measured ≈ 15 µm laterally. The increased defect density monolayer MoS<sup>2</sup> samples, presented in section , were produced the same as previously described in this section with the addition of a thermal annealing process under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere for 0.5 h, following the method as described by Zhu et al.<sup>22</sup> Varying annealing temperature and time allows for control of defect density. Exact annealing parameters, with final defect densities characterised by stoichiometric XPS, are shown in Table 2. # Acknowledgements The work was supported by EPSRC grant EP/R008272/1, Innovate UK/Ionoptika Ltd. through Knowledge Transfer Partnership 10000925. The work was performed in part at CORDE, the Collaborative R&D Environment established to provide access to physics related facilities at the Cavendish Laboratory, University of Cambridge and EPSRC award EP/T00634X/1. SML acknowledges support from EPSRC grant EP/X525686/1. # References - 1. von Jeinsen, N. A.; Lambrick, S. M.; Bergin, M.; Radi´c, A.; Liu, B.; Seremet, D.; Jardine, A. P.; Ward, D. J. 2D Helium Atom Diffraction from a Microscopic Spot. Physical Review Letters 2023, 131, 236202, Publisher: American Physical Society. - 5. Man, M. K. L.; Deckoff-Jones, S.; Winchester, A.; Shi, G.; Gupta, G.; Mohite, A. D.; Kar, S.; Kioupakis, E.; Talapatra, S.; Dani, K. M. Protecting the properties of monolayer MoS2 on silicon based substrates with an atomically thin buffer. Scientific Reports 2016, 6, 20890, Number: 1 Publisher: Nature Publishing Group. - 6. Radi´c, A.; von Jeinsen, N.; Wang, K.; Zhu, Y.; Sami, I.; Perez, V.; Ward, D.; Jardine, A.; Chhowwalla, M.; Lambrick, S. Defect density quantification in monolayer MoS<sup>2</sup> using helium atom micro-diffraction. arXiv 2024, - 7. Late, D. J.; Shirodkar, S. N.; Waghmare, U. V.; Dravid, V. P.; Rao, C. N. R. Thermal Expansion, Anharmonicity and Temperature-Dependent Raman Spectra of Single- and Few-Layer MoSe2 and WSe2. ChemPhysChem 2014, 15, 1592–1598. - 10. Zhang, L.; Lu, Z.; Song, Y.; Zhao, L.; Bhatia, B.; Bagnall, K. R.; Wang, E. N. Thermal Expansion Coefficient of Monolayer Molybdenum Disulfide Using Micro-Raman Spectroscopy. Nano Letters 2019, 19, 4745–4751. - 11. Anemone, G.; Taleb, A. A.; Politano, A.; Kuo, C.-N.; Lue, C. S.; Miranda, R.; Far´ıas, D. Setting the limit for the lateral thermal expansion of layered crystals via helium atom scattering. Physical Chemistry Chemical Physics 2022, 24, 13229–13233, Publisher: The Royal Society of Chemistry. - 14. Anemone, G.; Taleb, A. A.; Benedek, G.; Castellanos-Gomez, A.; Far´ıas, D. Electron–Phonon Coupling Constant of 2H-MoS2(0001) from Helium-Atom Scattering. The Journal of Physical Chemistry C 2019, 123, 3682–3686, Publisher: American Chemical Society. - 15. Anemone, G.; Casado Aguilar, P.; Garnica, M.; Calleja, F.; Al Taleb, A.; Kuo, C.- N.; Lue, C. S.; Politano, A.; V´azquez de Parga, A. L.; Benedek, G.; Far´ıas, D.; Miranda, R. Electron–phonon coupling in superconducting 1T-PdTe2. npj 2D Materials and Applications 2021, 5, 1–7, Number: 1 Publisher: Nature Publishing Group. - 16. Regan, E. C.; Wang, D.; Paik, E. Y.; Zeng, Y.; Zhang, L.; Zhu, J.; MacDonald, A. H.; Deng, H.; Wang, F. Emerging exciton physics in transition metal dichalcogenide heterobilayers. Nature Reviews Materials 2022, 7, 778–795. - 17. Yang, J.; Wang, Y.; Lagos, M. J.; Manichev, V.; Fullon, R.; Song, X.; Voiry, D.; Chakraborty, S.; Zhang, W.; Batson, P. E.; Feldman, L.; Gustafsson, T.; Chhowalla, M. Single Atomic Vacancy Catalysis. ACS Nano 2019, 13, 9958–9964, Publisher: American Chemical Society. - 18. Mitterreiter, E.; Schuler, B.; Micevic, A.; Hernang´omez-P´erez, D.; Barthelmi, K.; Cochrane, K. A.; Kiemle, J.; Sigger, F.; Klein, J.; Wong, E.; Barnard, E. S.; Watanabe, K.; Taniguchi, T.; Lorke, M.; Jahnke, F.; Finley, J. J.; Schwartzberg, A. M.; Qiu, D. Y.; Refaely-Abramson, S.; Holleitner, A. W. et al. The role of chalcogen vacancies for - 19. Barthelmi, K.; Klein, J.; H¨otger, A.; Sigl, L.; Sigger, F.; Mitterreiter, E.; Rey, S.; Gyger, S.; Lorke, M.; Florian, M.; Jahnke, F.; Taniguchi, T.; Watanabe, K.; Zwiller, V.; J¨ons, K. D.; Wurstbauer, U.; Kastl, C.; Weber-Bargioni, A.; Finley, J. J.; M¨uller, K. et al. Atomistic defects as singlephoton emitters in atomically thin MoS2. Applied Physics Letters 2020, 117 . - 22. Zhu, Y.; Lim, J.; Zhang, Z.; Wang, Y.; Sarkar, S.; Ramsden, H.; Li, Y.; Yan, H.; Phuyal, D.; Gauriot, N.; Rao, A.; Hoye, R. L. Z.; Eda, G.; Chhowalla, M. Room-Temperature Photoluminescence Mediated by Sulfur Vacancies in 2D Molybdenum Disulfide. ACS Nano 2023, 17, 13545– 13553, Publisher: American Chemical Society. - 23. Xu, K.; Holbrook, M.; Holtzman, L. N.; Pasupathy, A. N.; Barmak, K.; Hone, J. C.; Rosenberger, M. R. Validating the Use of Conductive Atomic Force Microscopy for Defect Quantification in 2D Materials. ACS Nano 2023, 17, 24743–24752, PMID: 38095969. - 24. Stern, H. L.; M. Gilardoni, C.; Gu, Q.; Eizagirre Barker, S.; Powell, O. F. J.; Deng, X.; Fraser, S. A.; Follet, L.; Li, C.; Ramsay, A. J.; Tan, H. H.; Aharonovich, I.; Atat¨ure, M. A quantum coherent spin in hexagonal boron nitride at ambient conditions. Nature Materials 2024, # Helium atom micro-diffraction as a characterisation tool for 2D materials - supporting information # Bulk MoS<sup>2</sup> diffraction pattern ![Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure 4: A 2D diffraction pattern taken on monolayer-MoS2/hBN/SiO2, with the identified centres of the diffraction peaks shown as red dots. All of the lattice spacings between adjacent peaks were measured and averaged give a lattice constant 3.14±0.07˚A.	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage ![Figure 1: Reflection mode optical (top-left) and real-space SHeM (bottom) images of bulk MoS2, hBN/SiO<sup>2</sup> and monolayer MoS2/hBN/SiO<sup>2</sup> (Shaded intersection). A 2D micro diffraction measurement (top-right) was taken to identify diffraction conditions for the real space imaging. An obvious change in contrast can be seen as the instrument is configured for](path) and species that one expects during standard device fabrication. Once under high-vacuum (2×10−<sup>8</sup> mbar) in the SHeM, a real-space image is taken of the as-prepared sample (Figure 2a) which demonstrates it is difficult to differentiate between MoS<sup>2</sup> and hBN, with the surrounding SiO<sup>2</sup> appearing less intense. Upon heating to 240 ◦C for 8 hours, another real-space image (figure 2b) reveals a stark difference in contrast between MoS2, hBN, and SiO2. The MoS<sup>2</sup> and hBN display significantly more intense scattering than the surrounding SiO<sup>2</sup> which can be quantitatively compared in Figure 2c. Having demonstrated the sensitivity of helium atom micro-diffraction to adsorbates, one can trivially extend the method to quantitatively map adsorbate densities across real-space images. By cleaning and measuring the sample with the presented procedure, one can identify the positions of diffraction peaks in reciprocal space (∆K). This is followed by acquiring a real-space image at a diffraction peak's maximum scattering condition in reciprocal space, achieved by translation of the sample in the zaxis. One can now dose the sample environment with a given contaminant and measure a realspace image to determine if there is a change adsorbate coverage. Through dosing, adsorbation behaviour of a given sample to varying species can also be investigated by monitoring a specific diffraction peak while dosing. ### Monolayer-substrate interactions It is well documented that the choice of substrate placed under a few-layer material has a significant effect on a range of material properties. One can measure the effect of the substrate on optoelectronic or structural properties of the sample via methods such as photoluminescence spectroscopy (PL), Raman, or low-energy electron microscopy/diffraction (LEEM/D), respectively.<sup>5</sup> To demonstrate the exclusive surface sensitivity of helium atom micro-diffraction, monolayer-MoS<sup>2</sup> was placed directly onto both an SiO<sup>2</sup> substrate and few-layer hBN which is in turn on SiO2, sample geometry is shown in figure 1. It has been demonstrated that certain atomically thin buffer layers, such as hBN, ![Figure 2: Real-space images and diffraction scans of monolayer-MoS2/hBN/SiO<sup>2</sup> (red), fewlayer hBN/SiO<sup>2</sup> (black) and SiO<sup>2</sup> (blue) before and after heating (left to right) to 240 ◦C for 8 hours to remove physisorbed species. For monolayer-MoS<sup>2</sup> and hBN we see ordered scattering emerging upon cleaning in contrast to the cosine-like scattering prior. SiO<sup>2</sup> remains cosine-like post-heating. Diffraction scans were acquired at the indicated points in Panels a,b. Real-space images were taken at a single ∆K, marked with a vertical line in Panel C.](path) LaAlO<sup>3</sup> and SrTiO3, can protect the optoelectronic properties of monolayers mounted on them from strongly interacting substrates like SiO2. <sup>5</sup> Figure 3 shows that the helium scattering from monolayer MoS<sup>2</sup> becomes almost entirely disordered when mounted directly on SiO<sup>2</sup> (gray), with faint signs of structure remaining at the expected ∆K diffraction peak positions. In contrast, placing the monolayer onto a few-layer hBN buffer protects its structure and produces diffraction that matches bulk MoS<sup>2</sup> to within experimental bounds, suggesting that their surface morphologies are the same. ![Figure 3: 1D diffraction scans showing that the difference in monolayer-MoS<sup>2</sup> structure when placed directly onto SiO<sup>2</sup> (gray) versus when a few-layer hBN buffer is used between the monolayer and SiO<sup>2</sup> (red). Bulk MoS<sup>2</sup> and direct measurement of SiO<sup>2</sup> are included as references for known ordered/disordered scattering, respectively. Figure reproduced with permission from Radi´c et al. <sup>6</sup>](path) can leverage the spatial resolution and exclusive surface sensitivity of the technique to investigate inter-layer coupling strength in otherwise difficult to measure systems such as van Der Waals heterostructures where comparable techniques like LEED/M, PL or Raman struggle due to the inherent transmission of the probe through the sample. #### Thermal expansion coefficient Structural information, such as the lattice constant, can be accurately measured using 2D diffraction scans, found to be within 1%. <sup>1</sup> 2D diffraction scans offer increased accuracy over 1D scans, like those in figure 2, because more diffraction provide a statistical advantage. Initially, we measure the 2D diffraction scan on the surface of bulk MoS2, shown in figure S1, and determine the lattice constant to be 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A, matching literature values within experimental uncertainty. We extend the method to mechanically exfoliated monolayer-MoS2/hBN/SiO2, with diffraction pattern shown in figure 4, to showcase the exclusive surface sensitivity of the technique and determine that the lattice constant is <sup>3</sup>.14±0.07˚A, which is within experimental error ![Figure 4: A 2D diffraction pattern taken on monolayer-MoS2/hBN/SiO2, with the identified centres of the diffraction peaks shown as red dots. All of the lattice spacings between adjacent peaks were measured and averaged give a lattice constant 3.<sup>14</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) The thermal expansion coefficient (TEC) of 2D materials is crucial to understanding their behaviour for successful integration into optoelectronic devices, these materials also provide direct measurement of the anharmonic vibrational modes of low-dimensional systems, 7,8 however, difficult to measure due to their optical transparency. 9,10 Using helium atom microdiffraction one can measure the lattice constant, and therefore TEC, of microscopic optically transparent materials, whether supported by a substrate or free-standing. The diffraction pattern was measured along the principle ⟨10⟩ azimuth of MoS<sup>2</sup> at temperatures from 60 −450 ◦C (the data is presented in figure 4), as with work on HAS no change in the lattice constant was observed, allowing us to put an upper bound on the expansion coefficient of 16 × 10−<sup>6</sup> K−<sup>1</sup> . Our measurement is consistent with the upper bound determined as α < 14 × 10−<sup>6</sup> K−<sup>1</sup> reported by Anemone et al.,<sup>11</sup> and α = 7.6×10−<sup>6</sup> K−<sup>1</sup> measured by Zhang et al. using micro-Raman spectroscopy.<sup>10</sup> Our method is currently limited in angular resolution (radial dimension of figure 4) by the size of the limiting aperture that lies between the sample and detector. The current configuration uses a nominal 0.5 mm diameter hole for an in-plane angular resolution ∼ 7.9 ◦ , representing a compromise between measurement time and helium signal. By changing the limiting aperture to a rectangle with dimensions 0.075 × 0.5 mm we can integrate the signal in the axis perpendicular to any change in lattice parameter, the radial direction in figure 4, while improving in-plane angular resolution. We estimate that moving from the current circular aperture to the proposed rectangular aperture decreases helium signal by a factor of 5, but improves in-plane angular resolution by a factor of 6.7 from 7.9 ◦ → 1.2 ◦ which would be sufficient to resolve the TEC over the current temperature range. Details of the current instrument's spatial and angular resolutions, alongside their definitions, can be found in section 'Experimental Methods - Helium atom micro-diffraction instrument details'. ### Electron-phonon coupling Electron-phonon coupling is a major decoherence mechanism in semiconductors, often causing electron-phonon scattering, and eventually energy dissipation, in turn harming optoelectronic device performance. Optimisation of electron-phonon coupling is therefore critical in device design, both in cases where it must be minimised, or maximised. Electron-phonon coupling is typically measured using inelastic optical techniques, such as Raman or Brillouin scattering, and are primarily limited to optically transparent materials and access to electronic coupling to optical phonons only. It has been shown that helium atom scattering is an effective tool for determining the coupling strength between 2D materials and their substrates via Debye-Waller attenuation of helium scattering.12–15 However, previous atom scattering techniques have been limited to millimetre-scale spot sizes which made devicescale samples inaccessible. Here we introduce helium atom micro-diffraction as a technique well suited to the measurement of electronphonon coupling in a range of materials where its non-damaging, surface sensitive and microscopic neutral probe can be used to access electronic couplings to both optical and acoustic phonon modes in 2D materials. The temperature dependence of reflected and diffracted helium intensities can be modelled by Debye-Waller attenuation, 12,13 which describes the increasing motion of the surface atoms with temperature and causes an increase in inelastic scattering. Thus, the ordered scattering intensity is reduced. The attenuation is described by where 2W is the D-W factor. As the helium scattering occurs from the valence electron density rather than from the ionic cores themselves, one can link the D-W factor to electron phonon coupling, as described by Al Taleb et al. <sup>13</sup> D-W attenuation has been measured previously for macro-scale samples for both LiF and bulk MoS2, however without the spatial resolution enabled by SHeM, monolayer MoS<sup>2</sup> flakes could not be measured.<sup>13</sup> ![Figure 5: Log diffracted intensity as a function of temperature for monolayer MoS<sup>2</sup> on few-layer hBN on glass. In theory all diffraction orders should yield the same D-W, and therefore electron-phonon coupling, constants but the signal-to-noise ratios of each peak are a function of relative peak heights and instrument geometry, resulting in small differences.](path) From figure 5 we can extract the D-W factor as the straight-line gradient, and from that the electron-phonon coupling constant λ, contained in table 1 for monolayer and bulk MoS2. The exponents are compared relative to each other, and to literature values for the bulk case, and found to be in agreement with the expected behaviour with parallel momentum transfer (∆K). Table 1: Debye-Waller factors and electronphonon coupling constants extracted from the data presented in figure 5 using the model in equation 1 and theoretical relations outlined by Anemone et al.,<sup>13</sup> respectively. We report the electron-phonon coupling constant in bulk MoS<sup>2</sup> as λbulk ≈ 0.51 × 10−<sup>3</sup> K−<sup>1</sup> , showing good agreement to the literature values λ ≈ 0.41 × 10−<sup>3</sup> K−<sup>1</sup> , 0.49 × 10−<sup>3</sup> K−<sup>1</sup> measured using a typical helium atom scattering instrument with few-millimetre spot size, reported by Anemone et al. <sup>13</sup> We also find λML ≈ 0.40 × 10−<sup>3</sup> K−<sup>1</sup> for ML-MoS<sup>2</sup> on fewlayer hBN. We find the electron-phonon coupling constant is ∼ 20 % smaller in the monolayer compared to bulk MoS2. By leveraging the current best reported ≈ 350 nm spot size, the method can be extended to perform spatial mapping of the Debye-Waller factor, and therefore electron-phonon coupling constant, without sample preparation or damage in thin films and delicate materials, a class of materials typically difficult to characterise using standard optical and electron beam techniques. #### Vacancy-type defect density Precise control of defect density in semiconductors is instrumental for both current and future semiconductor device development. In particular, the optoelectronic properties of two-dimensional TMD semiconductors such as MoS2, can be tuned using single-atom defects. <sup>16</sup> Applications of these materials includes catalysis<sup>17</sup> and a plethora of devices, 18–21 all of which can affected by the material's defect density. In many applications a balance must be made between a sufficiently high number of defects for the material to acquire the desired properties while the material remaining sufficiently ordered to not degrade electronic performance, <sup>22</sup> or even entirely degrade the lattice structure. However, quantification of defect densities in 2D materials remains a significant experimental challenge, where typically used methods are XPS<sup>22</sup> and STEM, with conductive AFM (CAFM) being explored recently. <sup>23</sup> As all of these methods commonly require complicated sample preparation processes, there is a characterisation shortcoming that is only going to grow more acute as devices using 2D materials start being produced on an industrial scale, and therefore enter the commercial sphere. In this work we demonstrate how helium atom micro-diffraction can be used to characterise the vacancy-type defect density on the surface of few-layer materials using data reproduced from Radi´c et al. with permission. <sup>6</sup> Three mechanically exfoliated monolayer flakes of MoS<sup>2</sup> with increasing defect densities, ranging from the intrinsic 0.1 × 10<sup>14</sup> cm−<sup>2</sup> to 1.8 × 10<sup>14</sup> cm−<sup>2</sup> , and ≈ 15 µm lateral size were produced using high-temperature annealing under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere as outlined by Zhu et al.<sup>22</sup> Exact annealing parameters, and the resulting defect densities inferred using photoluminescence spectroscopy, which has been calibrated against stoichiometric beam-line XPS, are contained in Table 2. Figure 6 shows that diffracted helium intensity decreases approximately linearly as a function of defect density within the single-vacancy limit. Detailed analysis and discussion of the results in figure 6 can be found in the full manuscript.<sup>6</sup> ![Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS<sup>2</sup> flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>](path) tion can be employed as a lab-based method to quantify point-defect density without specific sample preparation or damage. The method is agnostic to sample chemistry, and thickness, because the mechanism through which it measures defect density is purely geometric, relying upon the degree of order of the sample surface, meaning that the method can be trivially extended to any system whose macroscopic properties are mediated by surface defects or dopants. Examples of current systems include hBN,<sup>24</sup> graphene,<sup>25</sup> doped systems such as diamond,<sup>26</sup> alongside other TMDs. In conjunction with its electrostatically neutral and low energy beam, microscopic helium atom diffraction presents itself as an ideal technique for the investigation of these few-layer materials. Further improvements in the instruments' lateral resolution resolution while performing diffraction measurements, down to the previously reported ∼ 300 nm spot sizes,<sup>2</sup> would allow for sub-micron scale mapping of defect density across samples. The technique is not currently capable of acting as a standalone, stoichimetric measure of defect density, like XPS, once an accurate helium-defect scattering crosssection is calculated this will also be possible. ## Conclusion The findings reported here have demonstrated the effectiveness of helium atom microdiffraction as a powerful, and completely noninvasive characterisation tool for 2D materials. The technique's ability to probe only the outermost atomic layers with micron spatial resolution makes it uniquely suited to addressing the challenges of analysing 2D materials, where established techniques often struggle. The results and discussion have highlighted several key applications of helium atom microdiffraction, including the measurement of surface cleanliness and contamination without altering the sample, a critical feature for ensuring the integrity of materials used in device fabrication. We also investigated the impact of substrate on monolayer properties, confirming that the use of a few-layer hBN buffer can preserve the structural integrity of monolayer MoS2, whereas direct contact with SiO2 results in significant disorder.? We have also explored the potential of helium atom micro-diffraction for determining the thermal expansion coefficient and electronphonon coupling in monolayer MoS2, and have demonstrated its capability for precise structural and thermal analysis. The technique's ability to quantify defect densities without sample preparation or damage also highlights it as an obvious tool for optimizing 2D materials in ever expanding areas of application, from optoelectronics to catalysis and more. Future enhancements to helium atom microdiffraction, such as achieving sub-50 nm spatial resolution, and incorporating out-of-plane scattering capabilities, will further expand its applicability, particularly for complex heterostructures and device-grade materials. In summary, helium atom micro-diffraction promises to provide a robust and adaptable platform for characterisation and development of 2D materials, paving the way for advancements in nanotechnology and material science. ## Experimental Methods ## Helium atom micro-diffraction instrument details In the presented work we used a Scanning Helium Microscope (SHeM) with 5 µm spatial and 7.9 ◦ in-plane angular resolution at the specular condition, representing a factor of 2 improvement in spatial resolution over the first iteration published by von Jeinsen et al. <sup>1</sup> Spatial resolution is defined as the full-width at halfmaximum of the beam spot on the sample at the designed working distance. In-plane angular resolution is given by the angle subtended by the limiting detector aperture and beam spot on the sample, with the correcting factor <sup>√</sup> 1 2 applied to account for the 45◦ instrument geometry. ## Sample details Monolayers of MoS<sup>2</sup> were produced by mechanical exfoliation of a bulk MoS<sup>2</sup> crystal, purchased from 2D Semiconductors Ltd., and were deposited onto few-layer thick hexgonal boron nitride (hBN) which in turn lies on a SiO<sup>2</sup> substrate (high-precision glass microscope slide). Each monolayer measured ≈ 15 µm laterally. The increased defect density monolayer MoS<sup>2</sup> samples, presented in section , were produced the same as previously described in this section with the addition of a thermal annealing process under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere for 0.5 h, following the method as described by Zhu et al.<sup>22</sup> Varying annealing temperature and time allows for control of defect density. Exact annealing parameters, with final defect densities characterised by stoichiometric XPS, are shown in Table 2. # Acknowledgements The work was supported by EPSRC grant EP/R008272/1, Innovate UK/Ionoptika Ltd. through Knowledge Transfer Partnership 10000925. The work was performed in part at CORDE, the Collaborative R&D Environment established to provide access to physics related facilities at the Cavendish Laboratory, University of Cambridge and EPSRC award EP/T00634X/1. SML acknowledges support from EPSRC grant EP/X525686/1. # References - 1. von Jeinsen, N. A.; Lambrick, S. M.; Bergin, M.; Radi´c, A.; Liu, B.; Seremet, D.; Jardine, A. P.; Ward, D. J. 2D Helium Atom Diffraction from a Microscopic Spot. Physical Review Letters 2023, 131, 236202, Publisher: American Physical Society. - 5. Man, M. K. L.; Deckoff-Jones, S.; Winchester, A.; Shi, G.; Gupta, G.; Mohite, A. D.; Kar, S.; Kioupakis, E.; Talapatra, S.; Dani, K. M. 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Nature Materials 2024, # Helium atom micro-diffraction as a characterisation tool for 2D materials - supporting information # Bulk MoS<sup>2</sup> diffraction pattern ![Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure 1: Reflection mode optical (top-left) and real-space SHeM (bottom) images of bulk MoS2, hBN/SiO2 and monolayer MoS2/hBN/SiO2 (Shaded intersection). A 2D micro diffraction measurement (top-right) was taken to identify diffraction conditions for the real space imaging. An obvious change in contrast can be seen as the instrument is configured for	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage ![Figure 1: Reflection mode optical (top-left) and real-space SHeM (bottom) images of bulk MoS2, hBN/SiO<sup>2</sup> and monolayer MoS2/hBN/SiO<sup>2</sup> (Shaded intersection). A 2D micro diffraction measurement (top-right) was taken to identify diffraction conditions for the real space imaging. An obvious change in contrast can be seen as the instrument is configured for](path) and species that one expects during standard device fabrication. Once under high-vacuum (2×10−<sup>8</sup> mbar) in the SHeM, a real-space image is taken of the as-prepared sample (Figure 2a) which demonstrates it is difficult to differentiate between MoS<sup>2</sup> and hBN, with the surrounding SiO<sup>2</sup> appearing less intense. Upon heating to 240 ◦C for 8 hours, another real-space image (figure 2b) reveals a stark difference in contrast between MoS2, hBN, and SiO2. The MoS<sup>2</sup> and hBN display significantly more intense scattering than the surrounding SiO<sup>2</sup> which can be quantitatively compared in Figure 2c. Having demonstrated the sensitivity of helium atom micro-diffraction to adsorbates, one can trivially extend the method to quantitatively map adsorbate densities across real-space images. By cleaning and measuring the sample with the presented procedure, one can identify the positions of diffraction peaks in reciprocal space (∆K). This is followed by acquiring a real-space image at a diffraction peak's maximum scattering condition in reciprocal space, achieved by translation of the sample in the zaxis. One can now dose the sample environment with a given contaminant and measure a realspace image to determine if there is a change adsorbate coverage. Through dosing, adsorbation behaviour of a given sample to varying species can also be investigated by monitoring a specific diffraction peak while dosing. ### Monolayer-substrate interactions It is well documented that the choice of substrate placed under a few-layer material has a significant effect on a range of material properties. One can measure the effect of the substrate on optoelectronic or structural properties of the sample via methods such as photoluminescence spectroscopy (PL), Raman, or low-energy electron microscopy/diffraction (LEEM/D), respectively.<sup>5</sup> To demonstrate the exclusive surface sensitivity of helium atom micro-diffraction, monolayer-MoS<sup>2</sup> was placed directly onto both an SiO<sup>2</sup> substrate and few-layer hBN which is in turn on SiO2, sample geometry is shown in figure 1. It has been demonstrated that certain atomically thin buffer layers, such as hBN, ![Figure 2: Real-space images and diffraction scans of monolayer-MoS2/hBN/SiO<sup>2</sup> (red), fewlayer hBN/SiO<sup>2</sup> (black) and SiO<sup>2</sup> (blue) before and after heating (left to right) to 240 ◦C for 8 hours to remove physisorbed species. For monolayer-MoS<sup>2</sup> and hBN we see ordered scattering emerging upon cleaning in contrast to the cosine-like scattering prior. SiO<sup>2</sup> remains cosine-like post-heating. Diffraction scans were acquired at the indicated points in Panels a,b. Real-space images were taken at a single ∆K, marked with a vertical line in Panel C.](path) LaAlO<sup>3</sup> and SrTiO3, can protect the optoelectronic properties of monolayers mounted on them from strongly interacting substrates like SiO2. <sup>5</sup> Figure 3 shows that the helium scattering from monolayer MoS<sup>2</sup> becomes almost entirely disordered when mounted directly on SiO<sup>2</sup> (gray), with faint signs of structure remaining at the expected ∆K diffraction peak positions. In contrast, placing the monolayer onto a few-layer hBN buffer protects its structure and produces diffraction that matches bulk MoS<sup>2</sup> to within experimental bounds, suggesting that their surface morphologies are the same. ![Figure 3: 1D diffraction scans showing that the difference in monolayer-MoS<sup>2</sup> structure when placed directly onto SiO<sup>2</sup> (gray) versus when a few-layer hBN buffer is used between the monolayer and SiO<sup>2</sup> (red). Bulk MoS<sup>2</sup> and direct measurement of SiO<sup>2</sup> are included as references for known ordered/disordered scattering, respectively. Figure reproduced with permission from Radi´c et al. <sup>6</sup>](path) can leverage the spatial resolution and exclusive surface sensitivity of the technique to investigate inter-layer coupling strength in otherwise difficult to measure systems such as van Der Waals heterostructures where comparable techniques like LEED/M, PL or Raman struggle due to the inherent transmission of the probe through the sample. #### Thermal expansion coefficient Structural information, such as the lattice constant, can be accurately measured using 2D diffraction scans, found to be within 1%. <sup>1</sup> 2D diffraction scans offer increased accuracy over 1D scans, like those in figure 2, because more diffraction provide a statistical advantage. Initially, we measure the 2D diffraction scan on the surface of bulk MoS2, shown in figure S1, and determine the lattice constant to be 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A, matching literature values within experimental uncertainty. We extend the method to mechanically exfoliated monolayer-MoS2/hBN/SiO2, with diffraction pattern shown in figure 4, to showcase the exclusive surface sensitivity of the technique and determine that the lattice constant is <sup>3</sup>.14±0.07˚A, which is within experimental error ![Figure 4: A 2D diffraction pattern taken on monolayer-MoS2/hBN/SiO2, with the identified centres of the diffraction peaks shown as red dots. All of the lattice spacings between adjacent peaks were measured and averaged give a lattice constant 3.<sup>14</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) The thermal expansion coefficient (TEC) of 2D materials is crucial to understanding their behaviour for successful integration into optoelectronic devices, these materials also provide direct measurement of the anharmonic vibrational modes of low-dimensional systems, 7,8 however, difficult to measure due to their optical transparency. 9,10 Using helium atom microdiffraction one can measure the lattice constant, and therefore TEC, of microscopic optically transparent materials, whether supported by a substrate or free-standing. The diffraction pattern was measured along the principle ⟨10⟩ azimuth of MoS<sup>2</sup> at temperatures from 60 −450 ◦C (the data is presented in figure 4), as with work on HAS no change in the lattice constant was observed, allowing us to put an upper bound on the expansion coefficient of 16 × 10−<sup>6</sup> K−<sup>1</sup> . Our measurement is consistent with the upper bound determined as α < 14 × 10−<sup>6</sup> K−<sup>1</sup> reported by Anemone et al.,<sup>11</sup> and α = 7.6×10−<sup>6</sup> K−<sup>1</sup> measured by Zhang et al. using micro-Raman spectroscopy.<sup>10</sup> Our method is currently limited in angular resolution (radial dimension of figure 4) by the size of the limiting aperture that lies between the sample and detector. The current configuration uses a nominal 0.5 mm diameter hole for an in-plane angular resolution ∼ 7.9 ◦ , representing a compromise between measurement time and helium signal. By changing the limiting aperture to a rectangle with dimensions 0.075 × 0.5 mm we can integrate the signal in the axis perpendicular to any change in lattice parameter, the radial direction in figure 4, while improving in-plane angular resolution. We estimate that moving from the current circular aperture to the proposed rectangular aperture decreases helium signal by a factor of 5, but improves in-plane angular resolution by a factor of 6.7 from 7.9 ◦ → 1.2 ◦ which would be sufficient to resolve the TEC over the current temperature range. Details of the current instrument's spatial and angular resolutions, alongside their definitions, can be found in section 'Experimental Methods - Helium atom micro-diffraction instrument details'. ### Electron-phonon coupling Electron-phonon coupling is a major decoherence mechanism in semiconductors, often causing electron-phonon scattering, and eventually energy dissipation, in turn harming optoelectronic device performance. Optimisation of electron-phonon coupling is therefore critical in device design, both in cases where it must be minimised, or maximised. Electron-phonon coupling is typically measured using inelastic optical techniques, such as Raman or Brillouin scattering, and are primarily limited to optically transparent materials and access to electronic coupling to optical phonons only. It has been shown that helium atom scattering is an effective tool for determining the coupling strength between 2D materials and their substrates via Debye-Waller attenuation of helium scattering.12–15 However, previous atom scattering techniques have been limited to millimetre-scale spot sizes which made devicescale samples inaccessible. Here we introduce helium atom micro-diffraction as a technique well suited to the measurement of electronphonon coupling in a range of materials where its non-damaging, surface sensitive and microscopic neutral probe can be used to access electronic couplings to both optical and acoustic phonon modes in 2D materials. The temperature dependence of reflected and diffracted helium intensities can be modelled by Debye-Waller attenuation, 12,13 which describes the increasing motion of the surface atoms with temperature and causes an increase in inelastic scattering. Thus, the ordered scattering intensity is reduced. The attenuation is described by where 2W is the D-W factor. As the helium scattering occurs from the valence electron density rather than from the ionic cores themselves, one can link the D-W factor to electron phonon coupling, as described by Al Taleb et al. <sup>13</sup> D-W attenuation has been measured previously for macro-scale samples for both LiF and bulk MoS2, however without the spatial resolution enabled by SHeM, monolayer MoS<sup>2</sup> flakes could not be measured.<sup>13</sup> ![Figure 5: Log diffracted intensity as a function of temperature for monolayer MoS<sup>2</sup> on few-layer hBN on glass. In theory all diffraction orders should yield the same D-W, and therefore electron-phonon coupling, constants but the signal-to-noise ratios of each peak are a function of relative peak heights and instrument geometry, resulting in small differences.](path) From figure 5 we can extract the D-W factor as the straight-line gradient, and from that the electron-phonon coupling constant λ, contained in table 1 for monolayer and bulk MoS2. The exponents are compared relative to each other, and to literature values for the bulk case, and found to be in agreement with the expected behaviour with parallel momentum transfer (∆K). Table 1: Debye-Waller factors and electronphonon coupling constants extracted from the data presented in figure 5 using the model in equation 1 and theoretical relations outlined by Anemone et al.,<sup>13</sup> respectively. We report the electron-phonon coupling constant in bulk MoS<sup>2</sup> as λbulk ≈ 0.51 × 10−<sup>3</sup> K−<sup>1</sup> , showing good agreement to the literature values λ ≈ 0.41 × 10−<sup>3</sup> K−<sup>1</sup> , 0.49 × 10−<sup>3</sup> K−<sup>1</sup> measured using a typical helium atom scattering instrument with few-millimetre spot size, reported by Anemone et al. <sup>13</sup> We also find λML ≈ 0.40 × 10−<sup>3</sup> K−<sup>1</sup> for ML-MoS<sup>2</sup> on fewlayer hBN. We find the electron-phonon coupling constant is ∼ 20 % smaller in the monolayer compared to bulk MoS2. By leveraging the current best reported ≈ 350 nm spot size, the method can be extended to perform spatial mapping of the Debye-Waller factor, and therefore electron-phonon coupling constant, without sample preparation or damage in thin films and delicate materials, a class of materials typically difficult to characterise using standard optical and electron beam techniques. #### Vacancy-type defect density Precise control of defect density in semiconductors is instrumental for both current and future semiconductor device development. In particular, the optoelectronic properties of two-dimensional TMD semiconductors such as MoS2, can be tuned using single-atom defects. <sup>16</sup> Applications of these materials includes catalysis<sup>17</sup> and a plethora of devices, 18–21 all of which can affected by the material's defect density. In many applications a balance must be made between a sufficiently high number of defects for the material to acquire the desired properties while the material remaining sufficiently ordered to not degrade electronic performance, <sup>22</sup> or even entirely degrade the lattice structure. However, quantification of defect densities in 2D materials remains a significant experimental challenge, where typically used methods are XPS<sup>22</sup> and STEM, with conductive AFM (CAFM) being explored recently. <sup>23</sup> As all of these methods commonly require complicated sample preparation processes, there is a characterisation shortcoming that is only going to grow more acute as devices using 2D materials start being produced on an industrial scale, and therefore enter the commercial sphere. In this work we demonstrate how helium atom micro-diffraction can be used to characterise the vacancy-type defect density on the surface of few-layer materials using data reproduced from Radi´c et al. with permission. <sup>6</sup> Three mechanically exfoliated monolayer flakes of MoS<sup>2</sup> with increasing defect densities, ranging from the intrinsic 0.1 × 10<sup>14</sup> cm−<sup>2</sup> to 1.8 × 10<sup>14</sup> cm−<sup>2</sup> , and ≈ 15 µm lateral size were produced using high-temperature annealing under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere as outlined by Zhu et al.<sup>22</sup> Exact annealing parameters, and the resulting defect densities inferred using photoluminescence spectroscopy, which has been calibrated against stoichiometric beam-line XPS, are contained in Table 2. Figure 6 shows that diffracted helium intensity decreases approximately linearly as a function of defect density within the single-vacancy limit. Detailed analysis and discussion of the results in figure 6 can be found in the full manuscript.<sup>6</sup> ![Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS<sup>2</sup> flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>](path) tion can be employed as a lab-based method to quantify point-defect density without specific sample preparation or damage. The method is agnostic to sample chemistry, and thickness, because the mechanism through which it measures defect density is purely geometric, relying upon the degree of order of the sample surface, meaning that the method can be trivially extended to any system whose macroscopic properties are mediated by surface defects or dopants. Examples of current systems include hBN,<sup>24</sup> graphene,<sup>25</sup> doped systems such as diamond,<sup>26</sup> alongside other TMDs. In conjunction with its electrostatically neutral and low energy beam, microscopic helium atom diffraction presents itself as an ideal technique for the investigation of these few-layer materials. Further improvements in the instruments' lateral resolution resolution while performing diffraction measurements, down to the previously reported ∼ 300 nm spot sizes,<sup>2</sup> would allow for sub-micron scale mapping of defect density across samples. The technique is not currently capable of acting as a standalone, stoichimetric measure of defect density, like XPS, once an accurate helium-defect scattering crosssection is calculated this will also be possible. ## Conclusion The findings reported here have demonstrated the effectiveness of helium atom microdiffraction as a powerful, and completely noninvasive characterisation tool for 2D materials. The technique's ability to probe only the outermost atomic layers with micron spatial resolution makes it uniquely suited to addressing the challenges of analysing 2D materials, where established techniques often struggle. The results and discussion have highlighted several key applications of helium atom microdiffraction, including the measurement of surface cleanliness and contamination without altering the sample, a critical feature for ensuring the integrity of materials used in device fabrication. We also investigated the impact of substrate on monolayer properties, confirming that the use of a few-layer hBN buffer can preserve the structural integrity of monolayer MoS2, whereas direct contact with SiO2 results in significant disorder.? We have also explored the potential of helium atom micro-diffraction for determining the thermal expansion coefficient and electronphonon coupling in monolayer MoS2, and have demonstrated its capability for precise structural and thermal analysis. The technique's ability to quantify defect densities without sample preparation or damage also highlights it as an obvious tool for optimizing 2D materials in ever expanding areas of application, from optoelectronics to catalysis and more. Future enhancements to helium atom microdiffraction, such as achieving sub-50 nm spatial resolution, and incorporating out-of-plane scattering capabilities, will further expand its applicability, particularly for complex heterostructures and device-grade materials. In summary, helium atom micro-diffraction promises to provide a robust and adaptable platform for characterisation and development of 2D materials, paving the way for advancements in nanotechnology and material science. ## Experimental Methods ## Helium atom micro-diffraction instrument details In the presented work we used a Scanning Helium Microscope (SHeM) with 5 µm spatial and 7.9 ◦ in-plane angular resolution at the specular condition, representing a factor of 2 improvement in spatial resolution over the first iteration published by von Jeinsen et al. <sup>1</sup> Spatial resolution is defined as the full-width at halfmaximum of the beam spot on the sample at the designed working distance. In-plane angular resolution is given by the angle subtended by the limiting detector aperture and beam spot on the sample, with the correcting factor <sup>√</sup> 1 2 applied to account for the 45◦ instrument geometry. ## Sample details Monolayers of MoS<sup>2</sup> were produced by mechanical exfoliation of a bulk MoS<sup>2</sup> crystal, purchased from 2D Semiconductors Ltd., and were deposited onto few-layer thick hexgonal boron nitride (hBN) which in turn lies on a SiO<sup>2</sup> substrate (high-precision glass microscope slide). Each monolayer measured ≈ 15 µm laterally. The increased defect density monolayer MoS<sup>2</sup> samples, presented in section , were produced the same as previously described in this section with the addition of a thermal annealing process under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere for 0.5 h, following the method as described by Zhu et al.<sup>22</sup> Varying annealing temperature and time allows for control of defect density. Exact annealing parameters, with final defect densities characterised by stoichiometric XPS, are shown in Table 2. # Acknowledgements The work was supported by EPSRC grant EP/R008272/1, Innovate UK/Ionoptika Ltd. through Knowledge Transfer Partnership 10000925. The work was performed in part at CORDE, the Collaborative R&D Environment established to provide access to physics related facilities at the Cavendish Laboratory, University of Cambridge and EPSRC award EP/T00634X/1. SML acknowledges support from EPSRC grant EP/X525686/1. # References - 1. von Jeinsen, N. A.; Lambrick, S. M.; Bergin, M.; Radi´c, A.; Liu, B.; Seremet, D.; Jardine, A. P.; Ward, D. J. 2D Helium Atom Diffraction from a Microscopic Spot. Physical Review Letters 2023, 131, 236202, Publisher: American Physical Society. - 5. Man, M. K. L.; Deckoff-Jones, S.; Winchester, A.; Shi, G.; Gupta, G.; Mohite, A. D.; Kar, S.; Kioupakis, E.; Talapatra, S.; Dani, K. M. 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Nature Materials 2024, # Helium atom micro-diffraction as a characterisation tool for 2D materials - supporting information # Bulk MoS<sup>2</sup> diffraction pattern ![Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS² flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage ![Figure 1: Reflection mode optical (top-left) and real-space SHeM (bottom) images of bulk MoS2, hBN/SiO<sup>2</sup> and monolayer MoS2/hBN/SiO<sup>2</sup> (Shaded intersection). A 2D micro diffraction measurement (top-right) was taken to identify diffraction conditions for the real space imaging. An obvious change in contrast can be seen as the instrument is configured for](path) and species that one expects during standard device fabrication. Once under high-vacuum (2×10−<sup>8</sup> mbar) in the SHeM, a real-space image is taken of the as-prepared sample (Figure 2a) which demonstrates it is difficult to differentiate between MoS<sup>2</sup> and hBN, with the surrounding SiO<sup>2</sup> appearing less intense. Upon heating to 240 ◦C for 8 hours, another real-space image (figure 2b) reveals a stark difference in contrast between MoS2, hBN, and SiO2. The MoS<sup>2</sup> and hBN display significantly more intense scattering than the surrounding SiO<sup>2</sup> which can be quantitatively compared in Figure 2c. Having demonstrated the sensitivity of helium atom micro-diffraction to adsorbates, one can trivially extend the method to quantitatively map adsorbate densities across real-space images. By cleaning and measuring the sample with the presented procedure, one can identify the positions of diffraction peaks in reciprocal space (∆K). This is followed by acquiring a real-space image at a diffraction peak's maximum scattering condition in reciprocal space, achieved by translation of the sample in the zaxis. One can now dose the sample environment with a given contaminant and measure a realspace image to determine if there is a change adsorbate coverage. Through dosing, adsorbation behaviour of a given sample to varying species can also be investigated by monitoring a specific diffraction peak while dosing. ### Monolayer-substrate interactions It is well documented that the choice of substrate placed under a few-layer material has a significant effect on a range of material properties. One can measure the effect of the substrate on optoelectronic or structural properties of the sample via methods such as photoluminescence spectroscopy (PL), Raman, or low-energy electron microscopy/diffraction (LEEM/D), respectively.<sup>5</sup> To demonstrate the exclusive surface sensitivity of helium atom micro-diffraction, monolayer-MoS<sup>2</sup> was placed directly onto both an SiO<sup>2</sup> substrate and few-layer hBN which is in turn on SiO2, sample geometry is shown in figure 1. It has been demonstrated that certain atomically thin buffer layers, such as hBN, ![Figure 2: Real-space images and diffraction scans of monolayer-MoS2/hBN/SiO<sup>2</sup> (red), fewlayer hBN/SiO<sup>2</sup> (black) and SiO<sup>2</sup> (blue) before and after heating (left to right) to 240 ◦C for 8 hours to remove physisorbed species. For monolayer-MoS<sup>2</sup> and hBN we see ordered scattering emerging upon cleaning in contrast to the cosine-like scattering prior. SiO<sup>2</sup> remains cosine-like post-heating. Diffraction scans were acquired at the indicated points in Panels a,b. Real-space images were taken at a single ∆K, marked with a vertical line in Panel C.](path) LaAlO<sup>3</sup> and SrTiO3, can protect the optoelectronic properties of monolayers mounted on them from strongly interacting substrates like SiO2. <sup>5</sup> Figure 3 shows that the helium scattering from monolayer MoS<sup>2</sup> becomes almost entirely disordered when mounted directly on SiO<sup>2</sup> (gray), with faint signs of structure remaining at the expected ∆K diffraction peak positions. In contrast, placing the monolayer onto a few-layer hBN buffer protects its structure and produces diffraction that matches bulk MoS<sup>2</sup> to within experimental bounds, suggesting that their surface morphologies are the same. ![Figure 3: 1D diffraction scans showing that the difference in monolayer-MoS<sup>2</sup> structure when placed directly onto SiO<sup>2</sup> (gray) versus when a few-layer hBN buffer is used between the monolayer and SiO<sup>2</sup> (red). Bulk MoS<sup>2</sup> and direct measurement of SiO<sup>2</sup> are included as references for known ordered/disordered scattering, respectively. Figure reproduced with permission from Radi´c et al. <sup>6</sup>](path) can leverage the spatial resolution and exclusive surface sensitivity of the technique to investigate inter-layer coupling strength in otherwise difficult to measure systems such as van Der Waals heterostructures where comparable techniques like LEED/M, PL or Raman struggle due to the inherent transmission of the probe through the sample. #### Thermal expansion coefficient Structural information, such as the lattice constant, can be accurately measured using 2D diffraction scans, found to be within 1%. <sup>1</sup> 2D diffraction scans offer increased accuracy over 1D scans, like those in figure 2, because more diffraction provide a statistical advantage. Initially, we measure the 2D diffraction scan on the surface of bulk MoS2, shown in figure S1, and determine the lattice constant to be 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A, matching literature values within experimental uncertainty. We extend the method to mechanically exfoliated monolayer-MoS2/hBN/SiO2, with diffraction pattern shown in figure 4, to showcase the exclusive surface sensitivity of the technique and determine that the lattice constant is <sup>3</sup>.14±0.07˚A, which is within experimental error ![Figure 4: A 2D diffraction pattern taken on monolayer-MoS2/hBN/SiO2, with the identified centres of the diffraction peaks shown as red dots. All of the lattice spacings between adjacent peaks were measured and averaged give a lattice constant 3.<sup>14</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) The thermal expansion coefficient (TEC) of 2D materials is crucial to understanding their behaviour for successful integration into optoelectronic devices, these materials also provide direct measurement of the anharmonic vibrational modes of low-dimensional systems, 7,8 however, difficult to measure due to their optical transparency. 9,10 Using helium atom microdiffraction one can measure the lattice constant, and therefore TEC, of microscopic optically transparent materials, whether supported by a substrate or free-standing. The diffraction pattern was measured along the principle ⟨10⟩ azimuth of MoS<sup>2</sup> at temperatures from 60 −450 ◦C (the data is presented in figure 4), as with work on HAS no change in the lattice constant was observed, allowing us to put an upper bound on the expansion coefficient of 16 × 10−<sup>6</sup> K−<sup>1</sup> . Our measurement is consistent with the upper bound determined as α < 14 × 10−<sup>6</sup> K−<sup>1</sup> reported by Anemone et al.,<sup>11</sup> and α = 7.6×10−<sup>6</sup> K−<sup>1</sup> measured by Zhang et al. using micro-Raman spectroscopy.<sup>10</sup> Our method is currently limited in angular resolution (radial dimension of figure 4) by the size of the limiting aperture that lies between the sample and detector. The current configuration uses a nominal 0.5 mm diameter hole for an in-plane angular resolution ∼ 7.9 ◦ , representing a compromise between measurement time and helium signal. By changing the limiting aperture to a rectangle with dimensions 0.075 × 0.5 mm we can integrate the signal in the axis perpendicular to any change in lattice parameter, the radial direction in figure 4, while improving in-plane angular resolution. We estimate that moving from the current circular aperture to the proposed rectangular aperture decreases helium signal by a factor of 5, but improves in-plane angular resolution by a factor of 6.7 from 7.9 ◦ → 1.2 ◦ which would be sufficient to resolve the TEC over the current temperature range. Details of the current instrument's spatial and angular resolutions, alongside their definitions, can be found in section 'Experimental Methods - Helium atom micro-diffraction instrument details'. ### Electron-phonon coupling Electron-phonon coupling is a major decoherence mechanism in semiconductors, often causing electron-phonon scattering, and eventually energy dissipation, in turn harming optoelectronic device performance. Optimisation of electron-phonon coupling is therefore critical in device design, both in cases where it must be minimised, or maximised. Electron-phonon coupling is typically measured using inelastic optical techniques, such as Raman or Brillouin scattering, and are primarily limited to optically transparent materials and access to electronic coupling to optical phonons only. It has been shown that helium atom scattering is an effective tool for determining the coupling strength between 2D materials and their substrates via Debye-Waller attenuation of helium scattering.12–15 However, previous atom scattering techniques have been limited to millimetre-scale spot sizes which made devicescale samples inaccessible. Here we introduce helium atom micro-diffraction as a technique well suited to the measurement of electronphonon coupling in a range of materials where its non-damaging, surface sensitive and microscopic neutral probe can be used to access electronic couplings to both optical and acoustic phonon modes in 2D materials. The temperature dependence of reflected and diffracted helium intensities can be modelled by Debye-Waller attenuation, 12,13 which describes the increasing motion of the surface atoms with temperature and causes an increase in inelastic scattering. Thus, the ordered scattering intensity is reduced. The attenuation is described by where 2W is the D-W factor. As the helium scattering occurs from the valence electron density rather than from the ionic cores themselves, one can link the D-W factor to electron phonon coupling, as described by Al Taleb et al. <sup>13</sup> D-W attenuation has been measured previously for macro-scale samples for both LiF and bulk MoS2, however without the spatial resolution enabled by SHeM, monolayer MoS<sup>2</sup> flakes could not be measured.<sup>13</sup> ![Figure 5: Log diffracted intensity as a function of temperature for monolayer MoS<sup>2</sup> on few-layer hBN on glass. In theory all diffraction orders should yield the same D-W, and therefore electron-phonon coupling, constants but the signal-to-noise ratios of each peak are a function of relative peak heights and instrument geometry, resulting in small differences.](path) From figure 5 we can extract the D-W factor as the straight-line gradient, and from that the electron-phonon coupling constant λ, contained in table 1 for monolayer and bulk MoS2. The exponents are compared relative to each other, and to literature values for the bulk case, and found to be in agreement with the expected behaviour with parallel momentum transfer (∆K). Table 1: Debye-Waller factors and electronphonon coupling constants extracted from the data presented in figure 5 using the model in equation 1 and theoretical relations outlined by Anemone et al.,<sup>13</sup> respectively. We report the electron-phonon coupling constant in bulk MoS<sup>2</sup> as λbulk ≈ 0.51 × 10−<sup>3</sup> K−<sup>1</sup> , showing good agreement to the literature values λ ≈ 0.41 × 10−<sup>3</sup> K−<sup>1</sup> , 0.49 × 10−<sup>3</sup> K−<sup>1</sup> measured using a typical helium atom scattering instrument with few-millimetre spot size, reported by Anemone et al. <sup>13</sup> We also find λML ≈ 0.40 × 10−<sup>3</sup> K−<sup>1</sup> for ML-MoS<sup>2</sup> on fewlayer hBN. We find the electron-phonon coupling constant is ∼ 20 % smaller in the monolayer compared to bulk MoS2. By leveraging the current best reported ≈ 350 nm spot size, the method can be extended to perform spatial mapping of the Debye-Waller factor, and therefore electron-phonon coupling constant, without sample preparation or damage in thin films and delicate materials, a class of materials typically difficult to characterise using standard optical and electron beam techniques. #### Vacancy-type defect density Precise control of defect density in semiconductors is instrumental for both current and future semiconductor device development. In particular, the optoelectronic properties of two-dimensional TMD semiconductors such as MoS2, can be tuned using single-atom defects. <sup>16</sup> Applications of these materials includes catalysis<sup>17</sup> and a plethora of devices, 18–21 all of which can affected by the material's defect density. In many applications a balance must be made between a sufficiently high number of defects for the material to acquire the desired properties while the material remaining sufficiently ordered to not degrade electronic performance, <sup>22</sup> or even entirely degrade the lattice structure. However, quantification of defect densities in 2D materials remains a significant experimental challenge, where typically used methods are XPS<sup>22</sup> and STEM, with conductive AFM (CAFM) being explored recently. <sup>23</sup> As all of these methods commonly require complicated sample preparation processes, there is a characterisation shortcoming that is only going to grow more acute as devices using 2D materials start being produced on an industrial scale, and therefore enter the commercial sphere. In this work we demonstrate how helium atom micro-diffraction can be used to characterise the vacancy-type defect density on the surface of few-layer materials using data reproduced from Radi´c et al. with permission. <sup>6</sup> Three mechanically exfoliated monolayer flakes of MoS<sup>2</sup> with increasing defect densities, ranging from the intrinsic 0.1 × 10<sup>14</sup> cm−<sup>2</sup> to 1.8 × 10<sup>14</sup> cm−<sup>2</sup> , and ≈ 15 µm lateral size were produced using high-temperature annealing under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere as outlined by Zhu et al.<sup>22</sup> Exact annealing parameters, and the resulting defect densities inferred using photoluminescence spectroscopy, which has been calibrated against stoichiometric beam-line XPS, are contained in Table 2. Figure 6 shows that diffracted helium intensity decreases approximately linearly as a function of defect density within the single-vacancy limit. Detailed analysis and discussion of the results in figure 6 can be found in the full manuscript.<sup>6</sup> ![Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS<sup>2</sup> flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>](path) tion can be employed as a lab-based method to quantify point-defect density without specific sample preparation or damage. The method is agnostic to sample chemistry, and thickness, because the mechanism through which it measures defect density is purely geometric, relying upon the degree of order of the sample surface, meaning that the method can be trivially extended to any system whose macroscopic properties are mediated by surface defects or dopants. Examples of current systems include hBN,<sup>24</sup> graphene,<sup>25</sup> doped systems such as diamond,<sup>26</sup> alongside other TMDs. In conjunction with its electrostatically neutral and low energy beam, microscopic helium atom diffraction presents itself as an ideal technique for the investigation of these few-layer materials. Further improvements in the instruments' lateral resolution resolution while performing diffraction measurements, down to the previously reported ∼ 300 nm spot sizes,<sup>2</sup> would allow for sub-micron scale mapping of defect density across samples. The technique is not currently capable of acting as a standalone, stoichimetric measure of defect density, like XPS, once an accurate helium-defect scattering crosssection is calculated this will also be possible. ## Conclusion The findings reported here have demonstrated the effectiveness of helium atom microdiffraction as a powerful, and completely noninvasive characterisation tool for 2D materials. The technique's ability to probe only the outermost atomic layers with micron spatial resolution makes it uniquely suited to addressing the challenges of analysing 2D materials, where established techniques often struggle. The results and discussion have highlighted several key applications of helium atom microdiffraction, including the measurement of surface cleanliness and contamination without altering the sample, a critical feature for ensuring the integrity of materials used in device fabrication. We also investigated the impact of substrate on monolayer properties, confirming that the use of a few-layer hBN buffer can preserve the structural integrity of monolayer MoS2, whereas direct contact with SiO2 results in significant disorder.? We have also explored the potential of helium atom micro-diffraction for determining the thermal expansion coefficient and electronphonon coupling in monolayer MoS2, and have demonstrated its capability for precise structural and thermal analysis. The technique's ability to quantify defect densities without sample preparation or damage also highlights it as an obvious tool for optimizing 2D materials in ever expanding areas of application, from optoelectronics to catalysis and more. Future enhancements to helium atom microdiffraction, such as achieving sub-50 nm spatial resolution, and incorporating out-of-plane scattering capabilities, will further expand its applicability, particularly for complex heterostructures and device-grade materials. In summary, helium atom micro-diffraction promises to provide a robust and adaptable platform for characterisation and development of 2D materials, paving the way for advancements in nanotechnology and material science. ## Experimental Methods ## Helium atom micro-diffraction instrument details In the presented work we used a Scanning Helium Microscope (SHeM) with 5 µm spatial and 7.9 ◦ in-plane angular resolution at the specular condition, representing a factor of 2 improvement in spatial resolution over the first iteration published by von Jeinsen et al. <sup>1</sup> Spatial resolution is defined as the full-width at halfmaximum of the beam spot on the sample at the designed working distance. In-plane angular resolution is given by the angle subtended by the limiting detector aperture and beam spot on the sample, with the correcting factor <sup>√</sup> 1 2 applied to account for the 45◦ instrument geometry. ## Sample details Monolayers of MoS<sup>2</sup> were produced by mechanical exfoliation of a bulk MoS<sup>2</sup> crystal, purchased from 2D Semiconductors Ltd., and were deposited onto few-layer thick hexgonal boron nitride (hBN) which in turn lies on a SiO<sup>2</sup> substrate (high-precision glass microscope slide). Each monolayer measured ≈ 15 µm laterally. The increased defect density monolayer MoS<sup>2</sup> samples, presented in section , were produced the same as previously described in this section with the addition of a thermal annealing process under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere for 0.5 h, following the method as described by Zhu et al.<sup>22</sup> Varying annealing temperature and time allows for control of defect density. Exact annealing parameters, with final defect densities characterised by stoichiometric XPS, are shown in Table 2. # Acknowledgements The work was supported by EPSRC grant EP/R008272/1, Innovate UK/Ionoptika Ltd. through Knowledge Transfer Partnership 10000925. The work was performed in part at CORDE, the Collaborative R&D Environment established to provide access to physics related facilities at the Cavendish Laboratory, University of Cambridge and EPSRC award EP/T00634X/1. SML acknowledges support from EPSRC grant EP/X525686/1. # References - 1. von Jeinsen, N. A.; Lambrick, S. M.; Bergin, M.; Radi´c, A.; Liu, B.; Seremet, D.; Jardine, A. P.; Ward, D. J. 2D Helium Atom Diffraction from a Microscopic Spot. Physical Review Letters 2023, 131, 236202, Publisher: American Physical Society. - 5. Man, M. K. L.; Deckoff-Jones, S.; Winchester, A.; Shi, G.; Gupta, G.; Mohite, A. D.; Kar, S.; Kioupakis, E.; Talapatra, S.; Dani, K. M. 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Nature Materials 2024, # Helium atom micro-diffraction as a characterisation tool for 2D materials - supporting information # Bulk MoS<sup>2</sup> diffraction pattern ![Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure 3: 1D diffraction scans showing that the difference in monolayer-MoS² structure when placed directly onto SiO² (gray) versus when a few-layer hBN buffer is used between the monolayer and SiO² (red). Bulk MoS² and direct measurement of SiO² are included as references for known ordered/disordered scattering, respectively. Figure reproduced with permission from Radi´c et al. <sup>6</sup>	# Helium atom micro-diffraction as a characterisation tool for 2D materials ## Abstract We present helium atom micro-diffraction as an ideal technique for characterization of 2D materials due to its ultimate surface sensitivity combined with sub-micron spatial resolution. Thermal energy neutral helium scatters from the valence electron density, 2 <sup>−</sup><sup>3</sup> ˚A above the ionic cores of a surface, making the technique ideal for studying 2D materials, where other approaches can struggle due to small interaction cross-sections with few-layer samples. Submicron spatial resolution is key development in neutral atom scattering to allow measurements from device-scale samples. We present measurements of monolayer-substrate interactions, thermal expansion coefficients, the electronphonon coupling constant and vacancy-type defect density on monolayer-MoS2. We also discuss extensions to the presented methods which can be immediately implemented on existing instruments to perform spatial mapping of these material properties. ## Keywords search spanning fundamental physics and physical chemistry, to applications in devices including photovoltaics, batteries and transistors. Contained within the 2D material family are numerous classes of materials whose thermal, mechanical and optoelectronic properties vary over orders of magnitude and hence find applicability across the device landscape. All 2D materials, however, share the characteristic of having a thickness of only a few angstroms. As a result, established non-contact characterization techniques struggle to measure them due to a limited interaction cross-section with the probe. Even when using relatively low-energy probe particles, such as visible photons or lowenergy electrons, 2D materials are easily penetrated. Consequently, measurements must be corrected for interactions between the probe particles and the substrate. Techniques that are genuinely sensitive to only the outermost atoms of a surface are primarily contact methods, such as STM or AFM. An alternative technique is helium atom micro-diffraction presented by von Jeinsen et al.,<sup>1</sup> which uses a thermal energy beam of neutral <sup>4</sup>He atoms. It has an ultra-low incident energy (∼ 64 meV) at ambient temperature, giving the <sup>4</sup>He a de Broglie wavelength of 0.06 nm commensurate to atomic features, giving the technique ultimate sensitivity. The low beam energy means that the probe particles scatter from the outermost electron density, with a turning point 2−<sup>3</sup> ˚A above the ionic cores of the top layer atoms. Therefore, the atoms cannot penetrate the sample and interact directly with the bulk or substrate beneath the true surface atoms. Neutral <sup>4</sup>He is further advantageous as a probe because it is chemically inert and electrically neutral, making the technique entirely agnostic to sample chemistry, allowing for the measurement of a wide range of sensitive samples without need for coatings or specific sample preparation. With reported spatial resolution reaching ≈ 300 nm,<sup>2</sup> helium atom microdiffraction can also measure device scale samples. In this paper, helium atom micro-diffraction is performed with a Scanning Helium Microscope (SHeM), where we present SHeM's current imaging capabilities, in real and reciprocal space, enabling several characterisation methods of 2D materials including, but not limited to, spatial mapping of lattice parameters, contamination, monolayer-substrate interactions, the Debye-Waller factor, vacancy-type defect density and crystal phases. ## Results and discussion ### Real-space imaging SHeM's most basic analysis mode rasters the sample laterally in front of the beam to acquire real-space images, forming the basis of all advanced analysis discussed in coming sections. Figure 1 displays real-space images of bulk and monolayer MoS2/hBN/SiO2, hBN/SiO<sup>2</sup> correlated with reflection mode optical microscopy for reference. Real-spacing imaging is predominantly used for correlation with complementary microscopy techniques to target diffraction measurements, and to investigate topography of macroscopically non-trivial sample geometries.3,4 It is important to understand that each real-space image, rastered in (x, y), is taken at a single reciprocal-space, or ∆K, value, corresponding to a z-axis position in real-space. Therefore, for clean, single crystalline samples, real-space images will exhibit contrast which is diffractive and topographic simultaneously. By extension, the contribution of diffractive contrast to real-space images means that differing chemical structures or domain orientations can be qualitatively observed immediately from a real-space image. #### Surface contamination Helium atom micro-diffraction can be used to detect the presence of surface contaminants using both real-space imaging and diffraction measurements. Importantly, the properties of layered devices are adversely affected by intralayer contamination. Consequently, it is vital to be able to measure surface cleanliness, contamination, and purity directly on the specific samples intended for device construction. With a de Broglie wavelength of 0.06 nm at ambient temperatures, the technique is highly sensitive to atomic scale features, which critically includes adsorbates. With the chemically inert beam, and low kinetic energy of (64 meV), the probe neither induces reactions with, nor transfers sufficient momentum to, adsorbates to remove them from the surface. As such, the presented surface analysis measurement of the sample is entirely decoupled from any cleaning procedures one may use, contrary to other more conventional non-contact techniques which typically use probe particles of orders of magnitude more energy (e.g. photons or electrons). Contact techniques are also known to be able to desorb or displace surface species due to high scanning tip electric potentials or direct contact with contaminant species (e.g. STM and AFM) leading to insensitivity, making Helium atom micro-diffraction the best option. The sensitivity of helium scattering to adsorbates has been demonstrated in figure 2 by preparation of a sample containing monolayer-MoS<sup>2</sup> on few-layer hBN on SiO2, few-layer hBN on SiO<sup>2</sup> and exposed SiO<sup>2</sup> to provide comparison between two highly ordered, but different, and one amorphous structure when pre- and post-cleaning. The sample was prepared in a glovebox under argon atmosphere and transferred into the SHeM under nitrogen, thus representing typical adsorbate coverage ![Figure 1: Reflection mode optical (top-left) and real-space SHeM (bottom) images of bulk MoS2, hBN/SiO<sup>2</sup> and monolayer MoS2/hBN/SiO<sup>2</sup> (Shaded intersection). A 2D micro diffraction measurement (top-right) was taken to identify diffraction conditions for the real space imaging. An obvious change in contrast can be seen as the instrument is configured for](path) and species that one expects during standard device fabrication. Once under high-vacuum (2×10−<sup>8</sup> mbar) in the SHeM, a real-space image is taken of the as-prepared sample (Figure 2a) which demonstrates it is difficult to differentiate between MoS<sup>2</sup> and hBN, with the surrounding SiO<sup>2</sup> appearing less intense. Upon heating to 240 ◦C for 8 hours, another real-space image (figure 2b) reveals a stark difference in contrast between MoS2, hBN, and SiO2. The MoS<sup>2</sup> and hBN display significantly more intense scattering than the surrounding SiO<sup>2</sup> which can be quantitatively compared in Figure 2c. Having demonstrated the sensitivity of helium atom micro-diffraction to adsorbates, one can trivially extend the method to quantitatively map adsorbate densities across real-space images. By cleaning and measuring the sample with the presented procedure, one can identify the positions of diffraction peaks in reciprocal space (∆K). This is followed by acquiring a real-space image at a diffraction peak's maximum scattering condition in reciprocal space, achieved by translation of the sample in the zaxis. One can now dose the sample environment with a given contaminant and measure a realspace image to determine if there is a change adsorbate coverage. Through dosing, adsorbation behaviour of a given sample to varying species can also be investigated by monitoring a specific diffraction peak while dosing. ### Monolayer-substrate interactions It is well documented that the choice of substrate placed under a few-layer material has a significant effect on a range of material properties. One can measure the effect of the substrate on optoelectronic or structural properties of the sample via methods such as photoluminescence spectroscopy (PL), Raman, or low-energy electron microscopy/diffraction (LEEM/D), respectively.<sup>5</sup> To demonstrate the exclusive surface sensitivity of helium atom micro-diffraction, monolayer-MoS<sup>2</sup> was placed directly onto both an SiO<sup>2</sup> substrate and few-layer hBN which is in turn on SiO2, sample geometry is shown in figure 1. It has been demonstrated that certain atomically thin buffer layers, such as hBN, ![Figure 2: Real-space images and diffraction scans of monolayer-MoS2/hBN/SiO<sup>2</sup> (red), fewlayer hBN/SiO<sup>2</sup> (black) and SiO<sup>2</sup> (blue) before and after heating (left to right) to 240 ◦C for 8 hours to remove physisorbed species. For monolayer-MoS<sup>2</sup> and hBN we see ordered scattering emerging upon cleaning in contrast to the cosine-like scattering prior. SiO<sup>2</sup> remains cosine-like post-heating. Diffraction scans were acquired at the indicated points in Panels a,b. Real-space images were taken at a single ∆K, marked with a vertical line in Panel C.](path) LaAlO<sup>3</sup> and SrTiO3, can protect the optoelectronic properties of monolayers mounted on them from strongly interacting substrates like SiO2. <sup>5</sup> Figure 3 shows that the helium scattering from monolayer MoS<sup>2</sup> becomes almost entirely disordered when mounted directly on SiO<sup>2</sup> (gray), with faint signs of structure remaining at the expected ∆K diffraction peak positions. In contrast, placing the monolayer onto a few-layer hBN buffer protects its structure and produces diffraction that matches bulk MoS<sup>2</sup> to within experimental bounds, suggesting that their surface morphologies are the same. ![Figure 3: 1D diffraction scans showing that the difference in monolayer-MoS<sup>2</sup> structure when placed directly onto SiO<sup>2</sup> (gray) versus when a few-layer hBN buffer is used between the monolayer and SiO<sup>2</sup> (red). Bulk MoS<sup>2</sup> and direct measurement of SiO<sup>2</sup> are included as references for known ordered/disordered scattering, respectively. Figure reproduced with permission from Radi´c et al. <sup>6</sup>](path) can leverage the spatial resolution and exclusive surface sensitivity of the technique to investigate inter-layer coupling strength in otherwise difficult to measure systems such as van Der Waals heterostructures where comparable techniques like LEED/M, PL or Raman struggle due to the inherent transmission of the probe through the sample. #### Thermal expansion coefficient Structural information, such as the lattice constant, can be accurately measured using 2D diffraction scans, found to be within 1%. <sup>1</sup> 2D diffraction scans offer increased accuracy over 1D scans, like those in figure 2, because more diffraction provide a statistical advantage. Initially, we measure the 2D diffraction scan on the surface of bulk MoS2, shown in figure S1, and determine the lattice constant to be 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A, matching literature values within experimental uncertainty. We extend the method to mechanically exfoliated monolayer-MoS2/hBN/SiO2, with diffraction pattern shown in figure 4, to showcase the exclusive surface sensitivity of the technique and determine that the lattice constant is <sup>3</sup>.14±0.07˚A, which is within experimental error ![Figure 4: A 2D diffraction pattern taken on monolayer-MoS2/hBN/SiO2, with the identified centres of the diffraction peaks shown as red dots. All of the lattice spacings between adjacent peaks were measured and averaged give a lattice constant 3.<sup>14</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) The thermal expansion coefficient (TEC) of 2D materials is crucial to understanding their behaviour for successful integration into optoelectronic devices, these materials also provide direct measurement of the anharmonic vibrational modes of low-dimensional systems, 7,8 however, difficult to measure due to their optical transparency. 9,10 Using helium atom microdiffraction one can measure the lattice constant, and therefore TEC, of microscopic optically transparent materials, whether supported by a substrate or free-standing. The diffraction pattern was measured along the principle ⟨10⟩ azimuth of MoS<sup>2</sup> at temperatures from 60 −450 ◦C (the data is presented in figure 4), as with work on HAS no change in the lattice constant was observed, allowing us to put an upper bound on the expansion coefficient of 16 × 10−<sup>6</sup> K−<sup>1</sup> . Our measurement is consistent with the upper bound determined as α < 14 × 10−<sup>6</sup> K−<sup>1</sup> reported by Anemone et al.,<sup>11</sup> and α = 7.6×10−<sup>6</sup> K−<sup>1</sup> measured by Zhang et al. using micro-Raman spectroscopy.<sup>10</sup> Our method is currently limited in angular resolution (radial dimension of figure 4) by the size of the limiting aperture that lies between the sample and detector. The current configuration uses a nominal 0.5 mm diameter hole for an in-plane angular resolution ∼ 7.9 ◦ , representing a compromise between measurement time and helium signal. By changing the limiting aperture to a rectangle with dimensions 0.075 × 0.5 mm we can integrate the signal in the axis perpendicular to any change in lattice parameter, the radial direction in figure 4, while improving in-plane angular resolution. We estimate that moving from the current circular aperture to the proposed rectangular aperture decreases helium signal by a factor of 5, but improves in-plane angular resolution by a factor of 6.7 from 7.9 ◦ → 1.2 ◦ which would be sufficient to resolve the TEC over the current temperature range. Details of the current instrument's spatial and angular resolutions, alongside their definitions, can be found in section 'Experimental Methods - Helium atom micro-diffraction instrument details'. ### Electron-phonon coupling Electron-phonon coupling is a major decoherence mechanism in semiconductors, often causing electron-phonon scattering, and eventually energy dissipation, in turn harming optoelectronic device performance. Optimisation of electron-phonon coupling is therefore critical in device design, both in cases where it must be minimised, or maximised. Electron-phonon coupling is typically measured using inelastic optical techniques, such as Raman or Brillouin scattering, and are primarily limited to optically transparent materials and access to electronic coupling to optical phonons only. It has been shown that helium atom scattering is an effective tool for determining the coupling strength between 2D materials and their substrates via Debye-Waller attenuation of helium scattering.12–15 However, previous atom scattering techniques have been limited to millimetre-scale spot sizes which made devicescale samples inaccessible. Here we introduce helium atom micro-diffraction as a technique well suited to the measurement of electronphonon coupling in a range of materials where its non-damaging, surface sensitive and microscopic neutral probe can be used to access electronic couplings to both optical and acoustic phonon modes in 2D materials. The temperature dependence of reflected and diffracted helium intensities can be modelled by Debye-Waller attenuation, 12,13 which describes the increasing motion of the surface atoms with temperature and causes an increase in inelastic scattering. Thus, the ordered scattering intensity is reduced. The attenuation is described by where 2W is the D-W factor. As the helium scattering occurs from the valence electron density rather than from the ionic cores themselves, one can link the D-W factor to electron phonon coupling, as described by Al Taleb et al. <sup>13</sup> D-W attenuation has been measured previously for macro-scale samples for both LiF and bulk MoS2, however without the spatial resolution enabled by SHeM, monolayer MoS<sup>2</sup> flakes could not be measured.<sup>13</sup> ![Figure 5: Log diffracted intensity as a function of temperature for monolayer MoS<sup>2</sup> on few-layer hBN on glass. In theory all diffraction orders should yield the same D-W, and therefore electron-phonon coupling, constants but the signal-to-noise ratios of each peak are a function of relative peak heights and instrument geometry, resulting in small differences.](path) From figure 5 we can extract the D-W factor as the straight-line gradient, and from that the electron-phonon coupling constant λ, contained in table 1 for monolayer and bulk MoS2. The exponents are compared relative to each other, and to literature values for the bulk case, and found to be in agreement with the expected behaviour with parallel momentum transfer (∆K). Table 1: Debye-Waller factors and electronphonon coupling constants extracted from the data presented in figure 5 using the model in equation 1 and theoretical relations outlined by Anemone et al.,<sup>13</sup> respectively. We report the electron-phonon coupling constant in bulk MoS<sup>2</sup> as λbulk ≈ 0.51 × 10−<sup>3</sup> K−<sup>1</sup> , showing good agreement to the literature values λ ≈ 0.41 × 10−<sup>3</sup> K−<sup>1</sup> , 0.49 × 10−<sup>3</sup> K−<sup>1</sup> measured using a typical helium atom scattering instrument with few-millimetre spot size, reported by Anemone et al. <sup>13</sup> We also find λML ≈ 0.40 × 10−<sup>3</sup> K−<sup>1</sup> for ML-MoS<sup>2</sup> on fewlayer hBN. We find the electron-phonon coupling constant is ∼ 20 % smaller in the monolayer compared to bulk MoS2. By leveraging the current best reported ≈ 350 nm spot size, the method can be extended to perform spatial mapping of the Debye-Waller factor, and therefore electron-phonon coupling constant, without sample preparation or damage in thin films and delicate materials, a class of materials typically difficult to characterise using standard optical and electron beam techniques. #### Vacancy-type defect density Precise control of defect density in semiconductors is instrumental for both current and future semiconductor device development. In particular, the optoelectronic properties of two-dimensional TMD semiconductors such as MoS2, can be tuned using single-atom defects. <sup>16</sup> Applications of these materials includes catalysis<sup>17</sup> and a plethora of devices, 18–21 all of which can affected by the material's defect density. In many applications a balance must be made between a sufficiently high number of defects for the material to acquire the desired properties while the material remaining sufficiently ordered to not degrade electronic performance, <sup>22</sup> or even entirely degrade the lattice structure. However, quantification of defect densities in 2D materials remains a significant experimental challenge, where typically used methods are XPS<sup>22</sup> and STEM, with conductive AFM (CAFM) being explored recently. <sup>23</sup> As all of these methods commonly require complicated sample preparation processes, there is a characterisation shortcoming that is only going to grow more acute as devices using 2D materials start being produced on an industrial scale, and therefore enter the commercial sphere. In this work we demonstrate how helium atom micro-diffraction can be used to characterise the vacancy-type defect density on the surface of few-layer materials using data reproduced from Radi´c et al. with permission. <sup>6</sup> Three mechanically exfoliated monolayer flakes of MoS<sup>2</sup> with increasing defect densities, ranging from the intrinsic 0.1 × 10<sup>14</sup> cm−<sup>2</sup> to 1.8 × 10<sup>14</sup> cm−<sup>2</sup> , and ≈ 15 µm lateral size were produced using high-temperature annealing under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere as outlined by Zhu et al.<sup>22</sup> Exact annealing parameters, and the resulting defect densities inferred using photoluminescence spectroscopy, which has been calibrated against stoichiometric beam-line XPS, are contained in Table 2. Figure 6 shows that diffracted helium intensity decreases approximately linearly as a function of defect density within the single-vacancy limit. Detailed analysis and discussion of the results in figure 6 can be found in the full manuscript.<sup>6</sup> ![Figure 6: Helium atom micro-diffraction scans along a principle ⟨10⟩ azimuth of mechanically exfoliated monolayer MoS<sup>2</sup> flakes. Taking line cuts through the 2nd and 3rd order diffraction peaks shows that increasing defect density approximately linearly decreases diffracted intensity. Figure reproduced with permission from Radi´c et al.<sup>6</sup>](path) tion can be employed as a lab-based method to quantify point-defect density without specific sample preparation or damage. The method is agnostic to sample chemistry, and thickness, because the mechanism through which it measures defect density is purely geometric, relying upon the degree of order of the sample surface, meaning that the method can be trivially extended to any system whose macroscopic properties are mediated by surface defects or dopants. Examples of current systems include hBN,<sup>24</sup> graphene,<sup>25</sup> doped systems such as diamond,<sup>26</sup> alongside other TMDs. In conjunction with its electrostatically neutral and low energy beam, microscopic helium atom diffraction presents itself as an ideal technique for the investigation of these few-layer materials. Further improvements in the instruments' lateral resolution resolution while performing diffraction measurements, down to the previously reported ∼ 300 nm spot sizes,<sup>2</sup> would allow for sub-micron scale mapping of defect density across samples. The technique is not currently capable of acting as a standalone, stoichimetric measure of defect density, like XPS, once an accurate helium-defect scattering crosssection is calculated this will also be possible. ## Conclusion The findings reported here have demonstrated the effectiveness of helium atom microdiffraction as a powerful, and completely noninvasive characterisation tool for 2D materials. The technique's ability to probe only the outermost atomic layers with micron spatial resolution makes it uniquely suited to addressing the challenges of analysing 2D materials, where established techniques often struggle. The results and discussion have highlighted several key applications of helium atom microdiffraction, including the measurement of surface cleanliness and contamination without altering the sample, a critical feature for ensuring the integrity of materials used in device fabrication. We also investigated the impact of substrate on monolayer properties, confirming that the use of a few-layer hBN buffer can preserve the structural integrity of monolayer MoS2, whereas direct contact with SiO2 results in significant disorder.? We have also explored the potential of helium atom micro-diffraction for determining the thermal expansion coefficient and electronphonon coupling in monolayer MoS2, and have demonstrated its capability for precise structural and thermal analysis. The technique's ability to quantify defect densities without sample preparation or damage also highlights it as an obvious tool for optimizing 2D materials in ever expanding areas of application, from optoelectronics to catalysis and more. Future enhancements to helium atom microdiffraction, such as achieving sub-50 nm spatial resolution, and incorporating out-of-plane scattering capabilities, will further expand its applicability, particularly for complex heterostructures and device-grade materials. In summary, helium atom micro-diffraction promises to provide a robust and adaptable platform for characterisation and development of 2D materials, paving the way for advancements in nanotechnology and material science. ## Experimental Methods ## Helium atom micro-diffraction instrument details In the presented work we used a Scanning Helium Microscope (SHeM) with 5 µm spatial and 7.9 ◦ in-plane angular resolution at the specular condition, representing a factor of 2 improvement in spatial resolution over the first iteration published by von Jeinsen et al. <sup>1</sup> Spatial resolution is defined as the full-width at halfmaximum of the beam spot on the sample at the designed working distance. In-plane angular resolution is given by the angle subtended by the limiting detector aperture and beam spot on the sample, with the correcting factor <sup>√</sup> 1 2 applied to account for the 45◦ instrument geometry. ## Sample details Monolayers of MoS<sup>2</sup> were produced by mechanical exfoliation of a bulk MoS<sup>2</sup> crystal, purchased from 2D Semiconductors Ltd., and were deposited onto few-layer thick hexgonal boron nitride (hBN) which in turn lies on a SiO<sup>2</sup> substrate (high-precision glass microscope slide). Each monolayer measured ≈ 15 µm laterally. The increased defect density monolayer MoS<sup>2</sup> samples, presented in section , were produced the same as previously described in this section with the addition of a thermal annealing process under a mixed Ar/H<sup>2</sup> (95%/5%) atmosphere for 0.5 h, following the method as described by Zhu et al.<sup>22</sup> Varying annealing temperature and time allows for control of defect density. Exact annealing parameters, with final defect densities characterised by stoichiometric XPS, are shown in Table 2. # Acknowledgements The work was supported by EPSRC grant EP/R008272/1, Innovate UK/Ionoptika Ltd. through Knowledge Transfer Partnership 10000925. The work was performed in part at CORDE, the Collaborative R&D Environment established to provide access to physics related facilities at the Cavendish Laboratory, University of Cambridge and EPSRC award EP/T00634X/1. SML acknowledges support from EPSRC grant EP/X525686/1. # References - 1. von Jeinsen, N. A.; Lambrick, S. M.; Bergin, M.; Radi´c, A.; Liu, B.; Seremet, D.; Jardine, A. P.; Ward, D. J. 2D Helium Atom Diffraction from a Microscopic Spot. Physical Review Letters 2023, 131, 236202, Publisher: American Physical Society. - 5. Man, M. K. L.; Deckoff-Jones, S.; Winchester, A.; Shi, G.; Gupta, G.; Mohite, A. D.; Kar, S.; Kioupakis, E.; Talapatra, S.; Dani, K. M. 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Nature Materials 2024, # Helium atom micro-diffraction as a characterisation tool for 2D materials - supporting information # Bulk MoS<sup>2</sup> diffraction pattern ![Figure S1: A 2D diffraction pattern taken on the surface of bulk MoS<sup>2</sup> from which the lattice constant is measured as 3.<sup>15</sup> <sup>±</sup> <sup>0</sup>.07˚A.](path) # Temperature dependence of diffracted intensity in monolayer MoS<sup>2</sup> ![Figure S2: Micro helium atom diffraction scans of an MoS<sup>2</sup> monolayer with intrinsic defect density taken along the principle ⟨10⟩ azimuth as a function of temperature. A clear inverse relationship between diffracted intensity and temperature is visible, the Debye-Waller factor, followed by the electron-phonon couplig constant, can be extracted as shown in figure 5, with exact values for all diffraction peaks in table 1.](path)
Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure 1*: XRD pattern of dip coated samples of precursor 4	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. Morphology of dip coated samples was investigated with field emission scanning electron microscopy (FESEM). For rough one-layer samples, the layer thickness is that thin that the topography of substrate (FTO) can mostly be seen (see figure 6). Also, the film is not constant but showing cracks and bare FTO substrate at some places (see red circle figure 6 a and b). The samples show two different types of morphology. There is a worm-like the structure of particles (orange rectangles figure 6a and 6c) as well as round shaped particles (green circles/ rectangle figure c). They can be either in separate spaces like figure 28 shows or in between the worm-like structure one can find round shaped particles (see figure 6 c). Further investigation of the fraction of single round shaped and single worm like shaped particles could give information about their relevance to the PEC performance. Particle sizes of round shaped particles are from around 70nm of large particles to 20nm for small ones. Samples that show higher photocurrent density also show round shaped particles that display a bigger size, from 90nm up to 250nm (see figure 7). ![Figure 6: a. SEM Image of hematite film made of iron oleate precursor the green circles and orange rectangles indicate round shaped and worm-like shaped particles, respectively, b. Bare FTO substrate, c. SEM picture is showing worm like shaped (orange rectangle) and round shaped (green box) particles separately.](path) ![Figure7: Round shaped particles with a bigger particle size that can be found with samples of higher PEC performance](path) A second layer deposited on the rough samples leads to a higher thickness of the film. Therefore, the topography of the substrate cannot be seen anymore. Morphology on the surface does not change, still, there are round shaped and also worm-like particles. For spin-coated samples, the morphology is the same as for dip coated samples, consisting of worm-like and also round formed particles, single and even mixed. 3.2.3. Crystallographic properties of hematite thin films prepared from iron salt stearic acid precursor The XRD - patterns reveal the presence of hematite, no other phase could be identified. Compared to the XRD - trends of samples of iron oleate precursor, the intensity of the hematite phase is higher here. The [110] peak shows higher intensity than the [104] peak, which is the average peak of highest intensity for hematite (see figure 8). Therefore, there should be a preferred orientation of crystals along this crystallographic direction. It has also been described elsewhere for hematite films [5]. The calculated crystallite size for one and two-layer samples is not different, around 24 nm to 26 nm (see table 3). ![Figure 8: XRD patterns for dip coated samples of stearic acid precursor](path) #### 3.2.4. Morphology of hematite thin films prepared from iron salt stearic acid precursor Morphology of the samples investigated with SEM shows a very thin layer of worm-like shaped particles on FTO substrate for one-layer samples. There are no round shaped particles, the topography of FTO substrate can be seen (see figure 9a). For two-layer samples, the film thickness increases as the substrate topography are not visible anymore (see figure 9b). Still, there is the only worm like shaped particles that seem to emerge from the bulk. They show an aspect ratio between two and three. ![Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor](path) Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure 3: XRR patterns for samples of precursor 3 (a) and one (b) and two (c) layer samples of precursor 4, the red arrows in the insets indicate the increase of roughness of samples with higher decay.	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. #### 3.2 Crystallographic and morphological properties of nanoparticulate hematite films #### 3.2.1. Crystallographic properties of hematite thin films prepared from iron salt oleic acid precursor Characteristic rough samples of the weight dependent study and two-layer samples of precursor 3 and precursor four were investigated with x-ray diffraction to determine the crystallographic structure and crystallite size. The XRD patterns also show the tin oxide peaks of the substrate beside the peaks of hematite. No other phase could be identified (see figures 4 and 5). Samples with a higher weight of organic film show slightly increased peak intensity and also for two-layer samples the peak intensity is getting bigger. The crystallite sizes for rough one-layer dip coated samples of precursor three do not differ (see table 1). But for two-layer samples, the size increases slightly. On the contrary one - layer dip coated samples of precursor four show differences in crystallite sizes from 25 nm to 34 nm. By deposition of a second layer, the crystallite size also increases. The photocurrent density of samples of precursor 4 is higher with a bigger crystallite size. Furthermore, a one-layer sample with the same crystallite size of a two-layer sample displays the same current density (see table 2). The increase of crystallite size for two-layer samples can be explained with crystallite growth of the first layer due to additional heat treatment. ![Figure 4: XRD pattern of dip coated samples of precursor 3](path) ![Figure 1: XRD pattern of dip coated samples of precursor 4](path) Morphology of dip coated samples was investigated with field emission scanning electron microscopy (FESEM). For rough one-layer samples, the layer thickness is that thin that the topography of substrate (FTO) can mostly be seen (see figure 6). Also, the film is not constant but showing cracks and bare FTO substrate at some places (see red circle figure 6 a and b). The samples show two different types of morphology. There is a worm-like the structure of particles (orange rectangles figure 6a and 6c) as well as round shaped particles (green circles/ rectangle figure c). They can be either in separate spaces like figure 28 shows or in between the worm-like structure one can find round shaped particles (see figure 6 c). Further investigation of the fraction of single round shaped and single worm like shaped particles could give information about their relevance to the PEC performance. Particle sizes of round shaped particles are from around 70nm of large particles to 20nm for small ones. Samples that show higher photocurrent density also show round shaped particles that display a bigger size, from 90nm up to 250nm (see figure 7). ![Figure 6: a. SEM Image of hematite film made of iron oleate precursor the green circles and orange rectangles indicate round shaped and worm-like shaped particles, respectively, b. Bare FTO substrate, c. SEM picture is showing worm like shaped (orange rectangle) and round shaped (green box) particles separately.](path) ![Figure7: Round shaped particles with a bigger particle size that can be found with samples of higher PEC performance](path) A second layer deposited on the rough samples leads to a higher thickness of the film. Therefore, the topography of the substrate cannot be seen anymore. Morphology on the surface does not change, still, there are round shaped and also worm-like particles. For spin-coated samples, the morphology is the same as for dip coated samples, consisting of worm-like and also round formed particles, single and even mixed. 3.2.3. Crystallographic properties of hematite thin films prepared from iron salt stearic acid precursor The XRD - patterns reveal the presence of hematite, no other phase could be identified. Compared to the XRD - trends of samples of iron oleate precursor, the intensity of the hematite phase is higher here. The [110] peak shows higher intensity than the [104] peak, which is the average peak of highest intensity for hematite (see figure 8). Therefore, there should be a preferred orientation of crystals along this crystallographic direction. It has also been described elsewhere for hematite films [5]. The calculated crystallite size for one and two-layer samples is not different, around 24 nm to 26 nm (see table 3). ![Figure 8: XRD patterns for dip coated samples of stearic acid precursor](path) #### 3.2.4. Morphology of hematite thin films prepared from iron salt stearic acid precursor Morphology of the samples investigated with SEM shows a very thin layer of worm-like shaped particles on FTO substrate for one-layer samples. There are no round shaped particles, the topography of FTO substrate can be seen (see figure 9a). For two-layer samples, the film thickness increases as the substrate topography are not visible anymore (see figure 9b). Still, there is the only worm like shaped particles that seem to emerge from the bulk. They show an aspect ratio between two and three. ![Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor](path) Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure 8: XRD patterns for dip coated samples of stearic acid precursor	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. ![Figure1: a) XRR pattern of a smooth hematite film on FTO substrate, b): XRR pattern of a tungsten oxide film on Titania substrate](path) Therefore, the aim is to make a qualitative discussion of the measurements as a higher roughness of the surface increases the decay for angles bigger than the critical angle [1]. Unfortunately, the analysis is found to be sensitive to the adjustment of the slot between the sample and beam knife. Also, different ranges of the roughness of the samples complicate the right alignment. With the bigger size of the slot, the starting intensity of normalized curves is smaller; a shoulder comes up where there is already a decline in intensity for measurements with a lower slot. After this shoulder, the decay starts. However, the slope difference is small for a too big slot and the right size, whereas a too small slot would indicate less roughness because of less decay (see figure 2). For angles 2θ bigger than 1° there is an intensity difference for the measured samples that also should be due to different adjustments of the slot (see figure3). ![Figure 2: Different XRR decays of one sample due to various changes in the aperture between the sample and wedge](path) When comparing the measured curves (see figure 3) the difference in decline could be the difference of roughness of samples, still influenced by the adjustment of the slot. Lines can describe the different decays with the same slope that is taken in the range 0.6°≤2θ≤0.7°. The decays are qualitatively displayed as a function of the measured value Rrms of each sample (see inset figure 3). For each set of samples, a trend of higher decay with bigger Rrms can be estimated; it is indicated with a red arrow in the insets. As scattering is very high, it is still to question if the differences in decay represent the roughness of the samples. Slight variations in the adjustment of the slot and differences in alignment of the sample before a measurement could also be the reason for small measurement differences. Furthermore, it is doubtful if the measurement method is sensitive to such high roughnesses as the samples display. Investigation of samples with differently known roughness's on flat substrates should be used to validate the XRR results. ![Figure 3: XRR patterns for samples of precursor 3 (a) and one (b) and two (c) layer samples of precursor 4, the red arrows in the insets indicate the increase of roughness of samples with higher decay.](path) #### 3.2 Crystallographic and morphological properties of nanoparticulate hematite films #### 3.2.1. Crystallographic properties of hematite thin films prepared from iron salt oleic acid precursor Characteristic rough samples of the weight dependent study and two-layer samples of precursor 3 and precursor four were investigated with x-ray diffraction to determine the crystallographic structure and crystallite size. The XRD patterns also show the tin oxide peaks of the substrate beside the peaks of hematite. No other phase could be identified (see figures 4 and 5). Samples with a higher weight of organic film show slightly increased peak intensity and also for two-layer samples the peak intensity is getting bigger. The crystallite sizes for rough one-layer dip coated samples of precursor three do not differ (see table 1). But for two-layer samples, the size increases slightly. On the contrary one - layer dip coated samples of precursor four show differences in crystallite sizes from 25 nm to 34 nm. By deposition of a second layer, the crystallite size also increases. The photocurrent density of samples of precursor 4 is higher with a bigger crystallite size. Furthermore, a one-layer sample with the same crystallite size of a two-layer sample displays the same current density (see table 2). The increase of crystallite size for two-layer samples can be explained with crystallite growth of the first layer due to additional heat treatment. ![Figure 4: XRD pattern of dip coated samples of precursor 3](path) ![Figure 1: XRD pattern of dip coated samples of precursor 4](path) Morphology of dip coated samples was investigated with field emission scanning electron microscopy (FESEM). For rough one-layer samples, the layer thickness is that thin that the topography of substrate (FTO) can mostly be seen (see figure 6). Also, the film is not constant but showing cracks and bare FTO substrate at some places (see red circle figure 6 a and b). The samples show two different types of morphology. There is a worm-like the structure of particles (orange rectangles figure 6a and 6c) as well as round shaped particles (green circles/ rectangle figure c). They can be either in separate spaces like figure 28 shows or in between the worm-like structure one can find round shaped particles (see figure 6 c). Further investigation of the fraction of single round shaped and single worm like shaped particles could give information about their relevance to the PEC performance. Particle sizes of round shaped particles are from around 70nm of large particles to 20nm for small ones. Samples that show higher photocurrent density also show round shaped particles that display a bigger size, from 90nm up to 250nm (see figure 7). ![Figure 6: a. SEM Image of hematite film made of iron oleate precursor the green circles and orange rectangles indicate round shaped and worm-like shaped particles, respectively, b. Bare FTO substrate, c. SEM picture is showing worm like shaped (orange rectangle) and round shaped (green box) particles separately.](path) ![Figure7: Round shaped particles with a bigger particle size that can be found with samples of higher PEC performance](path) A second layer deposited on the rough samples leads to a higher thickness of the film. Therefore, the topography of the substrate cannot be seen anymore. Morphology on the surface does not change, still, there are round shaped and also worm-like particles. For spin-coated samples, the morphology is the same as for dip coated samples, consisting of worm-like and also round formed particles, single and even mixed. 3.2.3. Crystallographic properties of hematite thin films prepared from iron salt stearic acid precursor The XRD - patterns reveal the presence of hematite, no other phase could be identified. Compared to the XRD - trends of samples of iron oleate precursor, the intensity of the hematite phase is higher here. The [110] peak shows higher intensity than the [104] peak, which is the average peak of highest intensity for hematite (see figure 8). Therefore, there should be a preferred orientation of crystals along this crystallographic direction. It has also been described elsewhere for hematite films [5]. The calculated crystallite size for one and two-layer samples is not different, around 24 nm to 26 nm (see table 3). ![Figure 8: XRD patterns for dip coated samples of stearic acid precursor](path) #### 3.2.4. Morphology of hematite thin films prepared from iron salt stearic acid precursor Morphology of the samples investigated with SEM shows a very thin layer of worm-like shaped particles on FTO substrate for one-layer samples. There are no round shaped particles, the topography of FTO substrate can be seen (see figure 9a). For two-layer samples, the film thickness increases as the substrate topography are not visible anymore (see figure 9b). Still, there is the only worm like shaped particles that seem to emerge from the bulk. They show an aspect ratio between two and three. ![Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor](path) Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure7: Round shaped particles with a bigger particle size that can be found with samples of higher PEC performance	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. A second layer deposited on the rough samples leads to a higher thickness of the film. Therefore, the topography of the substrate cannot be seen anymore. Morphology on the surface does not change, still, there are round shaped and also worm-like particles. For spin-coated samples, the morphology is the same as for dip coated samples, consisting of worm-like and also round formed particles, single and even mixed. 3.2.3. Crystallographic properties of hematite thin films prepared from iron salt stearic acid precursor The XRD - patterns reveal the presence of hematite, no other phase could be identified. Compared to the XRD - trends of samples of iron oleate precursor, the intensity of the hematite phase is higher here. The [110] peak shows higher intensity than the [104] peak, which is the average peak of highest intensity for hematite (see figure 8). Therefore, there should be a preferred orientation of crystals along this crystallographic direction. It has also been described elsewhere for hematite films [5]. The calculated crystallite size for one and two-layer samples is not different, around 24 nm to 26 nm (see table 3). ![Figure 8: XRD patterns for dip coated samples of stearic acid precursor](path) #### 3.2.4. Morphology of hematite thin films prepared from iron salt stearic acid precursor Morphology of the samples investigated with SEM shows a very thin layer of worm-like shaped particles on FTO substrate for one-layer samples. There are no round shaped particles, the topography of FTO substrate can be seen (see figure 9a). For two-layer samples, the film thickness increases as the substrate topography are not visible anymore (see figure 9b). Still, there is the only worm like shaped particles that seem to emerge from the bulk. They show an aspect ratio between two and three. ![Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor](path) Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure 4*: XRD pattern of dip coated samples of precursor 3	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. ![Figure1: a) XRR pattern of a smooth hematite film on FTO substrate, b): XRR pattern of a tungsten oxide film on Titania substrate](path) Therefore, the aim is to make a qualitative discussion of the measurements as a higher roughness of the surface increases the decay for angles bigger than the critical angle [1]. Unfortunately, the analysis is found to be sensitive to the adjustment of the slot between the sample and beam knife. Also, different ranges of the roughness of the samples complicate the right alignment. With the bigger size of the slot, the starting intensity of normalized curves is smaller; a shoulder comes up where there is already a decline in intensity for measurements with a lower slot. After this shoulder, the decay starts. However, the slope difference is small for a too big slot and the right size, whereas a too small slot would indicate less roughness because of less decay (see figure 2). For angles 2θ bigger than 1° there is an intensity difference for the measured samples that also should be due to different adjustments of the slot (see figure3). ![Figure 2: Different XRR decays of one sample due to various changes in the aperture between the sample and wedge](path) When comparing the measured curves (see figure 3) the difference in decline could be the difference of roughness of samples, still influenced by the adjustment of the slot. Lines can describe the different decays with the same slope that is taken in the range 0.6°≤2θ≤0.7°. The decays are qualitatively displayed as a function of the measured value Rrms of each sample (see inset figure 3). For each set of samples, a trend of higher decay with bigger Rrms can be estimated; it is indicated with a red arrow in the insets. As scattering is very high, it is still to question if the differences in decay represent the roughness of the samples. Slight variations in the adjustment of the slot and differences in alignment of the sample before a measurement could also be the reason for small measurement differences. Furthermore, it is doubtful if the measurement method is sensitive to such high roughnesses as the samples display. Investigation of samples with differently known roughness's on flat substrates should be used to validate the XRR results. ![Figure 3: XRR patterns for samples of precursor 3 (a) and one (b) and two (c) layer samples of precursor 4, the red arrows in the insets indicate the increase of roughness of samples with higher decay.](path) #### 3.2 Crystallographic and morphological properties of nanoparticulate hematite films #### 3.2.1. Crystallographic properties of hematite thin films prepared from iron salt oleic acid precursor Characteristic rough samples of the weight dependent study and two-layer samples of precursor 3 and precursor four were investigated with x-ray diffraction to determine the crystallographic structure and crystallite size. The XRD patterns also show the tin oxide peaks of the substrate beside the peaks of hematite. No other phase could be identified (see figures 4 and 5). Samples with a higher weight of organic film show slightly increased peak intensity and also for two-layer samples the peak intensity is getting bigger. The crystallite sizes for rough one-layer dip coated samples of precursor three do not differ (see table 1). But for two-layer samples, the size increases slightly. On the contrary one - layer dip coated samples of precursor four show differences in crystallite sizes from 25 nm to 34 nm. By deposition of a second layer, the crystallite size also increases. The photocurrent density of samples of precursor 4 is higher with a bigger crystallite size. Furthermore, a one-layer sample with the same crystallite size of a two-layer sample displays the same current density (see table 2). The increase of crystallite size for two-layer samples can be explained with crystallite growth of the first layer due to additional heat treatment. ![Figure 4: XRD pattern of dip coated samples of precursor 3](path) ![Figure 1: XRD pattern of dip coated samples of precursor 4](path) Morphology of dip coated samples was investigated with field emission scanning electron microscopy (FESEM). For rough one-layer samples, the layer thickness is that thin that the topography of substrate (FTO) can mostly be seen (see figure 6). Also, the film is not constant but showing cracks and bare FTO substrate at some places (see red circle figure 6 a and b). The samples show two different types of morphology. There is a worm-like the structure of particles (orange rectangles figure 6a and 6c) as well as round shaped particles (green circles/ rectangle figure c). They can be either in separate spaces like figure 28 shows or in between the worm-like structure one can find round shaped particles (see figure 6 c). Further investigation of the fraction of single round shaped and single worm like shaped particles could give information about their relevance to the PEC performance. Particle sizes of round shaped particles are from around 70nm of large particles to 20nm for small ones. Samples that show higher photocurrent density also show round shaped particles that display a bigger size, from 90nm up to 250nm (see figure 7). ![Figure 6: a. SEM Image of hematite film made of iron oleate precursor the green circles and orange rectangles indicate round shaped and worm-like shaped particles, respectively, b. Bare FTO substrate, c. SEM picture is showing worm like shaped (orange rectangle) and round shaped (green box) particles separately.](path) ![Figure7: Round shaped particles with a bigger particle size that can be found with samples of higher PEC performance](path) A second layer deposited on the rough samples leads to a higher thickness of the film. Therefore, the topography of the substrate cannot be seen anymore. Morphology on the surface does not change, still, there are round shaped and also worm-like particles. For spin-coated samples, the morphology is the same as for dip coated samples, consisting of worm-like and also round formed particles, single and even mixed. 3.2.3. Crystallographic properties of hematite thin films prepared from iron salt stearic acid precursor The XRD - patterns reveal the presence of hematite, no other phase could be identified. Compared to the XRD - trends of samples of iron oleate precursor, the intensity of the hematite phase is higher here. The [110] peak shows higher intensity than the [104] peak, which is the average peak of highest intensity for hematite (see figure 8). Therefore, there should be a preferred orientation of crystals along this crystallographic direction. It has also been described elsewhere for hematite films [5]. The calculated crystallite size for one and two-layer samples is not different, around 24 nm to 26 nm (see table 3). ![Figure 8: XRD patterns for dip coated samples of stearic acid precursor](path) #### 3.2.4. Morphology of hematite thin films prepared from iron salt stearic acid precursor Morphology of the samples investigated with SEM shows a very thin layer of worm-like shaped particles on FTO substrate for one-layer samples. There are no round shaped particles, the topography of FTO substrate can be seen (see figure 9a). For two-layer samples, the film thickness increases as the substrate topography are not visible anymore (see figure 9b). Still, there is the only worm like shaped particles that seem to emerge from the bulk. They show an aspect ratio between two and three. ![Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor](path) Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure10: XRD patterns for dip coated samples of lauric acid precursor	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Figure 6*: a. SEM Image of hematite film made of iron oleate precursor the green circles and orange rectangles indicate round shaped and worm-like shaped particles, respectively, b. Bare FTO substrate, c. SEM picture is showing worm like shaped (orange rectangle) and round shaped (green box) particles separately.	# X-ray reflectometric studies of nanoparticulate hematite films to decouple the rough and smooth behaviors of it and crystallographic and morphological properties concerning fatty acid chain length ## Abstract In this study, the use of X-Ray reflectometry technique signifies the types of rough and smooth surfaces of hematite film prepared from different fatty acid derivatives of the iron salt. Followed by this, the film morphology and crystallographic properties concerning different fatty acid chain length have been discussed. <span id="page-0-0"></span><sup>1</sup> Corresponding author: Dr. Debajeet K. Bora, E-Mail[: debajeet1@hotmail.com:](mailto:debajeet1@hotmail.com) [debajeet.bora@jainuniversity.ac.in](mailto:debajeet.bora@jainuniversity.ac.in) #### 1. Introduction In X-ray reflectometry (XRR) thickness, roughness and density information of single layer and multilayer thin films are determined. The sample is illuminated with a monochromatic X-ray beam at low grazing angles, and the specular x-ray reflectivity is measured. At very low incident angles total reflection of x-rays occurs. For angles bigger than the critical angle of total reflection the x-ray beam also penetrates the sample and intensity of reflected x-rays decreases fast with increasing incidence angle. The critical angle depends on the density of the material. For layered samples, the x-rays can be reflected at the inner interfaces and interfere with reflected waves from the surface. Therefore, intensity oscillations, called Kiessig fringes, result in the XRR pattern. The period of fringes gives information about the thickness of layers. The period decreases with higher layer thickness. \_Higher surface roughness diminishes specular reflection and leads to a more significant decay of intensity with increasing incidence angle while roughness of interfaces dampens the intensity of Kiessig fringes. XRR is usually used for layer thicknesses from 5Å to 200nm and roughness between 0Å to 20Å [1]. Considering the crystallographic structure of Hematite (α-Fe2O3), it is an iron oxide consisting of hexagonal close-packed oxygen ions and ternary charged iron ions located at two-thirds of the octahedral interstices. It is isostructural with corundum [2] and shows oxygen deficiency [3]. To ensure electroneutrality in α-Fe2O<sup>3</sup> sintered at temperatures between 450°C to 800°C oxygen vacancies are compensated by the reduction of Fe3+ to Fe2+ ions causing n-type semiconducting behavior [3]. Conduction occurs due to hopping of electrons along Fe2+-O-Fe3+ [4]. A rough surface of the photoelectrode is necessary to achieve good photocurrent values. We have already validated the roughness and thickness of the thin film using profilometric analysis in part 1 of the work. Note that, here a three-layer coated hematite thin film with roughness between 600nm to 800nm and bulk thickness up to 700nm provided photocurrent densities of 0.6mA/cm<sup>2</sup> . The spin-coated samples were found to be the same in thickness, morphology, crystallographic structure and PEC performance with dip coated ones. The primary motivation of the current investigation is to decipher the surface characteristics of hematite thin film regarding roughness and smoothness and to have a glimpse of crystallographic and morphological properties concerning the change in the chain length of fatty acid or different types of fatty acid. 2. Materials and methods: The hematite thin film synthesis and deposition techniques using the fatty acid derivatives of iron salts are already described in part 1 of this work. Here, only the characterization tools and protocols used for studying the reflectometric, crystallographic and morphological properties will be described. #### 2.1. X-ray reflectometry For measurement, a Siemens D5000 diffractometer with Cu Kα1 radiation was used. To cut off unparallel beam fractions a collimator and LiF-crystal monochromator are installed in front of the detector instead of slits and filters generally used for X-ray diffraction. The sample is fixed to the sample holder by under pressure, and a beam-knife is installed above the sample leaving only a small slot between the sample surface and beam-knife. In this way, the angular divergence of the beam is reduced, and sufficient angular resolution is achieved. Before the measurement, the sample surface has to be aligned precisely to the primary beam direction. By varying the applied voltage and current of the x-ray tube, the intensity of the x-ray beam is adjusted to a maximum count rate of 360000cps in the region of total reflection. The measurement was done in θ/2θ-configuration in a range of 0.07° ≤ 2θ ≤ 2° with a scan speed of 2s/step and an increment of 0.002°. The intensity of the reflected beam is displayed logarithmical as a function of grazing angle 2θ. The measured curves were normalized to a grazing angle of 0.722° for easier comparison. #### 2.2. X-Ray Diffraction The intensity of the diffracted x-rays is detected at an angle 2θ concerning the direction of the incident beam. In the case of constructive interference, the intensity increases significantly. Each family of planes generates an intensity peak at a defined angle θ that is related to the lattice plane distance. From this, the crystal structure of the sample can be identified. For measurement, an X'Pert High Score Plus diffractometer by Panalytical with Cu Kα1 radiation was used. The sample was fixed and leveled to a sample holder and the intensity measured at an angle 2θ of 5°≤2θ≤80° with a step size of 0.0167° and scan speed of 0.0857°/s. The measured intensity is displayed as a function of incidence angle 2θ. The phase of hematite and phase purity was verified by comparing the measured XRD pattern with a reference pattern [34]. The crystallite size dXRD was calculated by Scherrer formula using the hematite [104] peak. The exact position of the peak and the value of full width at half maximum were achieved by analysis of the diffractogram using XRD evaluation software X'PERT HighScore Plus. 2.3 Scanning electron microscopy for morphological characterization: For the investigation, a Hitachi S-4800 field emission scanning electron microscope was used. Images based on secondary electrons are taken of the samples using different magnifications from 20000 to 100000 at an acceleration voltage of 10kV to determine the morphology. ### 3.1. X-ray reflectometric studies of hematite films to decouple the surface characteristics regarding roughness and smoothness X-ray reflectometry was performed on representative rough one- and two-layer samples of precursor 3 and 4 to further investigate the thickness and roughness of the samples. Here, FTO substrate roughness is too high, which results in the suppression of Kiessig fringes. It can be seen in the XRR pattern of a smooth hematite sample with a thickness of about 60nm and roughness of 25nm (see figure 1a). For comparison in figure 1b, the XRR pattern of a 15nm thick tungsten oxide film with a roughness of 15.1 nm on Titania substrate with the roughness of 1.29nm displays Kiessig fringes. So, conclusions on the hematite layer thickness cannot be made. ![Figure7: Round shaped particles with a bigger particle size that can be found with samples of higher PEC performance](path) A second layer deposited on the rough samples leads to a higher thickness of the film. Therefore, the topography of the substrate cannot be seen anymore. Morphology on the surface does not change, still, there are round shaped and also worm-like particles. For spin-coated samples, the morphology is the same as for dip coated samples, consisting of worm-like and also round formed particles, single and even mixed. 3.2.3. Crystallographic properties of hematite thin films prepared from iron salt stearic acid precursor The XRD - patterns reveal the presence of hematite, no other phase could be identified. Compared to the XRD - trends of samples of iron oleate precursor, the intensity of the hematite phase is higher here. The [110] peak shows higher intensity than the [104] peak, which is the average peak of highest intensity for hematite (see figure 8). Therefore, there should be a preferred orientation of crystals along this crystallographic direction. It has also been described elsewhere for hematite films [5]. The calculated crystallite size for one and two-layer samples is not different, around 24 nm to 26 nm (see table 3). ![Figure 8: XRD patterns for dip coated samples of stearic acid precursor](path) #### 3.2.4. Morphology of hematite thin films prepared from iron salt stearic acid precursor Morphology of the samples investigated with SEM shows a very thin layer of worm-like shaped particles on FTO substrate for one-layer samples. There are no round shaped particles, the topography of FTO substrate can be seen (see figure 9a). For two-layer samples, the film thickness increases as the substrate topography are not visible anymore (see figure 9b). Still, there is the only worm like shaped particles that seem to emerge from the bulk. They show an aspect ratio between two and three. ![Figure 2: SEM picture of one-layer (a) and two layers (b) hematite film made by dip coating of iron stearate precursor](path) Iron stearate precursor can also be used to prepare photoactive hematite electrodes. Significant differences to hematite films of iron oleate precursor are the preferred orientation of crystals and the absence of round shaped particles. As the reaction of the precursor at preparation was not finished, a more extended heat treatment could affect the PEC performance. Also annealing temperature and dwell time should be optimized to get better information about the PEC performance of hematite samples prepared of this precursor. # 3.2.5. Crystallographic and morphological properties of hematite thin films made from an iron salt lauric acid precursor The XRD patterns for one - layer samples reveal the existence of hematite only. Compared to the XRD patterns of samples of oleic acid and stearic acid derivative of the iron precursor, the intensity of hematite peaks is more significant. Also, the intensity of [110] peak found here is higher than the average peak [104] of highest intensity revealing preferred orientation along this crystallographic direction (see figure 10). The crystallite size is in the range of 27 nm to 30 nm (see table 4). Morphology seems to consist of worm-like shaped particles but already in an advanced state of sintering. The structure of a single worm like shaped particles can still be estimated. No round shaped particles were found (see figure 11). ![Figure10: XRD patterns for dip coated samples of lauric acid precursor](path) ![Figure 11: SEM picture of hematite film made by dip coating of iron laurate precursor](path) Iron laurate precursor can be used to prepare photoactive hematite electrodes. The samples show a preferred orientation like hematite films prepared of iron stearate precursor do. Morphology also exposes wormlike shaped particles like iron oleate and stearate precursors but more sintered together. Therefore, the investigation concerning dwell time is necessary. Examinations with XRD proved the phase purity of hematite. The crystallite size is nearly the same for the samples, around 30nm. Morphology of samples consists of round shaped and worm-like shaped particles. They can be separated, or the round shaped particles are in between the worm-like shaped particles. Dip and spin coating process show no difference in achieved PEC performance, roughness, thickness or morphology of samples where the only spin coating is more affected regarding less PEC performance by the use of diluted precursor solutions. Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Thickness, roughness, and PEC performance are similar to samples of one particular iron oleate precursor. #### 4. Conclusion Films of different precursors prepared with different fatty acids show hematite phase with a preferred orientation along the [110] crystallographic direction. Morphology only reveals a worm-like shaped structure and no round shaped particles. Crystallite size is equal to values of hematite films of iron oleate precursors. Different precursor compositions had no significant influence on hematite morphology and PEC performance but provided a preferred orientation of hematite [110] crystallographic direction. Reflectometric intensity decline difference signifies that the samples exhibit different roughness's. The roughness of the samples is said to be the main factor for good photocurrent as it allows for a high semiconductor/electrolyte interface, #### Acknowledgment #### References - 3. Gardner, R. F. G.; Sweet, F.; Tanner, D. W.: The electrical properties of alpha ferric oxide-II: Ferric oxide of high purity. In: Journal of Physics and Chemistry of Solids 24, 1963, pp. 1183 – 1186. - 5. Souza, F. L.; Lopes, K. P.; Nascente, P. A. P.; Leite, E. R.: Nanostructured hematite thin films produced by spin-coating deposition solution: Application in water splitting. In: Solar Energy Materials and Solar Cells 93, 2009, pp. 362 - 368
Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. ![Fig. 1 Supercell lattice model of (a)BTO and (b)BZCT (light blue for barium atoms, orange spheres for titanium atoms, purple spheres for cobalt atoms, magenta spheres for zinc atoms, yellow spheres for oxygen atoms)](path) It is evident that doping results in a reduction in the lattice parameter of BZCT, an increase in the c lattice parameter, a slight rise in the c/a ratio, and an enhancement of the tetragonal nature of the lattice. Concurrently, the β angle, which is associated with the Zn atoms, undergoes a slight decline, and the lattice symmetry is diminished. The lattice aberration induced by (Zn, Co) doping is particularly pronounced. This is attributed to the difference in ion size between the dopant elements, Zn and Co, and the substituted ions. Additionally, the cell volume of the doped system is smaller than that of the pure BTO, indicating that the dopantelements induce lattice shrinkage. Following atomic relaxation, the optimized structure exhibits a smaller lattice volume, enhanced tetragonality, and relatively weakened symmetry, which is a contributing factor to the enhanced intrinsic polarization observed in BZCT ferroelectrics. ## 3.2. Electronic properties density of states(DOS), can reflect a multitude of physical properties and bonding characteristics. The study of the electronic energy band structure provides useful information for realizing paraelectricity and ferroelectricity, including semiconductor behavior. In order to gain a deeper understanding of the underlying mechanism of lattice distortion, we selected the 100-plane and the 110-plane, which contain dopant elements in the form of Zn and Co atoms, for a charge density analysis. This is illustrated in Fig. 2, where the upward direction corresponds to the c-axis. As illustrated in Fig. 2, the Ti and Co ions in BZCT exhibit relaxation along the c-axis direction, deviating from the Ti-O facets in comparison to the pure BTO structure. Additionally, as illustrated in Fig. 2(c), there is an overlapping region of electron densities between the Zn ions and the adjacent O ions, indicating a robust interaction between Zn and O. This observation implies the presence of covalent bonding character in the Zn-O linkage. These results in the adjacent O2 atoms of the Zn ions undergoing reverse relaxation along the C-axis, which causes a significant zigzagging of the Ti-O plane in this layer of the O2 ions and an increase in the distortion of the oxygen octahedron. Consequently, this causes an increase in the displacement of Ti atoms from the center of the oxygen octahedron, which accounts for the observed enhancement in the intrinsic polarization of the doped system. Furthermore, Fig. 2(b) illustrates that the electronic charge densities of Ti ions and Co ions overlap with the oxygen ions, forming covalent bonds. This phenomenon is analogous to the bonding observed between Zn ions and corresponding O2 ions, as depicted in Fig. 2(c). Furthermore, the nature of the bonding between the Ti ions and the O2 ions is markedly different, with no overlap of electronic layers between Ti-O2. It can thusbe postulated that the promotion of oxygen octahedral distortion by the Ti-O planar fluctuation of BZCT,caused by Co-O chemical bonding and the formation of covalent bonds between Zn-O, represents the primary factors responsible for the enhancement of ferroelectricity and the improvement of other properties of BZCT. ![Fig. 2 (100)plane charge density plots of(a)BTO,(b)BZCT, and (c) (110)plane charge density plots of BZCT.](path) To study the electronic structure of BTO perovskites and their doping modification changes, the electronic energy band structures of pure BTO and BZCT in the Brillouin zone along the high symmetry direction were calculated. The Fermi energy level was set to zero, which is shown as a gray dashed line in the figure. From Fig. 3(a), it can be seen that the CBM is located at the G-point due to the dominance of the Ti-3d state, while the VBM is located at the G-point at FL(0 eV), which is guided by the O-2p-state. The valence band tops (VBM) and conduction band bottoms (CBM) of the BTOs are located at highly symmetric G-points, indicating that the BTOs are direct bandgap semiconductors with an energy band value of 1.7918 eV, which is in close agreement with the reported values of 1.723 eV[36] and 1.778 eV[37]. Fig. 3(b) shows the electrified energy band structure of BZCT, where it is observed that the valence band top (VBM), predominantly constituted by the Ti-3d state, is situated at the X point, whereas the conduction band bottom (CBM), primarily comprising the O-2p state, is located at the Y point. This suggests that ![Fig. 3 Band structure of (a)BTO,(b)BZCT without Hubbard U and band structure of (c)BTO,(d)BZCT withHubbard U](path) The previous experimental study indicated that the band gap of BTO is approximately 3.2 eV[38]. This discrepancy can be attributed to the fact that the generalized gradient approximation (GGA) methodology employed in the calculations of p-d repulsion for cations and anions, as well as the estimation of band gaps, often results in an underestimation of the latter[39]. The observed trend of decreasing band gaps for BTO is consistent with the typicalunderestimation of density functional theory observed for another perovskite material, SrTiO3[40]. To rectify this discrepancy, we employed the Hubbard+U algorithm to compute the revised electronic energy band structure of pure BTO and BZCT. The outcomes of this calculation are illustrated in Fig. 3(c) and Fig. 3(d). Following the correction by the Hubbard+U algorithm, the electronic energy band gap of pure BTO is 3.21 eV, which is in agreement with other experimental findings [38] ,[41]. The U-added algorithm of BZCT demonstrates that its energy bandwidth bandgap is 2.34 eV, a value that is smaller than that ofpure BTO. The electrons and holes can be excited by lower electron energy, and due to its status as a direct bandgap semiconductor, the material is conducive to carrier migration. Consequently, the BZCT canbe applied to optoelectronic materials. As this paper is concerned with the comparative alterations in properties prior to and following doping, along with the underlying mechanisms, the data will be used without the inclusion of U in the subsequent investigation. ![Fig. 4 The total and partial density of states of(a) BTO-TDOS, (b) BZCT-TDOS, (c) BZCT-PDOS.](path) The total density of states (TDOS) and partial wave density of states (PDOS) for BTO and BCTO were calculated and are presented in Fig. 4. In the case of pure BTO, the energy range for the density of states (DOS) was selected to be between -6 eV and 6 eV. In the valence band (VB), the O-2p state is the primary contributor. In the conduction band (CB) region, the primary contributions are made by the Ti-3d and O-2p states. The hybridization of Ti-3d and O-2p states in the valence band (VB) and conduction band (CB) regions is a key factor contributing to the ferroelectricity observed in pure BTO[42]. Furthermore, the introduction of Zn and Co elements into the BTO cell results in the emergence of a new peak situated in close proximity to the Fermi energy level, specifically at 0 eV. The DOS plots of Fig. 4(b)(c) demonstrate that the primary contributions to this new state are Co-3d and O-2p states. The emergence of this peak results in the Fermi energy level of BCZT being situated in close proximity to the valence band. Concurrently, Co introduces a new electronic state at the base of the conduction band of BTO, thereby reducing the band gap of the energy band of BZCT. The main contributors to the valence band and the conduction band of BCTO remain the Ti-3d and O-2p states, respectively. The impact of Zn doping is primarily manifested in the peak at -6eV in the Zn-3d valence band. In conclusion, the introduction of (Zn,Co) co-doping results in the emergence of new impurity energy levels within the material, leading to a downward shift in the conduction band and a shift in the Fermi energy levels towards the valence band. This phenomenon contributes to a reduction in the energy band gap of the entire system. Furthermore, the strong hybridization between Ti-3d and O-2p orbitals, as well as between Co-4d and O-2p orbitals, suggests that this is the factor responsible for the enhancement of ferroelectricity in the BZCT materials[42]. ## 3.3 Electrical properties Fig. 5 shows the energy difference and polarization function of pure BTO and BSZT, respectively. The energy polarization curve isitted by the phenomenological Landau–Ginzburg–Devonshire theory, and the equation is as follows. where ΔGis the energy difference between the ferroelectric phase and the paraelectric phase, and α, β, γ are coefficient constants. The potential curves for both BTO and BCZT are well-fitted by the image-only Landau equation, respectively. The two minima in the double-trap potential curves correspond to the two stable polarisation states, the P+ state and the P- state,and, for each of these minima,the depth of the trap with respect to P=0 corresponds to the effective barrier for the reversal of the polarization. The P=0 state represents the paraelectric phase, which has no Ti displacement. For both cases of pure BTO and BZCT, the P+ and P- states are identical. It is generally accepted that the magnitude of the double-trap depth is proportional to the magnitude of the ferroelectric polarisation[43]. Therefore, analyzing the variation of the ferroelectric double-well potential depth will help to predict the ferroelectric phase transition and the response to the external electric field. As can be seen from Table 2, the double-well potential depth of BTO is 0.397 meV. The double-potential depth of BZCT is 0.766 meV, which is larger than that of BTO, and we predictthat it is more difficult for BZCT to achieve polarisation switching. This is due to the larger ionic shift and stronger hybridization of Co-3d with O-2p orbitals and Ti-3d with O-2p orbitals. The spontaneous polarisation intensity of BZCT is 32.41 μC/cm<sup>2</sup> , which is higher than that of pure BTO, which is 26.86 μC/cm<sup>2</sup> . The changes in the spontaneous polarisation and the depth of the traps suggest that the spontaneous ferroelectricity of BCZT is higher than that of pure BTO ferroelectric is enhanced. ![Fig. 5 Double-well potentials of (a) pure BTO and (b) BZCT. The red dots are the total energy calculated by DFT and theblue line is from the phenomenological Landau model.](path) In the realm of energy storage devices, superior performance is primarily determined by two critical metrics: enhanced energy storage density and optimal energy storage efficiency. The underpinnings of these attributes are typically reflected in the material's high spontaneous polarization and significant breakdown strength, according to the existing studies. Fig. 6 illustrates the transport properties of four distinct BTO systems, for which we have ascertained the conductivities. These calculations were grounded in the semi-empirical framework of the Boltzmann transport theory. where ƒμ(T,ε) is the Fermi-Dirac distribution function, μ is the chemical potential, and T is the temperature. When a constant relaxation time isknown, it can be shown that the conductivity at RTA and RBA can be obtained for a certain chemical potential and temperature by calculating the energy band structure. Only their conductivities have been discussed qualitatively, so their relaxation time constants have not been estimated here. ![Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT](path) between the pure BTO and doped BTO systems. A comparative analysis of the conductivity and relaxation time between the individual systems is illustrated in figure, where the horizontal coordinates are the maximum and minimum values of the chemical potential with respect to the Fermi energy level, We can see that the (Zn,Co) co-doped BZCT shows lower conductivity than the other three systems in a more stable state, and fewer carriers are transferred in the BZCT; and when the electrons in the system are in a higher energy state, i.e. the BZCT is in an excited state or there is an external energy input, such as sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, promoting carrier migration, which has good potential applications in the field of photovoltaics, When the electrons in the system are in a higher energy state, i.e., the BZCT is in an excited state or there is an external energy input, e.g., sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, which promotes carrier migration, which has good potential applications in the field of photovoltaics. This promotes carrier migration, which has good potential applications in the field of photovoltaics and is also consistentwith the previous results of electronic energy band analysis. Conductivity describes the conductivity of the material, from the overall point of view, the conductivity of BZCT is weaker than the pure BTO system, indicating that the breakdown strength of BZCT is higher than that of pure BTO system, BZCT has a higher spontaneous polarization and higher breakdown strength than that of pure BTO[45], and BZCT is expected to be used in the field of energy storage. ## 3.4 Optical properties In recent years, the effective utilization of solar energy has attracted the attention of many researchers [46, 47]. Ferroelectric photovoltaic materials have been demonstrated to exhibit an excellent photovoltaic effect, with photogenerated voltage not limited by the forbidden bandwidth (band gap) ofthe material itself. Furthermore, the photogenerated current can be regulated by the built-in electric field[48]. Accordingly, we have investigated the impact of modified ferroelectric characteristics on the optical attributes of the materials and explored the prospective applications of perovskite materials in photorefractive, optoelectronic, and solar cell(photovoltaic) domains. ![Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.](path) In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. 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Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. Wang, H. Lu, H. Huyan, S. Lindemann, S. Ryu, H. Lee, T.H. Kim, In-plane quasi-single-domain BaTiO<sup>3</sup> via interfacial symmetry engineering, Nature Communications (2021). [8] Hoffmann, Michael, J., Glaum, Julia, Genenko, Yuri, A., Albe, Karsten, Mechanisms of aging and fatigue in ferroelectrics, Materials Science & Engineering B Solid State Materials for Advanced Technology (2015). [9] H. Shen, K. Xia, P. Wang, R. Tan, The electronic, structural, ferroelectric and optical properties of strontium and zirconium co-doped BaTiO3: First-principles calculations, Solid State Communications (2022). [10] D.S. Fu, S. Hao, J.L. Li, L.S. 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Fig. 3 Band structure of (a)BTO,(b)BZCT without Hubbard U and band structure of (c)BTO,(d)BZCT withHubbard U	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. The previous experimental study indicated that the band gap of BTO is approximately 3.2 eV[38]. This discrepancy can be attributed to the fact that the generalized gradient approximation (GGA) methodology employed in the calculations of p-d repulsion for cations and anions, as well as the estimation of band gaps, often results in an underestimation of the latter[39]. The observed trend of decreasing band gaps for BTO is consistent with the typicalunderestimation of density functional theory observed for another perovskite material, SrTiO3[40]. To rectify this discrepancy, we employed the Hubbard+U algorithm to compute the revised electronic energy band structure of pure BTO and BZCT. The outcomes of this calculation are illustrated in Fig. 3(c) and Fig. 3(d). Following the correction by the Hubbard+U algorithm, the electronic energy band gap of pure BTO is 3.21 eV, which is in agreement with other experimental findings [38] ,[41]. The U-added algorithm of BZCT demonstrates that its energy bandwidth bandgap is 2.34 eV, a value that is smaller than that ofpure BTO. The electrons and holes can be excited by lower electron energy, and due to its status as a direct bandgap semiconductor, the material is conducive to carrier migration. Consequently, the BZCT canbe applied to optoelectronic materials. As this paper is concerned with the comparative alterations in properties prior to and following doping, along with the underlying mechanisms, the data will be used without the inclusion of U in the subsequent investigation. ![Fig. 4 The total and partial density of states of(a) BTO-TDOS, (b) BZCT-TDOS, (c) BZCT-PDOS.](path) The total density of states (TDOS) and partial wave density of states (PDOS) for BTO and BCTO were calculated and are presented in Fig. 4. In the case of pure BTO, the energy range for the density of states (DOS) was selected to be between -6 eV and 6 eV. In the valence band (VB), the O-2p state is the primary contributor. In the conduction band (CB) region, the primary contributions are made by the Ti-3d and O-2p states. The hybridization of Ti-3d and O-2p states in the valence band (VB) and conduction band (CB) regions is a key factor contributing to the ferroelectricity observed in pure BTO[42]. Furthermore, the introduction of Zn and Co elements into the BTO cell results in the emergence of a new peak situated in close proximity to the Fermi energy level, specifically at 0 eV. The DOS plots of Fig. 4(b)(c) demonstrate that the primary contributions to this new state are Co-3d and O-2p states. The emergence of this peak results in the Fermi energy level of BCZT being situated in close proximity to the valence band. Concurrently, Co introduces a new electronic state at the base of the conduction band of BTO, thereby reducing the band gap of the energy band of BZCT. The main contributors to the valence band and the conduction band of BCTO remain the Ti-3d and O-2p states, respectively. The impact of Zn doping is primarily manifested in the peak at -6eV in the Zn-3d valence band. In conclusion, the introduction of (Zn,Co) co-doping results in the emergence of new impurity energy levels within the material, leading to a downward shift in the conduction band and a shift in the Fermi energy levels towards the valence band. This phenomenon contributes to a reduction in the energy band gap of the entire system. Furthermore, the strong hybridization between Ti-3d and O-2p orbitals, as well as between Co-4d and O-2p orbitals, suggests that this is the factor responsible for the enhancement of ferroelectricity in the BZCT materials[42]. ## 3.3 Electrical properties Fig. 5 shows the energy difference and polarization function of pure BTO and BSZT, respectively. The energy polarization curve isitted by the phenomenological Landau–Ginzburg–Devonshire theory, and the equation is as follows. where ΔGis the energy difference between the ferroelectric phase and the paraelectric phase, and α, β, γ are coefficient constants. The potential curves for both BTO and BCZT are well-fitted by the image-only Landau equation, respectively. The two minima in the double-trap potential curves correspond to the two stable polarisation states, the P+ state and the P- state,and, for each of these minima,the depth of the trap with respect to P=0 corresponds to the effective barrier for the reversal of the polarization. The P=0 state represents the paraelectric phase, which has no Ti displacement. For both cases of pure BTO and BZCT, the P+ and P- states are identical. It is generally accepted that the magnitude of the double-trap depth is proportional to the magnitude of the ferroelectric polarisation[43]. Therefore, analyzing the variation of the ferroelectric double-well potential depth will help to predict the ferroelectric phase transition and the response to the external electric field. As can be seen from Table 2, the double-well potential depth of BTO is 0.397 meV. The double-potential depth of BZCT is 0.766 meV, which is larger than that of BTO, and we predictthat it is more difficult for BZCT to achieve polarisation switching. This is due to the larger ionic shift and stronger hybridization of Co-3d with O-2p orbitals and Ti-3d with O-2p orbitals. The spontaneous polarisation intensity of BZCT is 32.41 μC/cm<sup>2</sup> , which is higher than that of pure BTO, which is 26.86 μC/cm<sup>2</sup> . The changes in the spontaneous polarisation and the depth of the traps suggest that the spontaneous ferroelectricity of BCZT is higher than that of pure BTO ferroelectric is enhanced. ![Fig. 5 Double-well potentials of (a) pure BTO and (b) BZCT. The red dots are the total energy calculated by DFT and theblue line is from the phenomenological Landau model.](path) In the realm of energy storage devices, superior performance is primarily determined by two critical metrics: enhanced energy storage density and optimal energy storage efficiency. The underpinnings of these attributes are typically reflected in the material's high spontaneous polarization and significant breakdown strength, according to the existing studies. Fig. 6 illustrates the transport properties of four distinct BTO systems, for which we have ascertained the conductivities. These calculations were grounded in the semi-empirical framework of the Boltzmann transport theory. where ƒμ(T,ε) is the Fermi-Dirac distribution function, μ is the chemical potential, and T is the temperature. When a constant relaxation time isknown, it can be shown that the conductivity at RTA and RBA can be obtained for a certain chemical potential and temperature by calculating the energy band structure. Only their conductivities have been discussed qualitatively, so their relaxation time constants have not been estimated here. ![Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT](path) between the pure BTO and doped BTO systems. A comparative analysis of the conductivity and relaxation time between the individual systems is illustrated in figure, where the horizontal coordinates are the maximum and minimum values of the chemical potential with respect to the Fermi energy level, We can see that the (Zn,Co) co-doped BZCT shows lower conductivity than the other three systems in a more stable state, and fewer carriers are transferred in the BZCT; and when the electrons in the system are in a higher energy state, i.e. the BZCT is in an excited state or there is an external energy input, such as sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, promoting carrier migration, which has good potential applications in the field of photovoltaics, When the electrons in the system are in a higher energy state, i.e., the BZCT is in an excited state or there is an external energy input, e.g., sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, which promotes carrier migration, which has good potential applications in the field of photovoltaics. This promotes carrier migration, which has good potential applications in the field of photovoltaics and is also consistentwith the previous results of electronic energy band analysis. Conductivity describes the conductivity of the material, from the overall point of view, the conductivity of BZCT is weaker than the pure BTO system, indicating that the breakdown strength of BZCT is higher than that of pure BTO system, BZCT has a higher spontaneous polarization and higher breakdown strength than that of pure BTO[45], and BZCT is expected to be used in the field of energy storage. ## 3.4 Optical properties In recent years, the effective utilization of solar energy has attracted the attention of many researchers [46, 47]. Ferroelectric photovoltaic materials have been demonstrated to exhibit an excellent photovoltaic effect, with photogenerated voltage not limited by the forbidden bandwidth (band gap) ofthe material itself. Furthermore, the photogenerated current can be regulated by the built-in electric field[48]. Accordingly, we have investigated the impact of modified ferroelectric characteristics on the optical attributes of the materials and explored the prospective applications of perovskite materials in photorefractive, optoelectronic, and solar cell(photovoltaic) domains. ![Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.](path) In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. 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Fig. 4 The total and partial density of states of(a) BTO-TDOS, (b) BZCT-TDOS, (c) BZCT-PDOS.	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. The total density of states (TDOS) and partial wave density of states (PDOS) for BTO and BCTO were calculated and are presented in Fig. 4. In the case of pure BTO, the energy range for the density of states (DOS) was selected to be between -6 eV and 6 eV. In the valence band (VB), the O-2p state is the primary contributor. In the conduction band (CB) region, the primary contributions are made by the Ti-3d and O-2p states. The hybridization of Ti-3d and O-2p states in the valence band (VB) and conduction band (CB) regions is a key factor contributing to the ferroelectricity observed in pure BTO[42]. Furthermore, the introduction of Zn and Co elements into the BTO cell results in the emergence of a new peak situated in close proximity to the Fermi energy level, specifically at 0 eV. The DOS plots of Fig. 4(b)(c) demonstrate that the primary contributions to this new state are Co-3d and O-2p states. The emergence of this peak results in the Fermi energy level of BCZT being situated in close proximity to the valence band. Concurrently, Co introduces a new electronic state at the base of the conduction band of BTO, thereby reducing the band gap of the energy band of BZCT. The main contributors to the valence band and the conduction band of BCTO remain the Ti-3d and O-2p states, respectively. The impact of Zn doping is primarily manifested in the peak at -6eV in the Zn-3d valence band. In conclusion, the introduction of (Zn,Co) co-doping results in the emergence of new impurity energy levels within the material, leading to a downward shift in the conduction band and a shift in the Fermi energy levels towards the valence band. This phenomenon contributes to a reduction in the energy band gap of the entire system. Furthermore, the strong hybridization between Ti-3d and O-2p orbitals, as well as between Co-4d and O-2p orbitals, suggests that this is the factor responsible for the enhancement of ferroelectricity in the BZCT materials[42]. ## 3.3 Electrical properties Fig. 5 shows the energy difference and polarization function of pure BTO and BSZT, respectively. The energy polarization curve isitted by the phenomenological Landau–Ginzburg–Devonshire theory, and the equation is as follows. where ΔGis the energy difference between the ferroelectric phase and the paraelectric phase, and α, β, γ are coefficient constants. The potential curves for both BTO and BCZT are well-fitted by the image-only Landau equation, respectively. The two minima in the double-trap potential curves correspond to the two stable polarisation states, the P+ state and the P- state,and, for each of these minima,the depth of the trap with respect to P=0 corresponds to the effective barrier for the reversal of the polarization. The P=0 state represents the paraelectric phase, which has no Ti displacement. For both cases of pure BTO and BZCT, the P+ and P- states are identical. It is generally accepted that the magnitude of the double-trap depth is proportional to the magnitude of the ferroelectric polarisation[43]. Therefore, analyzing the variation of the ferroelectric double-well potential depth will help to predict the ferroelectric phase transition and the response to the external electric field. As can be seen from Table 2, the double-well potential depth of BTO is 0.397 meV. The double-potential depth of BZCT is 0.766 meV, which is larger than that of BTO, and we predictthat it is more difficult for BZCT to achieve polarisation switching. This is due to the larger ionic shift and stronger hybridization of Co-3d with O-2p orbitals and Ti-3d with O-2p orbitals. The spontaneous polarisation intensity of BZCT is 32.41 μC/cm<sup>2</sup> , which is higher than that of pure BTO, which is 26.86 μC/cm<sup>2</sup> . The changes in the spontaneous polarisation and the depth of the traps suggest that the spontaneous ferroelectricity of BCZT is higher than that of pure BTO ferroelectric is enhanced. ![Fig. 5 Double-well potentials of (a) pure BTO and (b) BZCT. The red dots are the total energy calculated by DFT and theblue line is from the phenomenological Landau model.](path) In the realm of energy storage devices, superior performance is primarily determined by two critical metrics: enhanced energy storage density and optimal energy storage efficiency. The underpinnings of these attributes are typically reflected in the material's high spontaneous polarization and significant breakdown strength, according to the existing studies. Fig. 6 illustrates the transport properties of four distinct BTO systems, for which we have ascertained the conductivities. These calculations were grounded in the semi-empirical framework of the Boltzmann transport theory. where ƒμ(T,ε) is the Fermi-Dirac distribution function, μ is the chemical potential, and T is the temperature. When a constant relaxation time isknown, it can be shown that the conductivity at RTA and RBA can be obtained for a certain chemical potential and temperature by calculating the energy band structure. Only their conductivities have been discussed qualitatively, so their relaxation time constants have not been estimated here. ![Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT](path) between the pure BTO and doped BTO systems. A comparative analysis of the conductivity and relaxation time between the individual systems is illustrated in figure, where the horizontal coordinates are the maximum and minimum values of the chemical potential with respect to the Fermi energy level, We can see that the (Zn,Co) co-doped BZCT shows lower conductivity than the other three systems in a more stable state, and fewer carriers are transferred in the BZCT; and when the electrons in the system are in a higher energy state, i.e. the BZCT is in an excited state or there is an external energy input, such as sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, promoting carrier migration, which has good potential applications in the field of photovoltaics, When the electrons in the system are in a higher energy state, i.e., the BZCT is in an excited state or there is an external energy input, e.g., sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, which promotes carrier migration, which has good potential applications in the field of photovoltaics. This promotes carrier migration, which has good potential applications in the field of photovoltaics and is also consistentwith the previous results of electronic energy band analysis. Conductivity describes the conductivity of the material, from the overall point of view, the conductivity of BZCT is weaker than the pure BTO system, indicating that the breakdown strength of BZCT is higher than that of pure BTO system, BZCT has a higher spontaneous polarization and higher breakdown strength than that of pure BTO[45], and BZCT is expected to be used in the field of energy storage. ## 3.4 Optical properties In recent years, the effective utilization of solar energy has attracted the attention of many researchers [46, 47]. Ferroelectric photovoltaic materials have been demonstrated to exhibit an excellent photovoltaic effect, with photogenerated voltage not limited by the forbidden bandwidth (band gap) ofthe material itself. Furthermore, the photogenerated current can be regulated by the built-in electric field[48]. Accordingly, we have investigated the impact of modified ferroelectric characteristics on the optical attributes of the materials and explored the prospective applications of perovskite materials in photorefractive, optoelectronic, and solar cell(photovoltaic) domains. ![Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.](path) In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. 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Fig. 5 Double-well potentials of (a) pure BTO and (b) BZCT. The red dots are the total energy calculated by DFT and theblue line is from the phenomenological Landau model.	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. In the realm of energy storage devices, superior performance is primarily determined by two critical metrics: enhanced energy storage density and optimal energy storage efficiency. The underpinnings of these attributes are typically reflected in the material's high spontaneous polarization and significant breakdown strength, according to the existing studies. Fig. 6 illustrates the transport properties of four distinct BTO systems, for which we have ascertained the conductivities. These calculations were grounded in the semi-empirical framework of the Boltzmann transport theory. where ƒμ(T,ε) is the Fermi-Dirac distribution function, μ is the chemical potential, and T is the temperature. When a constant relaxation time isknown, it can be shown that the conductivity at RTA and RBA can be obtained for a certain chemical potential and temperature by calculating the energy band structure. Only their conductivities have been discussed qualitatively, so their relaxation time constants have not been estimated here. ![Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT](path) between the pure BTO and doped BTO systems. A comparative analysis of the conductivity and relaxation time between the individual systems is illustrated in figure, where the horizontal coordinates are the maximum and minimum values of the chemical potential with respect to the Fermi energy level, We can see that the (Zn,Co) co-doped BZCT shows lower conductivity than the other three systems in a more stable state, and fewer carriers are transferred in the BZCT; and when the electrons in the system are in a higher energy state, i.e. the BZCT is in an excited state or there is an external energy input, such as sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, promoting carrier migration, which has good potential applications in the field of photovoltaics, When the electrons in the system are in a higher energy state, i.e., the BZCT is in an excited state or there is an external energy input, e.g., sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, which promotes carrier migration, which has good potential applications in the field of photovoltaics. This promotes carrier migration, which has good potential applications in the field of photovoltaics and is also consistentwith the previous results of electronic energy band analysis. Conductivity describes the conductivity of the material, from the overall point of view, the conductivity of BZCT is weaker than the pure BTO system, indicating that the breakdown strength of BZCT is higher than that of pure BTO system, BZCT has a higher spontaneous polarization and higher breakdown strength than that of pure BTO[45], and BZCT is expected to be used in the field of energy storage. ## 3.4 Optical properties In recent years, the effective utilization of solar energy has attracted the attention of many researchers [46, 47]. Ferroelectric photovoltaic materials have been demonstrated to exhibit an excellent photovoltaic effect, with photogenerated voltage not limited by the forbidden bandwidth (band gap) ofthe material itself. Furthermore, the photogenerated current can be regulated by the built-in electric field[48]. Accordingly, we have investigated the impact of modified ferroelectric characteristics on the optical attributes of the materials and explored the prospective applications of perovskite materials in photorefractive, optoelectronic, and solar cell(photovoltaic) domains. ![Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.](path) In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. 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Fig. 1 Supercell lattice model of (a)BTO and (b)BZCT (light blue for barium atoms, orange spheres for titanium atoms, purple spheres for cobalt atoms, magenta spheres for zinc atoms, yellow spheres for oxygen atoms)	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. It is evident that doping results in a reduction in the lattice parameter of BZCT, an increase in the c lattice parameter, a slight rise in the c/a ratio, and an enhancement of the tetragonal nature of the lattice. Concurrently, the β angle, which is associated with the Zn atoms, undergoes a slight decline, and the lattice symmetry is diminished. The lattice aberration induced by (Zn, Co) doping is particularly pronounced. This is attributed to the difference in ion size between the dopant elements, Zn and Co, and the substituted ions. Additionally, the cell volume of the doped system is smaller than that of the pure BTO, indicating that the dopantelements induce lattice shrinkage. Following atomic relaxation, the optimized structure exhibits a smaller lattice volume, enhanced tetragonality, and relatively weakened symmetry, which is a contributing factor to the enhanced intrinsic polarization observed in BZCT ferroelectrics. ## 3.2. Electronic properties density of states(DOS), can reflect a multitude of physical properties and bonding characteristics. The study of the electronic energy band structure provides useful information for realizing paraelectricity and ferroelectricity, including semiconductor behavior. In order to gain a deeper understanding of the underlying mechanism of lattice distortion, we selected the 100-plane and the 110-plane, which contain dopant elements in the form of Zn and Co atoms, for a charge density analysis. This is illustrated in Fig. 2, where the upward direction corresponds to the c-axis. As illustrated in Fig. 2, the Ti and Co ions in BZCT exhibit relaxation along the c-axis direction, deviating from the Ti-O facets in comparison to the pure BTO structure. Additionally, as illustrated in Fig. 2(c), there is an overlapping region of electron densities between the Zn ions and the adjacent O ions, indicating a robust interaction between Zn and O. This observation implies the presence of covalent bonding character in the Zn-O linkage. These results in the adjacent O2 atoms of the Zn ions undergoing reverse relaxation along the C-axis, which causes a significant zigzagging of the Ti-O plane in this layer of the O2 ions and an increase in the distortion of the oxygen octahedron. Consequently, this causes an increase in the displacement of Ti atoms from the center of the oxygen octahedron, which accounts for the observed enhancement in the intrinsic polarization of the doped system. Furthermore, Fig. 2(b) illustrates that the electronic charge densities of Ti ions and Co ions overlap with the oxygen ions, forming covalent bonds. This phenomenon is analogous to the bonding observed between Zn ions and corresponding O2 ions, as depicted in Fig. 2(c). Furthermore, the nature of the bonding between the Ti ions and the O2 ions is markedly different, with no overlap of electronic layers between Ti-O2. It can thusbe postulated that the promotion of oxygen octahedral distortion by the Ti-O planar fluctuation of BZCT,caused by Co-O chemical bonding and the formation of covalent bonds between Zn-O, represents the primary factors responsible for the enhancement of ferroelectricity and the improvement of other properties of BZCT. ![Fig. 2 (100)plane charge density plots of(a)BTO,(b)BZCT, and (c) (110)plane charge density plots of BZCT.](path) To study the electronic structure of BTO perovskites and their doping modification changes, the electronic energy band structures of pure BTO and BZCT in the Brillouin zone along the high symmetry direction were calculated. The Fermi energy level was set to zero, which is shown as a gray dashed line in the figure. From Fig. 3(a), it can be seen that the CBM is located at the G-point due to the dominance of the Ti-3d state, while the VBM is located at the G-point at FL(0 eV), which is guided by the O-2p-state. The valence band tops (VBM) and conduction band bottoms (CBM) of the BTOs are located at highly symmetric G-points, indicating that the BTOs are direct bandgap semiconductors with an energy band value of 1.7918 eV, which is in close agreement with the reported values of 1.723 eV[36] and 1.778 eV[37]. Fig. 3(b) shows the electrified energy band structure of BZCT, where it is observed that the valence band top (VBM), predominantly constituted by the Ti-3d state, is situated at the X point, whereas the conduction band bottom (CBM), primarily comprising the O-2p state, is located at the Y point. This suggests that ![Fig. 3 Band structure of (a)BTO,(b)BZCT without Hubbard U and band structure of (c)BTO,(d)BZCT withHubbard U](path) The previous experimental study indicated that the band gap of BTO is approximately 3.2 eV[38]. This discrepancy can be attributed to the fact that the generalized gradient approximation (GGA) methodology employed in the calculations of p-d repulsion for cations and anions, as well as the estimation of band gaps, often results in an underestimation of the latter[39]. The observed trend of decreasing band gaps for BTO is consistent with the typicalunderestimation of density functional theory observed for another perovskite material, SrTiO3[40]. To rectify this discrepancy, we employed the Hubbard+U algorithm to compute the revised electronic energy band structure of pure BTO and BZCT. The outcomes of this calculation are illustrated in Fig. 3(c) and Fig. 3(d). Following the correction by the Hubbard+U algorithm, the electronic energy band gap of pure BTO is 3.21 eV, which is in agreement with other experimental findings [38] ,[41]. The U-added algorithm of BZCT demonstrates that its energy bandwidth bandgap is 2.34 eV, a value that is smaller than that ofpure BTO. The electrons and holes can be excited by lower electron energy, and due to its status as a direct bandgap semiconductor, the material is conducive to carrier migration. Consequently, the BZCT canbe applied to optoelectronic materials. As this paper is concerned with the comparative alterations in properties prior to and following doping, along with the underlying mechanisms, the data will be used without the inclusion of U in the subsequent investigation. ![Fig. 4 The total and partial density of states of(a) BTO-TDOS, (b) BZCT-TDOS, (c) BZCT-PDOS.](path) The total density of states (TDOS) and partial wave density of states (PDOS) for BTO and BCTO were calculated and are presented in Fig. 4. In the case of pure BTO, the energy range for the density of states (DOS) was selected to be between -6 eV and 6 eV. In the valence band (VB), the O-2p state is the primary contributor. In the conduction band (CB) region, the primary contributions are made by the Ti-3d and O-2p states. The hybridization of Ti-3d and O-2p states in the valence band (VB) and conduction band (CB) regions is a key factor contributing to the ferroelectricity observed in pure BTO[42]. Furthermore, the introduction of Zn and Co elements into the BTO cell results in the emergence of a new peak situated in close proximity to the Fermi energy level, specifically at 0 eV. The DOS plots of Fig. 4(b)(c) demonstrate that the primary contributions to this new state are Co-3d and O-2p states. The emergence of this peak results in the Fermi energy level of BCZT being situated in close proximity to the valence band. Concurrently, Co introduces a new electronic state at the base of the conduction band of BTO, thereby reducing the band gap of the energy band of BZCT. The main contributors to the valence band and the conduction band of BCTO remain the Ti-3d and O-2p states, respectively. The impact of Zn doping is primarily manifested in the peak at -6eV in the Zn-3d valence band. In conclusion, the introduction of (Zn,Co) co-doping results in the emergence of new impurity energy levels within the material, leading to a downward shift in the conduction band and a shift in the Fermi energy levels towards the valence band. This phenomenon contributes to a reduction in the energy band gap of the entire system. Furthermore, the strong hybridization between Ti-3d and O-2p orbitals, as well as between Co-4d and O-2p orbitals, suggests that this is the factor responsible for the enhancement of ferroelectricity in the BZCT materials[42]. ## 3.3 Electrical properties Fig. 5 shows the energy difference and polarization function of pure BTO and BSZT, respectively. The energy polarization curve isitted by the phenomenological Landau–Ginzburg–Devonshire theory, and the equation is as follows. where ΔGis the energy difference between the ferroelectric phase and the paraelectric phase, and α, β, γ are coefficient constants. The potential curves for both BTO and BCZT are well-fitted by the image-only Landau equation, respectively. The two minima in the double-trap potential curves correspond to the two stable polarisation states, the P+ state and the P- state,and, for each of these minima,the depth of the trap with respect to P=0 corresponds to the effective barrier for the reversal of the polarization. The P=0 state represents the paraelectric phase, which has no Ti displacement. For both cases of pure BTO and BZCT, the P+ and P- states are identical. It is generally accepted that the magnitude of the double-trap depth is proportional to the magnitude of the ferroelectric polarisation[43]. Therefore, analyzing the variation of the ferroelectric double-well potential depth will help to predict the ferroelectric phase transition and the response to the external electric field. As can be seen from Table 2, the double-well potential depth of BTO is 0.397 meV. The double-potential depth of BZCT is 0.766 meV, which is larger than that of BTO, and we predictthat it is more difficult for BZCT to achieve polarisation switching. This is due to the larger ionic shift and stronger hybridization of Co-3d with O-2p orbitals and Ti-3d with O-2p orbitals. The spontaneous polarisation intensity of BZCT is 32.41 μC/cm<sup>2</sup> , which is higher than that of pure BTO, which is 26.86 μC/cm<sup>2</sup> . The changes in the spontaneous polarisation and the depth of the traps suggest that the spontaneous ferroelectricity of BCZT is higher than that of pure BTO ferroelectric is enhanced. ![Fig. 5 Double-well potentials of (a) pure BTO and (b) BZCT. The red dots are the total energy calculated by DFT and theblue line is from the phenomenological Landau model.](path) In the realm of energy storage devices, superior performance is primarily determined by two critical metrics: enhanced energy storage density and optimal energy storage efficiency. The underpinnings of these attributes are typically reflected in the material's high spontaneous polarization and significant breakdown strength, according to the existing studies. Fig. 6 illustrates the transport properties of four distinct BTO systems, for which we have ascertained the conductivities. These calculations were grounded in the semi-empirical framework of the Boltzmann transport theory. where ƒμ(T,ε) is the Fermi-Dirac distribution function, μ is the chemical potential, and T is the temperature. When a constant relaxation time isknown, it can be shown that the conductivity at RTA and RBA can be obtained for a certain chemical potential and temperature by calculating the energy band structure. Only their conductivities have been discussed qualitatively, so their relaxation time constants have not been estimated here. ![Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT](path) between the pure BTO and doped BTO systems. A comparative analysis of the conductivity and relaxation time between the individual systems is illustrated in figure, where the horizontal coordinates are the maximum and minimum values of the chemical potential with respect to the Fermi energy level, We can see that the (Zn,Co) co-doped BZCT shows lower conductivity than the other three systems in a more stable state, and fewer carriers are transferred in the BZCT; and when the electrons in the system are in a higher energy state, i.e. the BZCT is in an excited state or there is an external energy input, such as sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, promoting carrier migration, which has good potential applications in the field of photovoltaics, When the electrons in the system are in a higher energy state, i.e., the BZCT is in an excited state or there is an external energy input, e.g., sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, which promotes carrier migration, which has good potential applications in the field of photovoltaics. This promotes carrier migration, which has good potential applications in the field of photovoltaics and is also consistentwith the previous results of electronic energy band analysis. Conductivity describes the conductivity of the material, from the overall point of view, the conductivity of BZCT is weaker than the pure BTO system, indicating that the breakdown strength of BZCT is higher than that of pure BTO system, BZCT has a higher spontaneous polarization and higher breakdown strength than that of pure BTO[45], and BZCT is expected to be used in the field of energy storage. ## 3.4 Optical properties In recent years, the effective utilization of solar energy has attracted the attention of many researchers [46, 47]. Ferroelectric photovoltaic materials have been demonstrated to exhibit an excellent photovoltaic effect, with photogenerated voltage not limited by the forbidden bandwidth (band gap) ofthe material itself. Furthermore, the photogenerated current can be regulated by the built-in electric field[48]. Accordingly, we have investigated the impact of modified ferroelectric characteristics on the optical attributes of the materials and explored the prospective applications of perovskite materials in photorefractive, optoelectronic, and solar cell(photovoltaic) domains. ![Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.](path) In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. 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Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. Wang, H. Lu, H. Huyan, S. Lindemann, S. Ryu, H. Lee, T.H. Kim, In-plane quasi-single-domain BaTiO<sup>3</sup> via interfacial symmetry engineering, Nature Communications (2021). [8] Hoffmann, Michael, J., Glaum, Julia, Genenko, Yuri, A., Albe, Karsten, Mechanisms of aging and fatigue in ferroelectrics, Materials Science & Engineering B Solid State Materials for Advanced Technology (2015). [9] H. Shen, K. Xia, P. Wang, R. Tan, The electronic, structural, ferroelectric and optical properties of strontium and zirconium co-doped BaTiO3: First-principles calculations, Solid State Communications (2022). [10] D.S. Fu, S. Hao, J.L. Li, L.S. 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Fig. 2 (100)plane charge density plots of(a)BTO,(b)BZCT, and (c) (110)plane charge density plots of BZCT.	# Theoretical Investigation of (Zn, Co) co-Doped BaTiO<sup>3</sup> for Advanced Energy and Photonic Applications ## 550025, Guizhou, China ## SAR 999077, China Abstract: In light of recent advancements in energy technology, there is an urgent need for lead-free barium titanate (BTO) -based materials that exhibit remarkable ferroelectric and photoelectric properties. Notwithstanding the considerable experimental advances, a theoretical understanding from the electron and atomic perspectives remains elusive. This study employs the generalized gradient approximation plane wave pseudopotential technique to investigate the structural, electronic, ferroelectric, and optical properties of (Zn, Co) co-doped BaTiO3(BZCT) based on density functional theory. The objective is to ascertain the extent of performance enhancement and the underlying mechanism of (Zn, Co) co-doping on barium titanate. Our findings reveal that the incorporation of (Zn, Co) into the BaTiO₃ lattice significantly augments the tetragonality of the unit cell. Moreover, the ferroelectric properties are enhanced, with a spontaneous polarization that is stronger than that observed in pure BTO,exhibiting excellent ferroelectricity. The results of the Hubbard+U algorithm indicate that the band gap of BZCT is reduced. Concurrently, the enhanced ferroelectric polarization increases the built-in electric field of the material, facilitating the separation of photogenerated carriers and improving optical absorption. Consequently, the optical absorption ability and photorefractive ability are effectively enhanced. BZCT, with its high spontaneous polarization and outstanding optical properties, can serve as a promising candidate material in the fields ofenergy storage and photovoltaics. # I Introduction. Barium titanate BaTiO3(BTO), a member of the ABO<sup>3</sup> perovskite oxide family, is a prototypical lead-free ferroelectric material[1]. The subjectof recent research has attracted a growing interest. Since its discovery in 1941, BTO has been thesubjectof considerable research interest[2].Due to its numerous promising physical properties, including a high dielectric constant, positive resistivity temperature coefficient, high voltage tunability, piezoelectricity, ferroelectricity, low leakage current,and low dielectric dispersion, BTO has emerged asa versatile material for various applications within the electronics industr[3-10]. However, it also presents certain practical limitations due to its relatively low Curie temperature, which is around 120°C, a narrow range of tetragonal phase stability, a broad energy band gap, and a high dielectric constant at the Curie point. These characteristics can restrict its application in certain high-temperature or high-frequency electronic devices[11]. Significant research has been conducted to enhance the ferroelectric and optical characteristics of barium titanate, with the objective of expanding its potential applications. The prevailing research approach currently entails the synthesis of novel systems through the replacement of Ba 2+ or Ti4+ with analogous ions of comparable dimensions, a process known as doping modification. A multitude of metal oxides have been employed in this manner to enhance the electronic and optical characteristics of the raw materials and expand their applications in optoelectronics[12, 13]. Given the evident significance of BaTiO3-based materials, a substantial body of experimental and theoretical research has been conducted to investigate the modulation of their physicochemical and electrical properties through doping. Lu Wang et al.[14] disrupted the long-range ferroelectric ordering by incorporating Li<sup>+</sup> and .Bi3+ into BaTiO<sup>3</sup> ceramics via solid-phase sintering. The resulting nanoclusters effectively suppressed the polarization and sustained the electrical strain, attaining the highest electrostriction coefficient of 0.0712m<sup>4</sup> /c 2 to date among all known electrostrictive materials. Lois et al.[15] discovered that the Zn-doped BaTiO<sup>3</sup> system not only exhibits a linear decrease in lattice constant with respect to the Zn content but also demonstrated that this doped system is capable of providing enhanced ferroelectric and dielectric properties compared to the pure BTO. Additionally, there was a notable reduction in the bandgap. Anju et al[16]. formed a solid solution of SmxBa1-xTiO<sup>3</sup> by doping BaTiO<sup>3</sup> with Sm3+ . The substitution of Sm3+ results in lattice distortion of the grains due to the difference in the size of the substitutional ions, which enhances the tetragonal nature of the grains. Sm3+ doping reduces dielectric loss and increases the dielectric constant, thereby enhancing the dielectric properties of the system.Additionally, Sm3+ doping contributes to an increase in the carrier concentration and the formation of defects and vacancies in the material, which in turn leads to an enhancement in the spontaneous polarization of the system. The properties of barium zirconate titanate (BZT) ceramics are significantly influenced by varying amounts of zirconium substitution, resulting in the emergence of desirable piezoelectric, ferroelectric, and other electro-mechanical properties[17, 18]. The substitution of Ca 2+ in the A-site and Zr 4+ in the B-site of ABO<sup>3</sup> perovskites results in the formation of (Ba, Ca)(Zr, Ti)O3, which alters the lattice parameters and causes a shift in the phase transition temperature and a broadening of the peak at the maximum value of the dielectric constant[19]. For the growing new energy industry, based on the high dielectric constant and the large spontaneous polarization of barium titanate, these barium titanate compounds have been greatly emphasized in a variety of applications, such as in photoelectrochemical systems used to increase the separation of carriers[20], in energy storage capacitors[21] or in the electronic ceramics industry[21-23]. However, in light of ongoing technological advancements, there is a growing need for ferroelectric materials that exhibit enhanced ferroelectric and optoelectronic properties. BaZnTiO<sup>3</sup> has been demonstrated to exhibit enhanced ferroelectric and dielectric properties relative to BTO, with minimal impact on the lattice constants[15]. Conversely, Co 4+ has been shown to markedly enhance the polarization properties of BTO[24]. Previously, the effects of co-doping BTO with Zn 2+ and Co 4+ ions on its ferroelectric and photovoltaic properties have not been extensively investigated. This is partly due to the limitations imposed by experimental conditions, which have hindered a detailed exploration of the material's electronic and band structures. To address this knowledge gap, we have utilized first-principles calculations to introduce Co²⁺ and Zn²⁺ ions into BTO crystals and assess their influence on the material's properties. This study is, to our knowledge, one of the first to systematically investigate the impact of (Zn, Co) co-doping on BTO ferroelectricity and the associated mechanisms, focusing on the local interactions, structural modifications, and the resulting electrical and optical properties. # II. Calculation details The doping system has been investigated using first-principles calculations and the supercell method. The density functional theory calculations are based on the Vienna ab initio simulation package (VASP)[25, 26]. The exchange-correlation energy of electrons was calculated under the generalized gradient approximation (GGA) using the Per- dew-Burke-Ernzerhof (PBE)method[27]. Select a 2 × 2 × 2 supercell containing 40 atoms, belonging to the P4mm space group, as shown in Fig. 1(a)[28]. And the initial lattice constants a=b= 3.99 Å and c= 4.01 Å. Based on the supercell, we introduced a Zn atom and a Co atom to replace the Ba and Ti atoms in the BTO supercell, respectively. As shown in Fig. 1(b). It is well known that DFT has problems in correctly describing the strong correlations between the d electrons, so the DFT + U method was used, and the 3d orbitals of the Ti atom and the 3d orbitals of the Co atom were corrected using the GGA + U method based on the method proposed in the literature[29] with the correction values of U = 9.4 eV and U = 5 eV, respectively[30, 31]. The cutoff value was chosen to be 500 eV. Using the Monkhorst-Pack method[32], a 5 × 5 × 5 grid of K-points centered on the gamma point was chosen for structure optimization and property calculations[33]. The convergence criterion for the interatomic interaction force is2 × 10 -2 eV/Å and for the system, energy is 1 × 10 -5 eV/Å. The spontaneous polarization is calculated using the standard Berry-phase method. # III, Results and Discussion ## 3.1. Geometry optimization. Fig. 1 depicts the lattice models of pure BTO and BZCT, wherein elemental substitution was conducted with a single Zn atom and Co atom at the Ba site and Ti site, respectively. These calculations were performed using 40 atoms. The optimized lattice parameters of pure BTO and BZCT are presented in Table 1. The impactof (Zn, Co) on the structural properties of BTO materials is evaluated by examining the lattice parameters, cell angle, and tetragonality factor (c/a). The calculations yielded the following values for the lattice parameters of pure BTO: a = b = 3.96 Å and c = 4.04 Å. These values are in good agreement with those reported in previous experimental studies[34] and theoretical works[35]. The differences between our calculated lattice parameters and the previously reported theoretical and experimental lattice parameters are 0.025 Å and 0.000Å, respectively, with an error of less than 3%. This indicates that our present work is reasonable. ![Fig. 1 Supercell lattice model of (a)BTO and (b)BZCT (light blue for barium atoms, orange spheres for titanium atoms, purple spheres for cobalt atoms, magenta spheres for zinc atoms, yellow spheres for oxygen atoms)](path) It is evident that doping results in a reduction in the lattice parameter of BZCT, an increase in the c lattice parameter, a slight rise in the c/a ratio, and an enhancement of the tetragonal nature of the lattice. Concurrently, the β angle, which is associated with the Zn atoms, undergoes a slight decline, and the lattice symmetry is diminished. The lattice aberration induced by (Zn, Co) doping is particularly pronounced. This is attributed to the difference in ion size between the dopant elements, Zn and Co, and the substituted ions. Additionally, the cell volume of the doped system is smaller than that of the pure BTO, indicating that the dopantelements induce lattice shrinkage. Following atomic relaxation, the optimized structure exhibits a smaller lattice volume, enhanced tetragonality, and relatively weakened symmetry, which is a contributing factor to the enhanced intrinsic polarization observed in BZCT ferroelectrics. ## 3.2. Electronic properties density of states(DOS), can reflect a multitude of physical properties and bonding characteristics. The study of the electronic energy band structure provides useful information for realizing paraelectricity and ferroelectricity, including semiconductor behavior. In order to gain a deeper understanding of the underlying mechanism of lattice distortion, we selected the 100-plane and the 110-plane, which contain dopant elements in the form of Zn and Co atoms, for a charge density analysis. This is illustrated in Fig. 2, where the upward direction corresponds to the c-axis. As illustrated in Fig. 2, the Ti and Co ions in BZCT exhibit relaxation along the c-axis direction, deviating from the Ti-O facets in comparison to the pure BTO structure. Additionally, as illustrated in Fig. 2(c), there is an overlapping region of electron densities between the Zn ions and the adjacent O ions, indicating a robust interaction between Zn and O. This observation implies the presence of covalent bonding character in the Zn-O linkage. These results in the adjacent O2 atoms of the Zn ions undergoing reverse relaxation along the C-axis, which causes a significant zigzagging of the Ti-O plane in this layer of the O2 ions and an increase in the distortion of the oxygen octahedron. Consequently, this causes an increase in the displacement of Ti atoms from the center of the oxygen octahedron, which accounts for the observed enhancement in the intrinsic polarization of the doped system. Furthermore, Fig. 2(b) illustrates that the electronic charge densities of Ti ions and Co ions overlap with the oxygen ions, forming covalent bonds. This phenomenon is analogous to the bonding observed between Zn ions and corresponding O2 ions, as depicted in Fig. 2(c). Furthermore, the nature of the bonding between the Ti ions and the O2 ions is markedly different, with no overlap of electronic layers between Ti-O2. It can thusbe postulated that the promotion of oxygen octahedral distortion by the Ti-O planar fluctuation of BZCT,caused by Co-O chemical bonding and the formation of covalent bonds between Zn-O, represents the primary factors responsible for the enhancement of ferroelectricity and the improvement of other properties of BZCT. ![Fig. 2 (100)plane charge density plots of(a)BTO,(b)BZCT, and (c) (110)plane charge density plots of BZCT.](path) To study the electronic structure of BTO perovskites and their doping modification changes, the electronic energy band structures of pure BTO and BZCT in the Brillouin zone along the high symmetry direction were calculated. The Fermi energy level was set to zero, which is shown as a gray dashed line in the figure. From Fig. 3(a), it can be seen that the CBM is located at the G-point due to the dominance of the Ti-3d state, while the VBM is located at the G-point at FL(0 eV), which is guided by the O-2p-state. The valence band tops (VBM) and conduction band bottoms (CBM) of the BTOs are located at highly symmetric G-points, indicating that the BTOs are direct bandgap semiconductors with an energy band value of 1.7918 eV, which is in close agreement with the reported values of 1.723 eV[36] and 1.778 eV[37]. Fig. 3(b) shows the electrified energy band structure of BZCT, where it is observed that the valence band top (VBM), predominantly constituted by the Ti-3d state, is situated at the X point, whereas the conduction band bottom (CBM), primarily comprising the O-2p state, is located at the Y point. This suggests that ![Fig. 3 Band structure of (a)BTO,(b)BZCT without Hubbard U and band structure of (c)BTO,(d)BZCT withHubbard U](path) The previous experimental study indicated that the band gap of BTO is approximately 3.2 eV[38]. This discrepancy can be attributed to the fact that the generalized gradient approximation (GGA) methodology employed in the calculations of p-d repulsion for cations and anions, as well as the estimation of band gaps, often results in an underestimation of the latter[39]. The observed trend of decreasing band gaps for BTO is consistent with the typicalunderestimation of density functional theory observed for another perovskite material, SrTiO3[40]. To rectify this discrepancy, we employed the Hubbard+U algorithm to compute the revised electronic energy band structure of pure BTO and BZCT. The outcomes of this calculation are illustrated in Fig. 3(c) and Fig. 3(d). Following the correction by the Hubbard+U algorithm, the electronic energy band gap of pure BTO is 3.21 eV, which is in agreement with other experimental findings [38] ,[41]. The U-added algorithm of BZCT demonstrates that its energy bandwidth bandgap is 2.34 eV, a value that is smaller than that ofpure BTO. The electrons and holes can be excited by lower electron energy, and due to its status as a direct bandgap semiconductor, the material is conducive to carrier migration. Consequently, the BZCT canbe applied to optoelectronic materials. As this paper is concerned with the comparative alterations in properties prior to and following doping, along with the underlying mechanisms, the data will be used without the inclusion of U in the subsequent investigation. ![Fig. 4 The total and partial density of states of(a) BTO-TDOS, (b) BZCT-TDOS, (c) BZCT-PDOS.](path) The total density of states (TDOS) and partial wave density of states (PDOS) for BTO and BCTO were calculated and are presented in Fig. 4. In the case of pure BTO, the energy range for the density of states (DOS) was selected to be between -6 eV and 6 eV. In the valence band (VB), the O-2p state is the primary contributor. In the conduction band (CB) region, the primary contributions are made by the Ti-3d and O-2p states. The hybridization of Ti-3d and O-2p states in the valence band (VB) and conduction band (CB) regions is a key factor contributing to the ferroelectricity observed in pure BTO[42]. Furthermore, the introduction of Zn and Co elements into the BTO cell results in the emergence of a new peak situated in close proximity to the Fermi energy level, specifically at 0 eV. The DOS plots of Fig. 4(b)(c) demonstrate that the primary contributions to this new state are Co-3d and O-2p states. The emergence of this peak results in the Fermi energy level of BCZT being situated in close proximity to the valence band. Concurrently, Co introduces a new electronic state at the base of the conduction band of BTO, thereby reducing the band gap of the energy band of BZCT. The main contributors to the valence band and the conduction band of BCTO remain the Ti-3d and O-2p states, respectively. The impact of Zn doping is primarily manifested in the peak at -6eV in the Zn-3d valence band. In conclusion, the introduction of (Zn,Co) co-doping results in the emergence of new impurity energy levels within the material, leading to a downward shift in the conduction band and a shift in the Fermi energy levels towards the valence band. This phenomenon contributes to a reduction in the energy band gap of the entire system. Furthermore, the strong hybridization between Ti-3d and O-2p orbitals, as well as between Co-4d and O-2p orbitals, suggests that this is the factor responsible for the enhancement of ferroelectricity in the BZCT materials[42]. ## 3.3 Electrical properties Fig. 5 shows the energy difference and polarization function of pure BTO and BSZT, respectively. The energy polarization curve isitted by the phenomenological Landau–Ginzburg–Devonshire theory, and the equation is as follows. where ΔGis the energy difference between the ferroelectric phase and the paraelectric phase, and α, β, γ are coefficient constants. The potential curves for both BTO and BCZT are well-fitted by the image-only Landau equation, respectively. The two minima in the double-trap potential curves correspond to the two stable polarisation states, the P+ state and the P- state,and, for each of these minima,the depth of the trap with respect to P=0 corresponds to the effective barrier for the reversal of the polarization. The P=0 state represents the paraelectric phase, which has no Ti displacement. For both cases of pure BTO and BZCT, the P+ and P- states are identical. It is generally accepted that the magnitude of the double-trap depth is proportional to the magnitude of the ferroelectric polarisation[43]. Therefore, analyzing the variation of the ferroelectric double-well potential depth will help to predict the ferroelectric phase transition and the response to the external electric field. As can be seen from Table 2, the double-well potential depth of BTO is 0.397 meV. The double-potential depth of BZCT is 0.766 meV, which is larger than that of BTO, and we predictthat it is more difficult for BZCT to achieve polarisation switching. This is due to the larger ionic shift and stronger hybridization of Co-3d with O-2p orbitals and Ti-3d with O-2p orbitals. The spontaneous polarisation intensity of BZCT is 32.41 μC/cm<sup>2</sup> , which is higher than that of pure BTO, which is 26.86 μC/cm<sup>2</sup> . The changes in the spontaneous polarisation and the depth of the traps suggest that the spontaneous ferroelectricity of BCZT is higher than that of pure BTO ferroelectric is enhanced. ![Fig. 5 Double-well potentials of (a) pure BTO and (b) BZCT. The red dots are the total energy calculated by DFT and theblue line is from the phenomenological Landau model.](path) In the realm of energy storage devices, superior performance is primarily determined by two critical metrics: enhanced energy storage density and optimal energy storage efficiency. The underpinnings of these attributes are typically reflected in the material's high spontaneous polarization and significant breakdown strength, according to the existing studies. Fig. 6 illustrates the transport properties of four distinct BTO systems, for which we have ascertained the conductivities. These calculations were grounded in the semi-empirical framework of the Boltzmann transport theory. where ƒμ(T,ε) is the Fermi-Dirac distribution function, μ is the chemical potential, and T is the temperature. When a constant relaxation time isknown, it can be shown that the conductivity at RTA and RBA can be obtained for a certain chemical potential and temperature by calculating the energy band structure. Only their conductivities have been discussed qualitatively, so their relaxation time constants have not been estimated here. ![Fig. 6 Electrical conductivity of BTO, BZTO, BTCO, and BZCT](path) between the pure BTO and doped BTO systems. A comparative analysis of the conductivity and relaxation time between the individual systems is illustrated in figure, where the horizontal coordinates are the maximum and minimum values of the chemical potential with respect to the Fermi energy level, We can see that the (Zn,Co) co-doped BZCT shows lower conductivity than the other three systems in a more stable state, and fewer carriers are transferred in the BZCT; and when the electrons in the system are in a higher energy state, i.e. the BZCT is in an excited state or there is an external energy input, such as sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, promoting carrier migration, which has good potential applications in the field of photovoltaics, When the electrons in the system are in a higher energy state, i.e., the BZCT is in an excited state or there is an external energy input, e.g., sunlight irradiation, the conductivity is increased, which leads to more electrons participating in the conducting process, which promotes carrier migration, which has good potential applications in the field of photovoltaics. This promotes carrier migration, which has good potential applications in the field of photovoltaics and is also consistentwith the previous results of electronic energy band analysis. Conductivity describes the conductivity of the material, from the overall point of view, the conductivity of BZCT is weaker than the pure BTO system, indicating that the breakdown strength of BZCT is higher than that of pure BTO system, BZCT has a higher spontaneous polarization and higher breakdown strength than that of pure BTO[45], and BZCT is expected to be used in the field of energy storage. ## 3.4 Optical properties In recent years, the effective utilization of solar energy has attracted the attention of many researchers [46, 47]. Ferroelectric photovoltaic materials have been demonstrated to exhibit an excellent photovoltaic effect, with photogenerated voltage not limited by the forbidden bandwidth (band gap) ofthe material itself. Furthermore, the photogenerated current can be regulated by the built-in electric field[48]. Accordingly, we have investigated the impact of modified ferroelectric characteristics on the optical attributes of the materials and explored the prospective applications of perovskite materials in photorefractive, optoelectronic, and solar cell(photovoltaic) domains. ![Fig. 7 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) real part of the complex dielectric function; (b) the imaginary part of the complex dielectric function; (c) refractive index.](path) In order to investigate the effect of (Zn, Co) co-doping on the optical properties of BTO, we calculated a series of optical properties of pure BTO, Zn-doped (BZTO), Co-doped (BTCO) and (Zn, Co) co-doped BTO(BZCT). These included the complex dielectric function, absorption function, loss function, reflectance, and refractive index, as illustrated in Fig. 7 and Fig. 8. The complex dielectric function;ε(ω) = ε1(ω) + iε2(ω) is divided into two parts: the real part of the complex dielectric function; denoted as ε1(ω),and the imaginary part of the complex dielectric function, denoted as ε2(ω). The real part of the complex dielectric function, denoted as ε1(ω), is indicative of the polarization properties of the material in question. As illustrated in Fig. 7(a), the real part of the complex dielectric function, denoted as ε1(ω), is dependent on the incident photon energy. With an increase in the incident photon energy, the value of ε1(ω) subsequently decreases, indicating a reduction in the material polarization property. As illustrated in Fig. 7(a), the polarization properties of BZCT (8.76) and BTCO (7.47) are superior to those of pure BTO(6.04) at0.0 eV, suggesting that Co doping has a beneficial impact on the polarization properties. The pure and doped systems approach the minimum value of ε1(ω) ata photon energy of approximately 19.0 eV, respectively, and exhibit a slight increase thereafter. In contrast, the imaginary part of the complex dielectric function, represented as ε2(ω), is associated with the energy dissipation observed within the system. As illustrated in Fig. 7(b), the imaginary component of the dielectric function, ε2(ω), of the doped system exhibits an increase for all doped systems between 0 and 2.89 eV, indicating that the energy dissipation of the doped system is elevated. In particular, the imaginary part ε2(ω) is higher for the (Zn, Co) and Co doping cases than for pure BTO and BZTO at lower incident photon energies, including the peak in the visible range of 1.65-3.10 eV. Furthermore, the energy dissipation is higher in the low-energy region for BZCT and BTCO. The pure BTO exhibits superiorenergy dissipation characteristics in the medium and high energy regions when compared to the doped system. The refractive indices (n(ω)) of pure BTO and doped systems are demonstrated in Fig. 7(c),. The refractive indices of the refractive spectra of these materials in the infrared, visible, and most of the ultraviolet ranges are greater than 1. Co-doping has been observed to increase the static refractive indices ofthe pure BTOs, with the static refractive indices of the BZTOs,BTCOs, and BZCTs being 2. 40, 2.73, and 2.92, respectively. When the refractive indices exceed 1, photons encountering the material are decelerated due to electron interaction, resulting in a higher refractive index[49]. Materials with a refractive index of 1 or greater are considered transparent to incident light. Therefore, BZCT is transparent to incident light below 11.03 eV and opaque to incident light above this value. In general, any process that increases the electron density of a material will also result in an increase in the refractive index [50]. The effective enhancement of BTO refractive index properties by Co/Zn co-doping indicates that BZCT may bea suitable material for use in photorefractionation. ![Fig. 8 Optical properties ofpure BTO and doped BaTiO<sup>3</sup> (a) absorption spectra; (b) loss function; (c) absorption spectrum in the region of 0-4.5 eV; (d) loss function in the region of 0-5 eV.](path) The absorption spectra α(ω)of BTO, BZTO, BTCO, and BZCT are illustrated in Fig. 8(a). At an incident photon energy of 19.21 eV, a pronounced absorption peak is observed for each system. In comparison to BTO, BZTO, BTCO, and BZCT, the latter exhibit lower absorption coefficients at the peak. The pure BTO system, in particular, demonstrates a pronounced absorption of electromagnetic radiation energy in the vicinity of 19 eV, which can be classified as occurring in the medium-energy region. In the photon energy range of 20 to 40 eV, the systems display comparable absorption spectra. In the low-energy range (0–10 eV), BZCT and BTCO display enhanced absorption characteristics. In comparison to the pure BTO system, the BZCT system demonstrates the capacity to markedly enhance the material's light absorption capability. The BZCT system exhibits superior light absorption capabilities in the visible and low UV regions, with a broader absorption spectral range than that of the pure BTO system. In the deep UV high-energy region of the absorption spectrum, the absorption peak of BZTO is stronger than that of BTCO. In conclusion, the introduction of a Co dopant in pure BTO shifts the absorption characteristics of the material to low frequency, while the addition of a Zn dopant increases the absorption characteristics of the material at high frequency. In light of the aforementioned findings, BZCT emerges as a promising candidate for utilization as an absorber layer in solar cells. When electromagnetic radiation is incident on a material, a portion of the energy is lost within the material[51]. The energy loss is quantified by a loss function, as illustrated in Fig. 8(b). The highest peaks of the loss functions were observed in the pure system, and the energy loss peaks of the intrinsic and doped materials were obtained at the corresponding photon energies of26-28 eV. It can be seen that the loss functions of the infrared region of BTCO and BZCT are also high. The energy loss of the doped system is less pronounced than that of the pure BTO system at all three peaks, indicating an effective enhancement of the opticalproperties by the doped system. # III. CONCLUSION The theoretical characterization of the crystalstructure, electronic properties, ferroelectric, and optical properties of the (Zn, Co) co-doped BTO system was conducted through first-principle calculations, and the intrinsic mechanism of (Zn, Co) co-doping on the improvement of BTO properties was investigated. The results ofthe structure optimization demonstrate that the co-doping of (Zn,Co) results in a reduction of structural symmetry and an enhancement of lattice tetragonality in BaTiO3. The band gap of the co-doped system of BZCT exhibits a notable reduction due to the addition of the impurities. The introduction of impurity energy levels in comparison to the pure BTO material resultsin a transition from a direct to an indirect band gap. The incorporation of Hubbard's energy corrects the band gap of the electronic energy bands from 3.20 eV to 2.34 eV. The calculation of the density of states indicates the formation of strong force orbital hybridization between the Zn-3d and O-2p states, as well as between the Co-3d and O-2p states. It can be concluded that the Co-3d and O-2p states are the primary contributors to the shift in the conduction band level at the Fermi energy. In the charge density, covalent bonds are formed between Zn and O and Co and O, which represents a significant factor and direct manifestation of the lattice distortion. The deviation of the Ti-O planes results in a notable change in the oxygen octahedron, leading to a more pronounced spontaneous polarization of BZCT. This enhanced ferroelectricity makes BZCT a promising material for energy storage devices.It is noteworthy that the enhanced ferroelectricity improves the optical properties ofBZCT, which exhibits enhanced light absorption in the visible and low ultraviolet regions. This excellent light absorption property makes BZCT a promising candidate in the field of ferroelectric photovoltaics and photocatalysis. The alterations in electrical and optical characteristics resulting from (Zn, Co) co-doping will expand the scope of applications for BaTiO<sup>3</sup> ferroelectric materials in the domains of energy storage and optics. Moreover, they will furnish a theoretical foundation for the investigation of the properties ofperovskite materials and device fabrication in the future. #### Authorship contribution statement Zheng Kang : Writing - Original Draft, Conceptualization, Data curation, Visualization, Formal analysis. Mei Wu: Writing - Original Draft, Visualization, Software. Yiyu Feng: Original draft, Visualization. Jiahao Li: Software, Visualization.Jieming Zhang: Validation, Original draft. Haiyi Tian: Software, #### Data availability Data will be made available on request. #### Acknowledgments #### IV. References [5] J.W. Lee, K. Eom, T.R. Paudel, B. 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Fig. 1 The schem matic view o of cg-N cry ystal structu ure.	# One Pot Synthesis of Cubic Gauche Polymeric Nitrogen ### Abstract The long sought cubic gauche polymeric nitrogen (cg-N) consisting of N-N single bonds has been synthesized by a simple route using sodium azide as a precursor at ambient conditions. The recrystallization process was designed to expose crystal faces with low activation energy that facilitates initiating the polymeric reaction at ambient conditions. The azide was considered as a precursor due to the low energy barrier in transforming double bonded N=N to single bonded cg-N. Raman spectrum measurements detected the emerging vibron peaks at 635 cm-1 for the polymerized sodium azide samples, demonstrating the formation of cg-N with N-N single bonds. Different from traditional high pressure technique and recently developed plasma enhanced chemical vapor deposition method, the route achieves the quantitative synthesis of cg-N at ambient conditions. The simple method to synthesize cg-N offers potential for further scale up production as well as practical applications of polymeric nitrogen based materials as high energy density materials. ### Introduction The N≡N triple bond is one of the strongest chemical bonds, with a very high bond energy of 954 kJ/mol [1] , while the N-N single bond is much weaker, with a bond energy of 160 kJ/mol [2, 3] . Due to the huge difference in energy between triply bonded dinitrogen and singly bonded nitrogen, the polymerized nitrogen materials with N-N single bonds are proposed as high energy density materials and expected to release a high amount of chemical energy while transforming into a triple bonded dinitrogen molecules. It was anticipated that the greatest utility of fully single-bonded nitrogen would produce a tenfold improvement in detonation pressure over HMX [4] and therefore polymerized nitrogen is highly sought as a high energy density material. It was predicted in 1985 that molecular nitrogen would polymerize into atomic solid at high pressure [5] . Later, the cubic gauche (cg-N) lattice with a single bonded diamond like structure was proposed[6] . Subsequently, large amounts of theoretical work reported on the high pressure approach to polymerized nitrogen with N-N single bonded forms [7- 23] . It was not until 2004 that Eremets et. al. prepared cg-N directly from molecular nitrogen at pressures above 110 GPa (1GPa≈10,000 atmospheric pressure) and temperatures above 2000 K using laser heated diamond cell technique [24] . The cubic gauche structure was confirmed by growing the single crystal in 2007[25] . However synthesis pressures above 110 GPa are highly required. Later on polymerized nitrogen with layered structure (LP-N) [26] , hexagonal layered structure (HLP-N) [27] , black phosphorus nitrogen (BP-N) [28] , and Panda nitrogen [29] were successfully prepared at 120 GPa, 244 GPa, 146 GPa & 161 GPa, respectively. Nevertheless, the high pressure is commonly regarded as a necessary route to synthesize polymerized nitrogen. All these polymerized nitrogen samples decomposed in the process of releasing pressure. Some strategies, such as chemical doping, were attempted at high pressure by introducing metals in order to stabilize the polymerized nitrogen lattice at lower pressure [30-34] but it remains challenging to recover the polymerized nitrogen samples at ambient pressure. In addition, introducing non-energetic element, especially the heavy metal seriously decreases energetic density of polymerized nitrogen materials, detrimental to the natural applications. In the past few years, plasma enhanced chemical vapor deposition (PECVD) technique has been developed to take advantage of the non-equilibrium plasma environment to synthesize polymerized nitrogen materials. The relative thermodynamic instability of polymerized nitrogen can be overcome by kinetic stability provided through energy band hybridization by a charge transfer mechanism, which makes the high energy lattice gain stability and avoid the highly energetic conversion to molecular N2 [35] . The cg-N and N8 cluster were successfully prepared with or without the assistance of carbon nanotube using sodium azide as precursor [35-41] . However the plasma generally can only penetrate several nanometers into the solid surface [42] , leading to limited quantity of cg-N. Besides, high energy plasma particles also could dissociate the cg-N produced in the experiment, which is backed up by the fact that the optimized deposition time is demanded for a very small amount of cg-N sample prepared by the PECVD method [38] . Theory proposes that the transition path to polymerized nitrogen might pass through different molecular structures with limited thermodynamic stability fields [43] . Additionally earlier experiments aimed at synthesizing polymerized nitrogen only reached the amorphous state at ambient temperature compression, indicating that increasing reaction temperature to activate the precursor might play a crucial role in the formation of the atomic structures [44, 45] . For example LiN5 and K2N6 were synthesized at lower pressures [30, 32, 33] with the assistance of laser heating. We here for the first time report a simple route to synthesize cg-N by using sodium azide as a precursor. The cg-N was successfully obtained through recrystallization reaction, followed by transforming azide ions with N=N double bonds into single bonded cg-N that can stabilize at ambient pressure conditions. To date it is the simplest synthesis route to realize cg-N that can be further developed for scale up production and potential practical applications. #### Materials & Methods High pure sodium azide (NaN3) (≥99.9%) was chosen as nitrogen source to synthesize cg-N. The recrystallization process was performed first. A solution of NaN3 (2 mol/L) was pre-prepared and dropped into a crucible. Then the crucible was transferred into a tubular furnace. The vacuum operation was performed to realize the recrystallization of the NaN3 powder. The reaction temperatures and times within the 200- 300°C and up to 10 hours were tried in the synthesis processes. The optimized synthesis condition with reaction temperature 240 ~ 260 <sup>o</sup> C and heating time 5 hours was employed to synthesize the cg-N. The sample was named as polymerized sodium azide followed by synthesis temperature, time, such as PSA-240o C-5h. According to aforementioned conditions, heating NaN3 without recrystallization process to prepare cg-N was tried and it was named as unpolymerized sodium azide (UPSA). The Raman spectra were recorded in range of 100~1500 cm-1 with a spectral resolu micro laser ( ution of 1 cm scope and m (λ0 =532 nm m-<sup>1</sup> using a multichanne m). an integrated el air cooled d laser Ram d CCD dete man system ector. The e (Renishaw excitation so w) with a co ource is an onfocal Ar ion #### Resul lts ![#### Fig. 1 The schem matic view o of cg-N cry ystal structu ure.](path) T I213 s be 1.4 the cry with N lone p diamo with o of Fig spectr metast The cg-N w space group 40 Å and th ystal structu N-N single pair electro ond (left in F others formi g. 1. The t rum calcula table at amb was proposed p with a latt he bond ang ure of cg-N, bonds at eq ons. Thus c Fig.1). The ing a three theoretical w ation [12] , in bient pressu d to adopt a ice constan le is 114.0° , in which e qual distanc cg-N exhib single bond dimensiona work on cg ndicating th ure. a body cent nt of 3.773 Å ° at ambient each nitroge ces and the its a near ded nitrogen al network s g-N didn't hat the enti tered cubic Å. The N-N t pressure c en atom bon e remaining tetrahedral n atoms form structure [47] show imag ire covalent Bravais lat N bond leng conditions [6 nds to three p orbital is structure s m fused ring ], as shown ginary freq t network o ttice, belong gth is predic 6, 46] . Fig. 1 nearest nei s occupied similar to t gs, which c in the right quency in p of cg-struc ging to cted to shows ghbors by the that in onnect t panel phonon ture is T precur single The r activa date th to obt charac The azide i rsor for syn e bonded cg recrystalliza ation energy he amount o tain ideal X cterize the c ion is comp nthesizing th g-N. In this ation proce y. Then the of cg-N obt X-ray diffrac chemical bo posed of N he cg-N du work the so ess is expe polymeric tained exper ction pattern ond transfor N=N doubl ue to the low odium azide ected to ex reaction ca rimentally a ns. Therefo rmation in t e bonds th w energy b e is used in xposing so an be initiat at ambient p ore the Ram the present hat can be barriers for n present sim me crystal ed at ambie pressure is man spectrum work that i used as an transformin mple experi l faces wit ent conditio still not suf m was adop is commonl n ideal ng into iments. th low ons. To fficient pted to ly used ![Fig. 2 The Raman spectra of polymerized sodium azide (PSA) synthesized at 200 ~300 <sup>o</sup> C with a reaction time of 3 hours, where the fingerprint peak at 635 cm-1 for cg-N is detected. The vibron intensity of cg-N shows optimized synthesis temperature between 240~260 <sup>o</sup> C.](path) Fig. 2 shows the Raman spectra of polymerized NaN3 (PSA) synthesized at 200~300 o C with a reaction time of 3 hours. The sharp peaks of NaN3 near 120, 1267 & 1358 cm-1 are assigned to the vibrational lattice mode, the first overtone of the IR active bending ν2 mode and the symmetric stretching ν1 mode of the azide ion, respectively, which matches well with the Raman spectra of free standing NaN3 [41] . For comparison the NaN3 raw material and a sample heated only at 250 <sup>o</sup> C for 3 hours without the recrystallization process were characterized and showed same Raman vibron profiles without any signals at 635 cm-1 (see Fig. S1). The lines at 635 cm-1 for polymerized sodium nitrogen sample (PSA) are comparable to the theoretical results extrapolated to ambient pressure because the Raman modes tend to soften with decreasing pressure [24, 46, 48] . It also agrees with our recent work on polymerized cg-N based on potassium azide. Therefore the emergent intense vibron at 635 cm-<sup>1</sup> is assigned to the pore breathing A symmetry and is regarded as the finger print peak of cg-N, which unambiguously indicates the successful synthesis of cg-N [46] . The synthesis conditions of cg-N were first systematically investigated by controlling the reaction temperature with a fixed reaction time of 3 hours. As shown in Fig. 2, the vibron intensity of cg-N increases as the reaction temperature raises from 200 <sup>o</sup> C to 240 <sup>o</sup> C compared to that at 1358 cm-<sup>1</sup> for the unreacted or remnant NaN3 in the PSA samples. As temperature increases up to 270 <sup>o</sup> C, the Raman intensity of cg-N remains comprehensive but decreases clearly at 300o C. Additionally, it is worth noting that the intensity of vibrational lattice mode at 120 cm-1 for the unreacted NaN3 in PSA-240 sample dramatically decreases compared to those prepared at other temperatures. The PSA sample exhibits uniformly dark blue color and should be related to the by product formed during the synthesis of cg-N and will be discussed below. Therefore, the optimized temperature range of 240~260 <sup>o</sup> C was adopted for cg-N preparation. A quantitative comparison was made by calculating the intensity ratios of the characteristic Raman peaks to measure the abundance of cg-N phase in the PSA samples. Table I summarizes the partial line intensities of NaN3 and cg-N. The amount of cg-N synthesized is reflected by the peak intensity at 635 cm-1 , while those at 120 and 1358 cm-<sup>1</sup> represent the unreacted NaN3. Herein, a simplified conversion degree from NaN3 to cg-N is defined by the peak intensity ratio. The intensities of NaN3 at 120 cm-1 and 1358 cm-1 , calculated by (I200-Ibg)/(I1358-Ibg) for the NaN3 raw material (SA-R), NaN3 heated only (UPSA) and polymerized NaN3 (PSA) samples at different temperatures are between 2 and 3 and remain little changed. For the SA-R and UPSA samples no trace of cg-N is detected. After recrystallization process the vibron intensity of cg-N enhances dramatically for PSA samples and the ratio of cg-N to NaN3 is 1.26-1.39 calculated by (I635-Ibg)/(I1358-Ibg), and 0.42~ 0.60 calculated by (I635-Ibg)/(I120-Ibg). Therefore it indicates that the crystallization process strongly promotes the formation of cg-N and the optimized synthesis temperature range is 240~260 <sup>o</sup> C. ![Fig. 3 The reaction time dependence of the conversion ratios of cg-N to NaN3 for the PSA samples prepared at 240, 250 & 260 <sup>o</sup> C. The optimized synthesis conditions for the PSA samples are 240~260 <sup>o</sup> C with a reaction time of 5 hours.](path) Further the preparation conditions were explored by synthesizing PSA samples at 240~260 <sup>o</sup> C with variable reaction time. The Raman vibron intensity of cg-N in the PSA-240 samples increases gradually as the reaction time extends up to 5 hours and then decreases, which is shown in Fig. S2. Samples prepared at 250 and 260 <sup>o</sup> C show the same results (Fig. S3). For comparison, the intensity ratios of the cg-N to unreacted NaN3 in theses PSA samples were calculated in Table SI. The synthesis time dependence of the ratios of cg-N to NaN3 at 240-260 <sup>o</sup> C was plotted in Fig. 3. All samples show consistent trends: the PSA samples with a reaction time of 5 hours show the highest conversion degree. Therefore, the optimized preparation conditions for polymerized sodium azide samples are a temperature range of 240~260 <sup>o</sup> C and a reaction time of 5 hours. #### Discussion been proposed, which suffer from the challenge that samples could not recover to ambient pressure and enhancing yield is difficult. Here a simple route was designed for synthesizing cg-N under atmospheric conditions considering the thermodynamic barriers related to transforming to N-N single bonds and activation energy of precursor. In our experiments it was found that recrystallization process played a crucial role to form cg-N. For azides the crystal face with low activation energy may facilitate initiating the transformation of the double bonds N=N in azide ions to single bonded cg-N. The optimized temperatures for cg-N preparation by NaN3 are primarily restricted by the melting points NaN3 (~275°C). These results align well with the fact that the cg-N forms on the crystal face with lower activation energy of azides precursors oriented by the recrystallization process. Regarding the formation mechanism of cg-N it was observed that the polymerized sodium azide turned dark blue in our experiments that should be related to the formation of by-product Na3N during the synthesis of cg-N. This is consistent with the reported results that pH value of the solution increases after synthesis reaction as Na3N displays alkalinity when dissolved in water [38] . A more detailed conversion mechanism is currently at investigation. ### Conclusion The polymeric nitrogen with cubic gauche structure was successfully synthesized by a simple route using sodium azide as a precursor. A high yield may be obtained by separating the by products formed on the surface of the produced cg-N. The route is promising for further scale up or applied to synthesize other metastable materials. #### Conflict of Interest There are no financial conflicts of interest to disclose. #### Acknowledgments The work was supported by the National Natural Science Foundation of China, Ministry of Science & Technology, and CAS Project for Young Scientists in Basic Research of China. ## Author Contributions Changqing Jin & Jun Zhang designed and supervised the whole research. Runteng Chen, Jun Zhang, Zelong Wang, Ke Lu, Yi Peng, Jianfa Zhao, Shaomin Feng and Changqing Jin performed the experiments. Jun Zhang & Changqing Jin wrote the manuscript with contributions from all authors. #### References
Fig. 2 The Raman spectra of polymerized sodium azide (PSA) synthesized at 200 ~300 <sup>o</sup> C with a reaction time of 3 hours, where the fingerprint peak at 635 cm-1 for cg-N is detected. The vibron intensity of cg-N shows optimized synthesis temperature between 240~260 <sup>o</sup> C.	# One Pot Synthesis of Cubic Gauche Polymeric Nitrogen ### Abstract The long sought cubic gauche polymeric nitrogen (cg-N) consisting of N-N single bonds has been synthesized by a simple route using sodium azide as a precursor at ambient conditions. The recrystallization process was designed to expose crystal faces with low activation energy that facilitates initiating the polymeric reaction at ambient conditions. The azide was considered as a precursor due to the low energy barrier in transforming double bonded N=N to single bonded cg-N. Raman spectrum measurements detected the emerging vibron peaks at 635 cm-1 for the polymerized sodium azide samples, demonstrating the formation of cg-N with N-N single bonds. Different from traditional high pressure technique and recently developed plasma enhanced chemical vapor deposition method, the route achieves the quantitative synthesis of cg-N at ambient conditions. The simple method to synthesize cg-N offers potential for further scale up production as well as practical applications of polymeric nitrogen based materials as high energy density materials. ### Introduction The N≡N triple bond is one of the strongest chemical bonds, with a very high bond energy of 954 kJ/mol [1] , while the N-N single bond is much weaker, with a bond energy of 160 kJ/mol [2, 3] . Due to the huge difference in energy between triply bonded dinitrogen and singly bonded nitrogen, the polymerized nitrogen materials with N-N single bonds are proposed as high energy density materials and expected to release a high amount of chemical energy while transforming into a triple bonded dinitrogen molecules. It was anticipated that the greatest utility of fully single-bonded nitrogen would produce a tenfold improvement in detonation pressure over HMX [4] and therefore polymerized nitrogen is highly sought as a high energy density material. It was predicted in 1985 that molecular nitrogen would polymerize into atomic solid at high pressure [5] . Later, the cubic gauche (cg-N) lattice with a single bonded diamond like structure was proposed[6] . Subsequently, large amounts of theoretical work reported on the high pressure approach to polymerized nitrogen with N-N single bonded forms [7- 23] . It was not until 2004 that Eremets et. al. prepared cg-N directly from molecular nitrogen at pressures above 110 GPa (1GPa≈10,000 atmospheric pressure) and temperatures above 2000 K using laser heated diamond cell technique [24] . The cubic gauche structure was confirmed by growing the single crystal in 2007[25] . However synthesis pressures above 110 GPa are highly required. Later on polymerized nitrogen with layered structure (LP-N) [26] , hexagonal layered structure (HLP-N) [27] , black phosphorus nitrogen (BP-N) [28] , and Panda nitrogen [29] were successfully prepared at 120 GPa, 244 GPa, 146 GPa & 161 GPa, respectively. Nevertheless, the high pressure is commonly regarded as a necessary route to synthesize polymerized nitrogen. All these polymerized nitrogen samples decomposed in the process of releasing pressure. Some strategies, such as chemical doping, were attempted at high pressure by introducing metals in order to stabilize the polymerized nitrogen lattice at lower pressure [30-34] but it remains challenging to recover the polymerized nitrogen samples at ambient pressure. In addition, introducing non-energetic element, especially the heavy metal seriously decreases energetic density of polymerized nitrogen materials, detrimental to the natural applications. In the past few years, plasma enhanced chemical vapor deposition (PECVD) technique has been developed to take advantage of the non-equilibrium plasma environment to synthesize polymerized nitrogen materials. The relative thermodynamic instability of polymerized nitrogen can be overcome by kinetic stability provided through energy band hybridization by a charge transfer mechanism, which makes the high energy lattice gain stability and avoid the highly energetic conversion to molecular N2 [35] . The cg-N and N8 cluster were successfully prepared with or without the assistance of carbon nanotube using sodium azide as precursor [35-41] . However the plasma generally can only penetrate several nanometers into the solid surface [42] , leading to limited quantity of cg-N. Besides, high energy plasma particles also could dissociate the cg-N produced in the experiment, which is backed up by the fact that the optimized deposition time is demanded for a very small amount of cg-N sample prepared by the PECVD method [38] . Theory proposes that the transition path to polymerized nitrogen might pass through different molecular structures with limited thermodynamic stability fields [43] . Additionally earlier experiments aimed at synthesizing polymerized nitrogen only reached the amorphous state at ambient temperature compression, indicating that increasing reaction temperature to activate the precursor might play a crucial role in the formation of the atomic structures [44, 45] . For example LiN5 and K2N6 were synthesized at lower pressures [30, 32, 33] with the assistance of laser heating. We here for the first time report a simple route to synthesize cg-N by using sodium azide as a precursor. The cg-N was successfully obtained through recrystallization reaction, followed by transforming azide ions with N=N double bonds into single bonded cg-N that can stabilize at ambient pressure conditions. To date it is the simplest synthesis route to realize cg-N that can be further developed for scale up production and potential practical applications. #### Materials & Methods High pure sodium azide (NaN3) (≥99.9%) was chosen as nitrogen source to synthesize cg-N. The recrystallization process was performed first. A solution of NaN3 (2 mol/L) was pre-prepared and dropped into a crucible. Then the crucible was transferred into a tubular furnace. The vacuum operation was performed to realize the recrystallization of the NaN3 powder. The reaction temperatures and times within the 200- 300°C and up to 10 hours were tried in the synthesis processes. The optimized synthesis condition with reaction temperature 240 ~ 260 <sup>o</sup> C and heating time 5 hours was employed to synthesize the cg-N. The sample was named as polymerized sodium azide followed by synthesis temperature, time, such as PSA-240o C-5h. According to aforementioned conditions, heating NaN3 without recrystallization process to prepare cg-N was tried and it was named as unpolymerized sodium azide (UPSA). The Raman spectra were recorded in range of 100~1500 cm-1 with a spectral resolu micro laser ( ution of 1 cm scope and m (λ0 =532 nm m-<sup>1</sup> using a multichanne m). an integrated el air cooled d laser Ram d CCD dete man system ector. The e (Renishaw excitation so w) with a co ource is an onfocal Ar ion #### Resul lts Fig. 2 shows the Raman spectra of polymerized NaN3 (PSA) synthesized at 200~300 o C with a reaction time of 3 hours. The sharp peaks of NaN3 near 120, 1267 & 1358 cm-1 are assigned to the vibrational lattice mode, the first overtone of the IR active bending ν2 mode and the symmetric stretching ν1 mode of the azide ion, respectively, which matches well with the Raman spectra of free standing NaN3 [41] . For comparison the NaN3 raw material and a sample heated only at 250 <sup>o</sup> C for 3 hours without the recrystallization process were characterized and showed same Raman vibron profiles without any signals at 635 cm-1 (see Fig. S1). The lines at 635 cm-1 for polymerized sodium nitrogen sample (PSA) are comparable to the theoretical results extrapolated to ambient pressure because the Raman modes tend to soften with decreasing pressure [24, 46, 48] . It also agrees with our recent work on polymerized cg-N based on potassium azide. Therefore the emergent intense vibron at 635 cm-<sup>1</sup> is assigned to the pore breathing A symmetry and is regarded as the finger print peak of cg-N, which unambiguously indicates the successful synthesis of cg-N [46] . The synthesis conditions of cg-N were first systematically investigated by controlling the reaction temperature with a fixed reaction time of 3 hours. As shown in Fig. 2, the vibron intensity of cg-N increases as the reaction temperature raises from 200 <sup>o</sup> C to 240 <sup>o</sup> C compared to that at 1358 cm-<sup>1</sup> for the unreacted or remnant NaN3 in the PSA samples. As temperature increases up to 270 <sup>o</sup> C, the Raman intensity of cg-N remains comprehensive but decreases clearly at 300o C. Additionally, it is worth noting that the intensity of vibrational lattice mode at 120 cm-1 for the unreacted NaN3 in PSA-240 sample dramatically decreases compared to those prepared at other temperatures. The PSA sample exhibits uniformly dark blue color and should be related to the by product formed during the synthesis of cg-N and will be discussed below. Therefore, the optimized temperature range of 240~260 <sup>o</sup> C was adopted for cg-N preparation. A quantitative comparison was made by calculating the intensity ratios of the characteristic Raman peaks to measure the abundance of cg-N phase in the PSA samples. Table I summarizes the partial line intensities of NaN3 and cg-N. The amount of cg-N synthesized is reflected by the peak intensity at 635 cm-1 , while those at 120 and 1358 cm-<sup>1</sup> represent the unreacted NaN3. Herein, a simplified conversion degree from NaN3 to cg-N is defined by the peak intensity ratio. The intensities of NaN3 at 120 cm-1 and 1358 cm-1 , calculated by (I200-Ibg)/(I1358-Ibg) for the NaN3 raw material (SA-R), NaN3 heated only (UPSA) and polymerized NaN3 (PSA) samples at different temperatures are between 2 and 3 and remain little changed. For the SA-R and UPSA samples no trace of cg-N is detected. After recrystallization process the vibron intensity of cg-N enhances dramatically for PSA samples and the ratio of cg-N to NaN3 is 1.26-1.39 calculated by (I635-Ibg)/(I1358-Ibg), and 0.42~ 0.60 calculated by (I635-Ibg)/(I120-Ibg). Therefore it indicates that the crystallization process strongly promotes the formation of cg-N and the optimized synthesis temperature range is 240~260 <sup>o</sup> C. ![Fig. 3 The reaction time dependence of the conversion ratios of cg-N to NaN3 for the PSA samples prepared at 240, 250 & 260 <sup>o</sup> C. The optimized synthesis conditions for the PSA samples are 240~260 <sup>o</sup> C with a reaction time of 5 hours.](path) Further the preparation conditions were explored by synthesizing PSA samples at 240~260 <sup>o</sup> C with variable reaction time. The Raman vibron intensity of cg-N in the PSA-240 samples increases gradually as the reaction time extends up to 5 hours and then decreases, which is shown in Fig. S2. Samples prepared at 250 and 260 <sup>o</sup> C show the same results (Fig. S3). For comparison, the intensity ratios of the cg-N to unreacted NaN3 in theses PSA samples were calculated in Table SI. The synthesis time dependence of the ratios of cg-N to NaN3 at 240-260 <sup>o</sup> C was plotted in Fig. 3. All samples show consistent trends: the PSA samples with a reaction time of 5 hours show the highest conversion degree. Therefore, the optimized preparation conditions for polymerized sodium azide samples are a temperature range of 240~260 <sup>o</sup> C and a reaction time of 5 hours. #### Discussion been proposed, which suffer from the challenge that samples could not recover to ambient pressure and enhancing yield is difficult. Here a simple route was designed for synthesizing cg-N under atmospheric conditions considering the thermodynamic barriers related to transforming to N-N single bonds and activation energy of precursor. In our experiments it was found that recrystallization process played a crucial role to form cg-N. For azides the crystal face with low activation energy may facilitate initiating the transformation of the double bonds N=N in azide ions to single bonded cg-N. The optimized temperatures for cg-N preparation by NaN3 are primarily restricted by the melting points NaN3 (~275°C). These results align well with the fact that the cg-N forms on the crystal face with lower activation energy of azides precursors oriented by the recrystallization process. Regarding the formation mechanism of cg-N it was observed that the polymerized sodium azide turned dark blue in our experiments that should be related to the formation of by-product Na3N during the synthesis of cg-N. This is consistent with the reported results that pH value of the solution increases after synthesis reaction as Na3N displays alkalinity when dissolved in water [38] . A more detailed conversion mechanism is currently at investigation. ### Conclusion The polymeric nitrogen with cubic gauche structure was successfully synthesized by a simple route using sodium azide as a precursor. A high yield may be obtained by separating the by products formed on the surface of the produced cg-N. The route is promising for further scale up or applied to synthesize other metastable materials. #### Conflict of Interest There are no financial conflicts of interest to disclose. #### Acknowledgments The work was supported by the National Natural Science Foundation of China, Ministry of Science & Technology, and CAS Project for Young Scientists in Basic Research of China. ## Author Contributions Changqing Jin & Jun Zhang designed and supervised the whole research. Runteng Chen, Jun Zhang, Zelong Wang, Ke Lu, Yi Peng, Jianfa Zhao, Shaomin Feng and Changqing Jin performed the experiments. Jun Zhang & Changqing Jin wrote the manuscript with contributions from all authors. #### References
Figure 3: Neutron scattering data taken in the [110] x [001] scattering plane of the 100 nm thick BiFeO3 thin film taken at a magnetic field of 10 T and a temperature of 150 K. Figure (a) and (b) show the RSMs around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc Bragg reflections, respectively. The black lines represent the corresponding linescans in [112�]pc direction as shown in Figs. (c) and (d), respectively. Gaussian lineshapes were fitted to the experimental data, as indicated by the dashed and solid lines.	# Continuous Collapse of the Spin Cycloid in BiFeO3 Thin Films under an Applied Magnetic Field probed by Neutron Scattering <sup>1</sup> School of Physics, The University of New South Wales, NSW 2052, Australia 2 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA. Ningbo Institute of Materials Technology & Engineering, Chinese Academy of Sciences, Ningbo 315201, China 4 Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA 5 School of Engineering, Brown University, Providence, RI, 02912, USA. 6 Department of Physics, University of California, Berkeley, CA, 94720, USA. Md. Firoz Pervez, ORCID: [0000-0002-1453-8704](https://orcid.org/0000-0002-1453-8704) Clemens Ulrich, ORCID: 0000-0002-6829-9374 Hongrui Zhang, ORCID: [0000-0001-7896-019X](https://orcid.org/0000-0001-7896-019X) Yen-Lin Huang, ORCID: 0000-0002-6129-8547 Lucas Caretta, ORCID: 0000-0001-7229-7980 Ramamoorthy Ramesh, ORCID: 0000-0003-0524-1332 ## ABSTRACT Bismuth ferrite (BiFeO3) is one of the rare materials that exhibits multiferroic properties already at room-temperature. Therefore, it offers tremendous potential for future technological applications, such as memory and logic. However, a weak magnetoelectric coupling together with the presence of a noncollinear cycloidal spin order restricts various practical applications of BiFeO3. Therefore, there is a large interest in the search for suitable methods for the modulation of the spin cycloid in BiFeO3. By performing neutron diffraction experiments using a triple-axis instrument we have determined that the spin cycloid can be systematically suppressed by applying a high magnetic field of 10 T in a BiFeO3 thin film of about 100 nm grown on a (110)-oriented SrTiO3 substrate. As predicted by previous theoretical calculations, we observed that the required critical magnetic field to suppress the spin cycloid in a BiFeO3 thin film was lower as compared to the previously reported critical magnetic field for bulk BiFeO3 single crystals. Our experiment reveals that the spin cycloid continuously expands with increasing magnetic field before the complete transformation into a G-type antiferromagnetic spin order. Such tuning of the length of the spin cycloid up to a complete suppression offers new functionalities for future technological applications as in spintronics or magnonics. ## 1. INTROCUCTION Multiferroics are a fascinating class of materials that manifest multiple, simultaneous ferroic orders such as ferroelectric and magnetic polarizations. Magnetoelectric coupling between the ferroelectric and magnetic orders, including ferromagnetic, ferrimagnetic, antiferromagnetic or more complex noncollinear spin structures has been observed in single or multiphase multiferroic materials. Often this magnetoelectric coupling emerges from the simultaneous breaking of spatial inversion symmetry and time-reversal symmetry [1-2]. A magnetoelectric coupling in multiferroics allows for an efficient method of the electric control of the magnetic order as well as the magnetic control of electric polarization. Therefore, the magnetoelectric coupling offers an enormous potential for the application in next-generation spintronics and magnonic devices (e.g., memory devices, magnetic switches, magnetic sensors, high-frequency magnetic devices, spin valve devices, etc.) at much lower power consumption and faster operation as compared to the present conventional electronic devices [1-13]. Bismuth ferrite (BiFeO3) is one of the rare room temperature type-I, single-phase multiferroic oxides where primarily independent magnetic and ferroelectric order emerge [1,9]. However, a weak direct coupling between magnetic and ferroelectric order is present in BiFeO<sup>3</sup> [7,14,15]. Bulk BiFeO<sup>3</sup> is a rhombohedrally distorted cubic perovskite with the polar space group 3 − 3 6 [16] and a lattice parameter of 3.968 Å in the pseudocubic (pc) notation. This notation will be used throughout this article. In the case of BiFeO<sup>3</sup> thin films, epitaxial tensile or compressive strain results in a further distortion of the in-plane and out-of-plane lattice parameters. As consequence, different crystal structures including tetragonal (4), monoclinic ( ), monoclinic ( ), rhombohedral (3) and orthorhombic ( 2) are possible in BiFeO<sup>3</sup> thin films [17,18]. In the BiFeO<sup>3</sup> single crystals, the displacement of the A-site Bi 3+ ions and the lone pair selectrons cause the rhombohedral distortion which leads to the spontaneous ferroelectric polarization of about 100 μC cm−2 along the direction of the body diagonal [111]pc. This spontaneous ferroelectric polarization persists up to high temperatures (TC~1123 K) [19,20]. Below 643 K BiFeO<sup>3</sup> possesses a complex noncollinear magnetic structure arising from the Fe3+ ions along with nearly zero average magnetization [21]. The Fe3+ magnetic moments of the nearest neighbors of the adjacent (111)pc planes are ordered in a predominantly G-type antiferromagnetic spin structure. The Dzyaloshinskii-Moriya interaction, which is caused by both, the spin-orbit coupling and a broken inversion symmetry, tends to favor a particular spin rotation in BiFeO<sup>3</sup> which leads to a cycloidal spin order with a propagation length of about λ = 63 nm. In bulk BiFeO<sup>3</sup> the spin cycloid propagates along one of three crystallographic directions: [1�10]pc, [101�]pc or [011�]pc where the rotation axis of the spins is perpendicular to the plane defined by the propagation vector and the direction of the electric polarization along [111]pc [20-25]. Through the existence of s spin cycloid in BiFeO3 thin films was long time under debate, in 2010 Ke et al. found a type-I spin cycloid (~64 nm ) propagating along the [11�0]pc direction in a partially relaxed BiFeO<sup>3</sup> thin film with a thickness of about 800 nm deposited on a (100)-oriented SrTiO<sup>3</sup> substrate [26]. In 2011, Ratcliff et al. discovered the presence of a spin cycloid (~62 nm) in a 1 μm thin BiFeO<sup>3</sup> film deposited on a (110)-SrTiO<sup>3</sup> substrate [27], propagating along a unique [112�]pc direction, which is different from the direction [11�0]pc in the bulk single crystals [24,28]. In a neutron scattering study, Bertinshaw et al. discovered a spin cycloid in a BiFeO<sup>3</sup> thin film of just 100 nm when grown on a (110)-oriented SrTiO<sup>3</sup> substrate with a thin SrRuO<sup>3</sup> intermediate layer [29]. This report also showed that the length of the spin cycloid extends and diverges to infinity at the magnetic phase transition temperature of 650 ± 10 K. Finally, the report of Burns et al. revealed a dependence of the length of the spin cycloid on the film thickness. They observed that the length of spin cycloid increases for decreasing film thicknesses down to 20 nm [30]. Various factors can influence the spin cycloid in BiFeO3 thin films such as temperature, external electric and magnetic fields, film thickness, epitaxial strain, an intermediate layer, doping, etc. [8,10,26-34]. One of the key experimental strategies for understanding the fundamental physical interactions and domain states in magnetic materials is applying an external magnetic field. Several experimental techniques, including magnetization measurements and electric polarization measurements [7,14,15,35-38], electron spin resonance experiments (ESR) [39] and Raman light scattering experiments [40] have been performed to investigate the influence of an applied magnetic field (pulsed and static) on BiFeO3. They showed that applying a high magnetic field can suppress spin cycloid in BiFeO<sup>3</sup> single crystals or polycrystalline ceramics at a magnetic field of 16 T to 20 T [7,14,15,35-38,41,42]. A theoretical study by Gareeva et al. predicted the lowering of the required critical magnetic field for strained BiFeO<sup>3</sup> thin films as compared to BiFeO<sup>3</sup> single crystals [43,44]. Moreover, the Raman experiments by Agbelele et al. on BiFeO<sup>3</sup> thin films [40] reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T−6 T for BiFeO3 thin films grown on different substrates. Neutron scattering experiments would unambiguously reveal the nature of magnetic phase transition and would determine the precise magnetic spin structure in BiFeO3 thin films. Therefore, we have performed neutron diffraction experiments on a 100 nm thick BiFeO<sup>3</sup> film deposited by the PLD technique on a (110)-oriented SrTiO3 substrate. The experiment was performed using a triple-axis spectrometer at high magnetic fields of up to 10 T, in order to investigate the overall effect of a magnetic field on the spin structure in BiFeO3 thin films. ## 2. EXPERIMENTAL DETAILS The BiFeO<sup>3</sup> thin film sample with a film thickness of ~100 nm was grown by the Pulsed Laser Deposition on a (110)-oriented single-sided polished, 10 mm × 10 mm × 0.5 mm SrTiO<sup>3</sup> substrate (SHINKOSHA CO., LTD.). As an SrRuO3 intermediate layers can become magnetic at low temperature, the sample was grown without an intermediate layer. The lattice mismatch between SrTiO3 (apc = 3.905 Å [45]) and BiFeO3 (apc= 3.968 Å [16]) leads to an out-of-plane elongated lattice parameter, resulting in a monoclinically distorted structure of the BiFeO<sup>3</sup> films [18,27,29,30,46,47]. The triple-axis spectrometer instrument TAIPAN [48], located at the Australian Centre for Neutron Scattering (ACNS) at the Australian Nuclear Science and Technology Organisation (ANSTO) in Sydney, Australia, has demonstrated to be the ideal choice for the measurement of magnetic Bragg peaks of transition metal oxide thin films due to its enhanced resolution and the improved signal-to-background ratio as compared to conventional neutron diffraction instruments [29,30,49]. In order to apply a high magnetic field, a 12 T superconducting magnet was used. The magnetic field was applied in the direction perpendicular to the scattering plane, i.e. along the [11�0] in-plane direction of the film. In order to suppress contaminations from second order reflections from the SrTiO3 substrate at the � 1 2 1 2 1 2 � Bragg peak position, two pyrolytic-graphite, PG (002), filters with a total thickness of 60 mm were placed behind the sample. An incident energy of 14.86 eV (λ = 2.3462 Å) was used in the standard elastic scattering mode. This was provided by a PG-monochromator and a PG-analyzer with vertical focusing but horizontally flat configuration. In order to enhance the resolution and to suppress the background, 40′′ collimators were placed in the neutron beam before and after the sample (see also Ref. [29,30,49]). A linescan was performed around the � 1 2 1 2 1 2 � Bragg peak of the second order contamination of the SrTiO3 substrate without the two PG (002) filters to determine the instrumental resolution of a Full Width at Half Maximum of 0.0032 reciprocal lattice units (r.l.u.). The substrate SrTiO3 possesses a crystallographic phase transition at 105 K. In order to avoid corresponding effects on the BiFeO<sup>3</sup> thin film, the neutron scattering measurements were performed at the temperature of 150 K [49], which also provides a stronger magnetic signal from the BiFeO3thin film compared to room temperature [29]. ## 3. RESULTS AND DISCUSSION Figure 1 shows the neutron diffraction data of the 100 nm thick BiFeO3 film taken at a temperature of 150 K and zero magnetic field. In order to access the magnetic Bragg peaks arising from the spin cycloid, the film was grown on a (110)-oriented SrTiO3 substrate and mounted in the [110] x [001] scattering plane. Figure 1(a) shows the neutron scattering reciprocal space map (RSM) around the � 1 2 1 2 1 2 �pc Bragg reflection. The peak at precisely � 1 2 1 2 1 2 �pc arises from the second order contamination of the (111) structural Bragg reflection of the SrTiO3 substrate and is still visible despite the use of two PG (002) filters. The two Bragg peaks at around QHK = 0.485 can be attributed to magnetic scattering from the spin cycloid of the BiFeO3 thin film. The reduced QHK value is a consequence of the epitaxial strain, which leads to an increase of the out-of-plane lattice parameter [29,30]. The distance between the peak maxima indicates the length of the spin cycloid (λ = 2π/δ) and the direction of their separation, as shown by the black line, corresponds to the propagation direction of the spin cycloid along [112�]pc. This propagation direction was also observed by previous neutron diffraction experiments on BiFeO3 thin films [26,27,29,30]. Theoretical studies have predicted that a spin cycloid can possibly propagate either along the [112�]pc, [12�1]pc or [2�11]pc directions in a BiFeO3 thin film deposited on a (110)-oriented SrTiO<sup>3</sup> substrate as compared to the [1�10]pc, [101�]pc or [011�]pc directions in bulk BiFeO3 [27,50]. The change in the propagation direction is a consequence of the in-plane strain of the BiFeO3 film. Figure 1(b) shows the RSM around the � 1 2 1 2 1 2 ̅ �pc. Only one magnetic Bragg peak appears. Due to the orientation of the instrumental resolution ellipsoid the two magnetic Bragg peaks of the spin cycloid are overlapping in this measurement geometry (see Suppl. Mater of Ref. [30]). Figures 1(c) and 1(d) show the corresponding line scans along the black lines in Figs. 1(a) and (b), respectively. The red lines are Gaussian lineshapes fitted to the experimental data. For the fit the linewidth was kept constant for both magnetic Bragg peaks. The separation of the two magnetic Bragg peaks in Fig. 1(c) is = Δ = Δ = Δ = 0.00415, which corresponds to a length of the spin cycloid of 65.7 ± 1.0 nm. This is in good agreement to the value of about 63 nm obtained for bulk BiFeO3 [20-25] and for BiFeO3 thin films [27,29,30]. Figure 1(d) shows the corresponding linescan over the � 1 2 1 2 1 2 ̅ �pc RSM as indicated by the black line in Fig. 1(b). Note, to avoid the presence of the λ/2 contamination from the SrTiO3 substrate, the linescan was not taken precisely through the peak maximum but slightly shifted as indicated by the black line in Fig. 1(b). Only one sharp peak is observed. Its integrated intensity corresponds roughly to the integrated intensity of the two magnetic Bragg peaks in Fig. 1(b) [27,29,30]. The magnetic field dependent data are shown in Fig. 2. The data were taken around the � 1 2 1 2 1 2 �pc Bragg reflection following the black line of the RSM as shown in Fig. 1(a), i.e. along the [112�]pc direction. The magnetic field was increased in 1 T steps to a strength of 10 T (Fig. 2(a)) and then decreased to zero field again (Fig. 2(b)). The obtained data were analyzed by fitting two Gaussian lineshapes to the experimental data, where the linewidth and peak intensity of each of the two Gaussian peaks were kept constant using the values obtained for the 0 T measurement. With increasing magnetic field the distance between the two magnetic Bragg peaks decreases, which converge into one single peak at a magnetic field of 10 T (see Fig. 2(a)). The intensity of the peak maximum increases with increasing magnetic field to a value of about twice the maximum intensity at 0 T. However, the integrated intensity over both peaks remains almost constant. The merging into one single Bragg peak is an indication that the length of the spin cycloid increases continuously and finally diverges to infinity at a magnetic field of 10 T. The resulting spin structure is G-type antiferromagnetic, i.e. the spin cycloid collapses into a G-type antiferromagnetic spin arrangement. Upon decreasing magnetic field the peak splits again into two well separated peaks and the overall height of the peak maximum decreases with decreasing magnetic field see Fig. 2(b)). The initial lineshape, both in peak intensity and peak splitting has recovered when reaching 0 T again, i.e. the length of the spin cycloid reaches the same value of 65.7 ± 1.0 nm after the magnetic field cycle. In order to confirm that only one magnetic Bragg peak remains at 10 T, RSMs were taken around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc positions at 10 T and (see Figs. 3(a) and 3(b), respectively). The corresponding linescans along the [112�]pc direction through the peak maxima are shown in Figs. 3(c) and 3(d). At zero magnetic field two magnetic Bragg peaks were present (see Figs. 1(a) and 1(c)) which indicates the presence of a spin cycloid in the BiFeO3 thin film. At a magnetic field of 10 T the two magnetic Bragg peaks have converged into one single peak. The intensity of its peak maximum is almost twice the intensity of an initial single Bragg peak. Note, due to the orientation of the elliptical resolution function for the � 1 2 1 2 1 2 ̅ �pc Bragg reflections, in Figs (b) and (d) only one single peak is expected even in the presence of a peak splitting. Figure 4 shows the overall result of the magnetic field dependence of the neutron diffraction signal of the 100 nm thick BiFeO3 thin film. The magnetic field dependence of the peak maximum is shown in Fig. 4(a) for an increasing (blue symbols) and a decreasing (red symbols) magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. In Fig. 4(b) the magnetic field dependence of the length of the spin cycloid as extracted from the splitting of the two magnetic Bragg peaks, is presented. The distance between both Bragg peaks decreases continuously and converges towards zero at 10 T. This indicates that the length of the spin cycloid increases systematically and diverges to infinity at a magnetic field of about 10 T. The spin cycloid expands and finally collapses into a commensurate G-type antiferromagnetic spin structure. Since only one set of magnetic Bragg peaks was measured in this experiment, no conclusion can be drawn if the final antiferromagnetic structure is canted. It should be noted that both, the peak intensity and the length of the spin cycloid recover with decreasing magnetic field and exhibited a slight hysteresis behavior between 4 T and 10 T. For bulk BiFeO3 an abrupt phase transition was observed between about 16 T and 20 T. Based on measurements of the electric polarization, magnetization, and ESR in combination with theoretical modelling, the resulting phase was attributed to a canted antiferromagnetic structure [7,14,37-39]. In addition, an intermediate phase was observed by ESR and neutron diffraction experiments between about 10 T and 20 T [39,42]. The magnetic structure of this phase was described as conical. It is important to note that the neutron diffraction experiments on BiFeO3 single crystals under high magnetic field did not indicate any change in the incommensurable splitting in both phases, i.e. the length of the spin cycloid did not change with magnetic field [41,42]. This is in contrast to our neutron diffraction experiments on a 100 nm BiFeO3 thin film where the length of the spin cycloid increases systematically and diverges at 10 T. The present experiment on a BiFeO3 thin film does not provide any information about the conical intermediate phase. With a maximum of 10 T this phase was perhaps not reached. In addition, a conical spin structure would cause two magnetic Bragg peaks in the direction perpendicular to the scattering plane. Due to the finite out-of-plane instrumental resolution, they would appear as a single magnetic Bragg peak in the centre. Since the incommensurate splitting of the two magnetic Bragg peaks did converge to zero, we would consider this scenario as unlikely. An external magnetic field aligns the spins along the direction of the field. As the spin cycloid is partly anisotropic on a large scale, the applied magnetic field direction and magnetization direction are not same [15]. With increasing the magnetic field the Zeeman interaction strength can become comparable to the DM interaction. At a critical magnetic field this can lead to a suppression of the spin cycloidal. This can be expressed through the Landau-Ginzburg formalism of its free energy [25,38,39]. The Lifshitz invariant (a term in the Landau-Ginzburg theory that describes a coupling between the magnetization and its spatial variation due to the cycloidal spin structure) is increased by applying a magnetic field. For BiFeO<sup>3</sup> this can lead to the transformation of spin cycloid order into a simpler collinear antiferromagnetic state as observed in bulk BiFeO3 [7,14,37-39,41.42]. Theoretical studies have suggested that a lower critical magnetic field is required to suppress the cycloidal spin order in epitaxially grown BiFeO<sup>3</sup> thin films [43,44]. This was confirmed by the Raman experiments of Agbelele et al. in BiFeO<sup>3</sup> thin films [40], where they reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T − 6 T (depending on the chosen substrate) for BiFeO<sup>3</sup> thin films with a thickness of 70 nm grown on different (001)-oriented substrates, i.e. DyScO3, GdScO3, and SmScO3. In comparison to our BiFeO3 thin film grown on (110)-oriented SrTiO3 substrate they had a significantly smaller epitaxial strain (see Refs. [30] and [40], respectively). Therefore, our critical value of 10 T for the collapse of the spin cycloid in a BeFeO3 thin film is in accordance with the results of Agbelele et al. on differently strained BiFeO3 thin films [40]. ## 4. CONCLUSION In summary, our neutron diffraction experiment using a triple-axis spectrometer revealed a magnetic field-dependent change of the spin cycloid in a 100 nm BiFeO<sup>3</sup> film deposited by PLD on a (110)-oriented SrTiO<sup>3</sup> substrate. The experimental result show a continuous change of the spin cycloidal upon applying a magnetic field. The length of the spin cycloid expands systematically and diverges to infinity at an applied magnetic field of 10 T where the spin cycloidal transforms into a commensurate G-type antiferromagnetic spin structure. A transition to a canted antiferromagnetic spin structure was observed before for polycrystalline BiFeO<sup>3</sup> and single crystals but at much higher magnetic fields in the range of 16 T to 20 T. In contrast, for bulk BiFeO3 the incommensurable splitting of the magnetic Bragg peaks did not change with the magnetic field. Raman light scattering experiments showed that BiFeO3 thin films with lower epitaxial strain when grown on different (001)-oriented substrates had a transition from the cycloidal phase to a collinear spin structure at a lower magnetic field of 4-6 T, depending on the substrate. Our neutron diffraction experiments directly proof that the length of the cycloid systematically extends and transitions continuously into a G-type antiferromagnetic structure. This provides important information about the electronic correlations and microscopic physical effects behind the origin of the spin cycloid in BiFeO3. Furthermore, our results demonstrate that the application of a magnetic field offers a further route for the manipulation of the spin cycloid in multiferroics. Therefore, our experimental result will help to develop strategies to improve the linear magnetoelectric coupling in BiFeO<sup>3</sup> thin films with larger electric polarization, which is crucial for future technological applications. ## ACKNOWLEDGEMENT We would like to thank ANSTO for providing the neutron scattering beamtime at the instrument TAIPAN through the proposals P7934 and P9961, and Guochu Deng for his support as beamline scientist. ## REFERENCES - [1] M. Fiebig, T. [Lottermoser,](https://www.nature.com/articles/natrevmats201646#auth-Thomas-Lottermoser-Aff1) [D. Meier,](https://www.nature.com/articles/natrevmats201646#auth-Dennis-Meier-Aff1) and [M. Trassin,](javascript:;) Nature Rev. Mater. 1, 16046 (2016), doi: 10.1038/natrevmats.2016.46 - [8] W. Ratcliff, J. W. Lynn, V. [Kiryukhin,](https://www.nature.com/articles/npjquantmats20163#auth-Valery-Kiryukhin-Aff2) P. Jain, and [M. R. 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Series. 340, 012003 (2012), doi: 10.1088/1742- 6596/340/1/012003 ![Figure 4: (a) Intensity of the pek maximum of the neuron scattering data as a function of magnetic field as shown in Fig. 2. The blue symbols indicate the data obtained for an increasing magnetic field and the red symbols present the data of a decreasing magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. (b) length of the spin cycloid extracted from the splitting of the two magnetic Bragg peaks. The length of the spin cycloid is 65.7 ± 1.0 nm at zero magnetic field, expands systematically with increasing magnetic field and diverges towards infinity at a magnetic field of 10 T. The inset shows the zoomed-in data of the cycloidal length and the dashed horizontal line indicates the spin cycloid length of 63 nm obtained previously on bulk and thin film BiFeO3 samples. The solid lines serve as a guides to the eye.](path)
Figure 2: Neutron scattering data of the magnetic field dependence of the two incommensurate magnetic Bragg peaks in a 100 nm thick BiFeO3 thin films. The linescans follow the black line shown in Fig. 1(a) and were measured at a temperature of 150 K in the magnetic field range from 0 T to 10 T, i.e. in (a) for an increasing magnetic field and in (b) for a decreasing magnetic field back to 0 T. The dashed and solid lines correspond to the result of a fit of two Gaussian lineshapes to the experimental data. For the fit the linewidth was kept constant at the value determined at zero magnetic field and the intensity of both Bragg peaks was kept constant at each magnetic field.	# Continuous Collapse of the Spin Cycloid in BiFeO3 Thin Films under an Applied Magnetic Field probed by Neutron Scattering <sup>1</sup> School of Physics, The University of New South Wales, NSW 2052, Australia 2 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA. Ningbo Institute of Materials Technology & Engineering, Chinese Academy of Sciences, Ningbo 315201, China 4 Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA 5 School of Engineering, Brown University, Providence, RI, 02912, USA. 6 Department of Physics, University of California, Berkeley, CA, 94720, USA. Md. Firoz Pervez, ORCID: [0000-0002-1453-8704](https://orcid.org/0000-0002-1453-8704) Clemens Ulrich, ORCID: 0000-0002-6829-9374 Hongrui Zhang, ORCID: [0000-0001-7896-019X](https://orcid.org/0000-0001-7896-019X) Yen-Lin Huang, ORCID: 0000-0002-6129-8547 Lucas Caretta, ORCID: 0000-0001-7229-7980 Ramamoorthy Ramesh, ORCID: 0000-0003-0524-1332 ## ABSTRACT Bismuth ferrite (BiFeO3) is one of the rare materials that exhibits multiferroic properties already at room-temperature. Therefore, it offers tremendous potential for future technological applications, such as memory and logic. However, a weak magnetoelectric coupling together with the presence of a noncollinear cycloidal spin order restricts various practical applications of BiFeO3. Therefore, there is a large interest in the search for suitable methods for the modulation of the spin cycloid in BiFeO3. By performing neutron diffraction experiments using a triple-axis instrument we have determined that the spin cycloid can be systematically suppressed by applying a high magnetic field of 10 T in a BiFeO3 thin film of about 100 nm grown on a (110)-oriented SrTiO3 substrate. As predicted by previous theoretical calculations, we observed that the required critical magnetic field to suppress the spin cycloid in a BiFeO3 thin film was lower as compared to the previously reported critical magnetic field for bulk BiFeO3 single crystals. Our experiment reveals that the spin cycloid continuously expands with increasing magnetic field before the complete transformation into a G-type antiferromagnetic spin order. Such tuning of the length of the spin cycloid up to a complete suppression offers new functionalities for future technological applications as in spintronics or magnonics. ## 1. INTROCUCTION Multiferroics are a fascinating class of materials that manifest multiple, simultaneous ferroic orders such as ferroelectric and magnetic polarizations. Magnetoelectric coupling between the ferroelectric and magnetic orders, including ferromagnetic, ferrimagnetic, antiferromagnetic or more complex noncollinear spin structures has been observed in single or multiphase multiferroic materials. Often this magnetoelectric coupling emerges from the simultaneous breaking of spatial inversion symmetry and time-reversal symmetry [1-2]. A magnetoelectric coupling in multiferroics allows for an efficient method of the electric control of the magnetic order as well as the magnetic control of electric polarization. Therefore, the magnetoelectric coupling offers an enormous potential for the application in next-generation spintronics and magnonic devices (e.g., memory devices, magnetic switches, magnetic sensors, high-frequency magnetic devices, spin valve devices, etc.) at much lower power consumption and faster operation as compared to the present conventional electronic devices [1-13]. Bismuth ferrite (BiFeO3) is one of the rare room temperature type-I, single-phase multiferroic oxides where primarily independent magnetic and ferroelectric order emerge [1,9]. However, a weak direct coupling between magnetic and ferroelectric order is present in BiFeO<sup>3</sup> [7,14,15]. Bulk BiFeO<sup>3</sup> is a rhombohedrally distorted cubic perovskite with the polar space group 3 − 3 6 [16] and a lattice parameter of 3.968 Å in the pseudocubic (pc) notation. This notation will be used throughout this article. In the case of BiFeO<sup>3</sup> thin films, epitaxial tensile or compressive strain results in a further distortion of the in-plane and out-of-plane lattice parameters. As consequence, different crystal structures including tetragonal (4), monoclinic ( ), monoclinic ( ), rhombohedral (3) and orthorhombic ( 2) are possible in BiFeO<sup>3</sup> thin films [17,18]. In the BiFeO<sup>3</sup> single crystals, the displacement of the A-site Bi 3+ ions and the lone pair selectrons cause the rhombohedral distortion which leads to the spontaneous ferroelectric polarization of about 100 μC cm−2 along the direction of the body diagonal [111]pc. This spontaneous ferroelectric polarization persists up to high temperatures (TC~1123 K) [19,20]. Below 643 K BiFeO<sup>3</sup> possesses a complex noncollinear magnetic structure arising from the Fe3+ ions along with nearly zero average magnetization [21]. The Fe3+ magnetic moments of the nearest neighbors of the adjacent (111)pc planes are ordered in a predominantly G-type antiferromagnetic spin structure. The Dzyaloshinskii-Moriya interaction, which is caused by both, the spin-orbit coupling and a broken inversion symmetry, tends to favor a particular spin rotation in BiFeO<sup>3</sup> which leads to a cycloidal spin order with a propagation length of about λ = 63 nm. In bulk BiFeO<sup>3</sup> the spin cycloid propagates along one of three crystallographic directions: [1�10]pc, [101�]pc or [011�]pc where the rotation axis of the spins is perpendicular to the plane defined by the propagation vector and the direction of the electric polarization along [111]pc [20-25]. Through the existence of s spin cycloid in BiFeO3 thin films was long time under debate, in 2010 Ke et al. found a type-I spin cycloid (~64 nm ) propagating along the [11�0]pc direction in a partially relaxed BiFeO<sup>3</sup> thin film with a thickness of about 800 nm deposited on a (100)-oriented SrTiO<sup>3</sup> substrate [26]. In 2011, Ratcliff et al. discovered the presence of a spin cycloid (~62 nm) in a 1 μm thin BiFeO<sup>3</sup> film deposited on a (110)-SrTiO<sup>3</sup> substrate [27], propagating along a unique [112�]pc direction, which is different from the direction [11�0]pc in the bulk single crystals [24,28]. In a neutron scattering study, Bertinshaw et al. discovered a spin cycloid in a BiFeO<sup>3</sup> thin film of just 100 nm when grown on a (110)-oriented SrTiO<sup>3</sup> substrate with a thin SrRuO<sup>3</sup> intermediate layer [29]. This report also showed that the length of the spin cycloid extends and diverges to infinity at the magnetic phase transition temperature of 650 ± 10 K. Finally, the report of Burns et al. revealed a dependence of the length of the spin cycloid on the film thickness. They observed that the length of spin cycloid increases for decreasing film thicknesses down to 20 nm [30]. Various factors can influence the spin cycloid in BiFeO3 thin films such as temperature, external electric and magnetic fields, film thickness, epitaxial strain, an intermediate layer, doping, etc. [8,10,26-34]. One of the key experimental strategies for understanding the fundamental physical interactions and domain states in magnetic materials is applying an external magnetic field. Several experimental techniques, including magnetization measurements and electric polarization measurements [7,14,15,35-38], electron spin resonance experiments (ESR) [39] and Raman light scattering experiments [40] have been performed to investigate the influence of an applied magnetic field (pulsed and static) on BiFeO3. They showed that applying a high magnetic field can suppress spin cycloid in BiFeO<sup>3</sup> single crystals or polycrystalline ceramics at a magnetic field of 16 T to 20 T [7,14,15,35-38,41,42]. A theoretical study by Gareeva et al. predicted the lowering of the required critical magnetic field for strained BiFeO<sup>3</sup> thin films as compared to BiFeO<sup>3</sup> single crystals [43,44]. Moreover, the Raman experiments by Agbelele et al. on BiFeO<sup>3</sup> thin films [40] reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T−6 T for BiFeO3 thin films grown on different substrates. Neutron scattering experiments would unambiguously reveal the nature of magnetic phase transition and would determine the precise magnetic spin structure in BiFeO3 thin films. Therefore, we have performed neutron diffraction experiments on a 100 nm thick BiFeO<sup>3</sup> film deposited by the PLD technique on a (110)-oriented SrTiO3 substrate. The experiment was performed using a triple-axis spectrometer at high magnetic fields of up to 10 T, in order to investigate the overall effect of a magnetic field on the spin structure in BiFeO3 thin films. ## 2. EXPERIMENTAL DETAILS The BiFeO<sup>3</sup> thin film sample with a film thickness of ~100 nm was grown by the Pulsed Laser Deposition on a (110)-oriented single-sided polished, 10 mm × 10 mm × 0.5 mm SrTiO<sup>3</sup> substrate (SHINKOSHA CO., LTD.). As an SrRuO3 intermediate layers can become magnetic at low temperature, the sample was grown without an intermediate layer. The lattice mismatch between SrTiO3 (apc = 3.905 Å [45]) and BiFeO3 (apc= 3.968 Å [16]) leads to an out-of-plane elongated lattice parameter, resulting in a monoclinically distorted structure of the BiFeO<sup>3</sup> films [18,27,29,30,46,47]. The triple-axis spectrometer instrument TAIPAN [48], located at the Australian Centre for Neutron Scattering (ACNS) at the Australian Nuclear Science and Technology Organisation (ANSTO) in Sydney, Australia, has demonstrated to be the ideal choice for the measurement of magnetic Bragg peaks of transition metal oxide thin films due to its enhanced resolution and the improved signal-to-background ratio as compared to conventional neutron diffraction instruments [29,30,49]. In order to apply a high magnetic field, a 12 T superconducting magnet was used. The magnetic field was applied in the direction perpendicular to the scattering plane, i.e. along the [11�0] in-plane direction of the film. In order to suppress contaminations from second order reflections from the SrTiO3 substrate at the � 1 2 1 2 1 2 � Bragg peak position, two pyrolytic-graphite, PG (002), filters with a total thickness of 60 mm were placed behind the sample. An incident energy of 14.86 eV (λ = 2.3462 Å) was used in the standard elastic scattering mode. This was provided by a PG-monochromator and a PG-analyzer with vertical focusing but horizontally flat configuration. In order to enhance the resolution and to suppress the background, 40′′ collimators were placed in the neutron beam before and after the sample (see also Ref. [29,30,49]). A linescan was performed around the � 1 2 1 2 1 2 � Bragg peak of the second order contamination of the SrTiO3 substrate without the two PG (002) filters to determine the instrumental resolution of a Full Width at Half Maximum of 0.0032 reciprocal lattice units (r.l.u.). The substrate SrTiO3 possesses a crystallographic phase transition at 105 K. In order to avoid corresponding effects on the BiFeO<sup>3</sup> thin film, the neutron scattering measurements were performed at the temperature of 150 K [49], which also provides a stronger magnetic signal from the BiFeO3thin film compared to room temperature [29]. ## 3. RESULTS AND DISCUSSION Figure 1 shows the neutron diffraction data of the 100 nm thick BiFeO3 film taken at a temperature of 150 K and zero magnetic field. In order to access the magnetic Bragg peaks arising from the spin cycloid, the film was grown on a (110)-oriented SrTiO3 substrate and mounted in the [110] x [001] scattering plane. Figure 1(a) shows the neutron scattering reciprocal space map (RSM) around the � 1 2 1 2 1 2 �pc Bragg reflection. The peak at precisely � 1 2 1 2 1 2 �pc arises from the second order contamination of the (111) structural Bragg reflection of the SrTiO3 substrate and is still visible despite the use of two PG (002) filters. The two Bragg peaks at around QHK = 0.485 can be attributed to magnetic scattering from the spin cycloid of the BiFeO3 thin film. The reduced QHK value is a consequence of the epitaxial strain, which leads to an increase of the out-of-plane lattice parameter [29,30]. The distance between the peak maxima indicates the length of the spin cycloid (λ = 2π/δ) and the direction of their separation, as shown by the black line, corresponds to the propagation direction of the spin cycloid along [112�]pc. This propagation direction was also observed by previous neutron diffraction experiments on BiFeO3 thin films [26,27,29,30]. Theoretical studies have predicted that a spin cycloid can possibly propagate either along the [112�]pc, [12�1]pc or [2�11]pc directions in a BiFeO3 thin film deposited on a (110)-oriented SrTiO<sup>3</sup> substrate as compared to the [1�10]pc, [101�]pc or [011�]pc directions in bulk BiFeO3 [27,50]. The change in the propagation direction is a consequence of the in-plane strain of the BiFeO3 film. Figure 1(b) shows the RSM around the � 1 2 1 2 1 2 ̅ �pc. Only one magnetic Bragg peak appears. Due to the orientation of the instrumental resolution ellipsoid the two magnetic Bragg peaks of the spin cycloid are overlapping in this measurement geometry (see Suppl. Mater of Ref. [30]). Figures 1(c) and 1(d) show the corresponding line scans along the black lines in Figs. 1(a) and (b), respectively. The red lines are Gaussian lineshapes fitted to the experimental data. For the fit the linewidth was kept constant for both magnetic Bragg peaks. The separation of the two magnetic Bragg peaks in Fig. 1(c) is = Δ = Δ = Δ = 0.00415, which corresponds to a length of the spin cycloid of 65.7 ± 1.0 nm. This is in good agreement to the value of about 63 nm obtained for bulk BiFeO3 [20-25] and for BiFeO3 thin films [27,29,30]. Figure 1(d) shows the corresponding linescan over the � 1 2 1 2 1 2 ̅ �pc RSM as indicated by the black line in Fig. 1(b). Note, to avoid the presence of the λ/2 contamination from the SrTiO3 substrate, the linescan was not taken precisely through the peak maximum but slightly shifted as indicated by the black line in Fig. 1(b). Only one sharp peak is observed. Its integrated intensity corresponds roughly to the integrated intensity of the two magnetic Bragg peaks in Fig. 1(b) [27,29,30]. The magnetic field dependent data are shown in Fig. 2. The data were taken around the � 1 2 1 2 1 2 �pc Bragg reflection following the black line of the RSM as shown in Fig. 1(a), i.e. along the [112�]pc direction. The magnetic field was increased in 1 T steps to a strength of 10 T (Fig. 2(a)) and then decreased to zero field again (Fig. 2(b)). The obtained data were analyzed by fitting two Gaussian lineshapes to the experimental data, where the linewidth and peak intensity of each of the two Gaussian peaks were kept constant using the values obtained for the 0 T measurement. With increasing magnetic field the distance between the two magnetic Bragg peaks decreases, which converge into one single peak at a magnetic field of 10 T (see Fig. 2(a)). The intensity of the peak maximum increases with increasing magnetic field to a value of about twice the maximum intensity at 0 T. However, the integrated intensity over both peaks remains almost constant. The merging into one single Bragg peak is an indication that the length of the spin cycloid increases continuously and finally diverges to infinity at a magnetic field of 10 T. The resulting spin structure is G-type antiferromagnetic, i.e. the spin cycloid collapses into a G-type antiferromagnetic spin arrangement. Upon decreasing magnetic field the peak splits again into two well separated peaks and the overall height of the peak maximum decreases with decreasing magnetic field see Fig. 2(b)). The initial lineshape, both in peak intensity and peak splitting has recovered when reaching 0 T again, i.e. the length of the spin cycloid reaches the same value of 65.7 ± 1.0 nm after the magnetic field cycle. In order to confirm that only one magnetic Bragg peak remains at 10 T, RSMs were taken around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc positions at 10 T and (see Figs. 3(a) and 3(b), respectively). The corresponding linescans along the [112�]pc direction through the peak maxima are shown in Figs. 3(c) and 3(d). At zero magnetic field two magnetic Bragg peaks were present (see Figs. 1(a) and 1(c)) which indicates the presence of a spin cycloid in the BiFeO3 thin film. At a magnetic field of 10 T the two magnetic Bragg peaks have converged into one single peak. The intensity of its peak maximum is almost twice the intensity of an initial single Bragg peak. Note, due to the orientation of the elliptical resolution function for the � 1 2 1 2 1 2 ̅ �pc Bragg reflections, in Figs (b) and (d) only one single peak is expected even in the presence of a peak splitting. Figure 4 shows the overall result of the magnetic field dependence of the neutron diffraction signal of the 100 nm thick BiFeO3 thin film. The magnetic field dependence of the peak maximum is shown in Fig. 4(a) for an increasing (blue symbols) and a decreasing (red symbols) magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. In Fig. 4(b) the magnetic field dependence of the length of the spin cycloid as extracted from the splitting of the two magnetic Bragg peaks, is presented. The distance between both Bragg peaks decreases continuously and converges towards zero at 10 T. This indicates that the length of the spin cycloid increases systematically and diverges to infinity at a magnetic field of about 10 T. The spin cycloid expands and finally collapses into a commensurate G-type antiferromagnetic spin structure. Since only one set of magnetic Bragg peaks was measured in this experiment, no conclusion can be drawn if the final antiferromagnetic structure is canted. It should be noted that both, the peak intensity and the length of the spin cycloid recover with decreasing magnetic field and exhibited a slight hysteresis behavior between 4 T and 10 T. For bulk BiFeO3 an abrupt phase transition was observed between about 16 T and 20 T. Based on measurements of the electric polarization, magnetization, and ESR in combination with theoretical modelling, the resulting phase was attributed to a canted antiferromagnetic structure [7,14,37-39]. In addition, an intermediate phase was observed by ESR and neutron diffraction experiments between about 10 T and 20 T [39,42]. The magnetic structure of this phase was described as conical. It is important to note that the neutron diffraction experiments on BiFeO3 single crystals under high magnetic field did not indicate any change in the incommensurable splitting in both phases, i.e. the length of the spin cycloid did not change with magnetic field [41,42]. This is in contrast to our neutron diffraction experiments on a 100 nm BiFeO3 thin film where the length of the spin cycloid increases systematically and diverges at 10 T. The present experiment on a BiFeO3 thin film does not provide any information about the conical intermediate phase. With a maximum of 10 T this phase was perhaps not reached. In addition, a conical spin structure would cause two magnetic Bragg peaks in the direction perpendicular to the scattering plane. Due to the finite out-of-plane instrumental resolution, they would appear as a single magnetic Bragg peak in the centre. Since the incommensurate splitting of the two magnetic Bragg peaks did converge to zero, we would consider this scenario as unlikely. An external magnetic field aligns the spins along the direction of the field. As the spin cycloid is partly anisotropic on a large scale, the applied magnetic field direction and magnetization direction are not same [15]. With increasing the magnetic field the Zeeman interaction strength can become comparable to the DM interaction. At a critical magnetic field this can lead to a suppression of the spin cycloidal. This can be expressed through the Landau-Ginzburg formalism of its free energy [25,38,39]. The Lifshitz invariant (a term in the Landau-Ginzburg theory that describes a coupling between the magnetization and its spatial variation due to the cycloidal spin structure) is increased by applying a magnetic field. For BiFeO<sup>3</sup> this can lead to the transformation of spin cycloid order into a simpler collinear antiferromagnetic state as observed in bulk BiFeO3 [7,14,37-39,41.42]. Theoretical studies have suggested that a lower critical magnetic field is required to suppress the cycloidal spin order in epitaxially grown BiFeO<sup>3</sup> thin films [43,44]. This was confirmed by the Raman experiments of Agbelele et al. in BiFeO<sup>3</sup> thin films [40], where they reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T − 6 T (depending on the chosen substrate) for BiFeO<sup>3</sup> thin films with a thickness of 70 nm grown on different (001)-oriented substrates, i.e. DyScO3, GdScO3, and SmScO3. In comparison to our BiFeO3 thin film grown on (110)-oriented SrTiO3 substrate they had a significantly smaller epitaxial strain (see Refs. [30] and [40], respectively). Therefore, our critical value of 10 T for the collapse of the spin cycloid in a BeFeO3 thin film is in accordance with the results of Agbelele et al. on differently strained BiFeO3 thin films [40]. ## 4. CONCLUSION In summary, our neutron diffraction experiment using a triple-axis spectrometer revealed a magnetic field-dependent change of the spin cycloid in a 100 nm BiFeO<sup>3</sup> film deposited by PLD on a (110)-oriented SrTiO<sup>3</sup> substrate. The experimental result show a continuous change of the spin cycloidal upon applying a magnetic field. The length of the spin cycloid expands systematically and diverges to infinity at an applied magnetic field of 10 T where the spin cycloidal transforms into a commensurate G-type antiferromagnetic spin structure. A transition to a canted antiferromagnetic spin structure was observed before for polycrystalline BiFeO<sup>3</sup> and single crystals but at much higher magnetic fields in the range of 16 T to 20 T. In contrast, for bulk BiFeO3 the incommensurable splitting of the magnetic Bragg peaks did not change with the magnetic field. Raman light scattering experiments showed that BiFeO3 thin films with lower epitaxial strain when grown on different (001)-oriented substrates had a transition from the cycloidal phase to a collinear spin structure at a lower magnetic field of 4-6 T, depending on the substrate. Our neutron diffraction experiments directly proof that the length of the cycloid systematically extends and transitions continuously into a G-type antiferromagnetic structure. This provides important information about the electronic correlations and microscopic physical effects behind the origin of the spin cycloid in BiFeO3. Furthermore, our results demonstrate that the application of a magnetic field offers a further route for the manipulation of the spin cycloid in multiferroics. Therefore, our experimental result will help to develop strategies to improve the linear magnetoelectric coupling in BiFeO<sup>3</sup> thin films with larger electric polarization, which is crucial for future technological applications. ## ACKNOWLEDGEMENT We would like to thank ANSTO for providing the neutron scattering beamtime at the instrument TAIPAN through the proposals P7934 and P9961, and Guochu Deng for his support as beamline scientist. ## REFERENCES - [1] M. Fiebig, T. [Lottermoser,](https://www.nature.com/articles/natrevmats201646#auth-Thomas-Lottermoser-Aff1) [D. Meier,](https://www.nature.com/articles/natrevmats201646#auth-Dennis-Meier-Aff1) and [M. Trassin,](javascript:;) Nature Rev. Mater. 1, 16046 (2016), doi: 10.1038/natrevmats.2016.46 - [8] W. Ratcliff, J. W. Lynn, V. [Kiryukhin,](https://www.nature.com/articles/npjquantmats20163#auth-Valery-Kiryukhin-Aff2) P. Jain, and [M. R. 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Series. 340, 012003 (2012), doi: 10.1088/1742- 6596/340/1/012003 ![Figure 3: Neutron scattering data taken in the [110] x [001] scattering plane of the 100 nm thick BiFeO3 thin film taken at a magnetic field of 10 T and a temperature of 150 K. Figure (a) and (b) show the RSMs around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc Bragg reflections, respectively. The black lines represent the corresponding linescans in [112�]pc direction as shown in Figs. (c) and (d), respectively. Gaussian lineshapes were fitted to the experimental data, as indicated by the dashed and solid lines.](path) ![Figure 4: (a) Intensity of the pek maximum of the neuron scattering data as a function of magnetic field as shown in Fig. 2. The blue symbols indicate the data obtained for an increasing magnetic field and the red symbols present the data of a decreasing magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. (b) length of the spin cycloid extracted from the splitting of the two magnetic Bragg peaks. The length of the spin cycloid is 65.7 ± 1.0 nm at zero magnetic field, expands systematically with increasing magnetic field and diverges towards infinity at a magnetic field of 10 T. The inset shows the zoomed-in data of the cycloidal length and the dashed horizontal line indicates the spin cycloid length of 63 nm obtained previously on bulk and thin film BiFeO3 samples. The solid lines serve as a guides to the eye.](path)
Figure 4: (a) Intensity of the pek maximum of the neuron scattering data as a function of magnetic field as shown in Fig. 2. The blue symbols indicate the data obtained for an increasing magnetic field and the red symbols present the data of a decreasing magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. (b) length of the spin cycloid extracted from the splitting of the two magnetic Bragg peaks. The length of the spin cycloid is 65.7 ± 1.0 nm at zero magnetic field, expands systematically with increasing magnetic field and diverges towards infinity at a magnetic field of 10 T. The inset shows the zoomed-in data of the cycloidal length and the dashed horizontal line indicates the spin cycloid length of 63 nm obtained previously on bulk and thin film BiFeO3 samples. The solid lines serve as a guides to the eye.	# Continuous Collapse of the Spin Cycloid in BiFeO3 Thin Films under an Applied Magnetic Field probed by Neutron Scattering <sup>1</sup> School of Physics, The University of New South Wales, NSW 2052, Australia 2 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA. Ningbo Institute of Materials Technology & Engineering, Chinese Academy of Sciences, Ningbo 315201, China 4 Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA 5 School of Engineering, Brown University, Providence, RI, 02912, USA. 6 Department of Physics, University of California, Berkeley, CA, 94720, USA. Md. Firoz Pervez, ORCID: [0000-0002-1453-8704](https://orcid.org/0000-0002-1453-8704) Clemens Ulrich, ORCID: 0000-0002-6829-9374 Hongrui Zhang, ORCID: [0000-0001-7896-019X](https://orcid.org/0000-0001-7896-019X) Yen-Lin Huang, ORCID: 0000-0002-6129-8547 Lucas Caretta, ORCID: 0000-0001-7229-7980 Ramamoorthy Ramesh, ORCID: 0000-0003-0524-1332 ## ABSTRACT Bismuth ferrite (BiFeO3) is one of the rare materials that exhibits multiferroic properties already at room-temperature. Therefore, it offers tremendous potential for future technological applications, such as memory and logic. However, a weak magnetoelectric coupling together with the presence of a noncollinear cycloidal spin order restricts various practical applications of BiFeO3. Therefore, there is a large interest in the search for suitable methods for the modulation of the spin cycloid in BiFeO3. By performing neutron diffraction experiments using a triple-axis instrument we have determined that the spin cycloid can be systematically suppressed by applying a high magnetic field of 10 T in a BiFeO3 thin film of about 100 nm grown on a (110)-oriented SrTiO3 substrate. As predicted by previous theoretical calculations, we observed that the required critical magnetic field to suppress the spin cycloid in a BiFeO3 thin film was lower as compared to the previously reported critical magnetic field for bulk BiFeO3 single crystals. Our experiment reveals that the spin cycloid continuously expands with increasing magnetic field before the complete transformation into a G-type antiferromagnetic spin order. Such tuning of the length of the spin cycloid up to a complete suppression offers new functionalities for future technological applications as in spintronics or magnonics. ## 1. INTROCUCTION Multiferroics are a fascinating class of materials that manifest multiple, simultaneous ferroic orders such as ferroelectric and magnetic polarizations. Magnetoelectric coupling between the ferroelectric and magnetic orders, including ferromagnetic, ferrimagnetic, antiferromagnetic or more complex noncollinear spin structures has been observed in single or multiphase multiferroic materials. Often this magnetoelectric coupling emerges from the simultaneous breaking of spatial inversion symmetry and time-reversal symmetry [1-2]. A magnetoelectric coupling in multiferroics allows for an efficient method of the electric control of the magnetic order as well as the magnetic control of electric polarization. Therefore, the magnetoelectric coupling offers an enormous potential for the application in next-generation spintronics and magnonic devices (e.g., memory devices, magnetic switches, magnetic sensors, high-frequency magnetic devices, spin valve devices, etc.) at much lower power consumption and faster operation as compared to the present conventional electronic devices [1-13]. Bismuth ferrite (BiFeO3) is one of the rare room temperature type-I, single-phase multiferroic oxides where primarily independent magnetic and ferroelectric order emerge [1,9]. However, a weak direct coupling between magnetic and ferroelectric order is present in BiFeO<sup>3</sup> [7,14,15]. Bulk BiFeO<sup>3</sup> is a rhombohedrally distorted cubic perovskite with the polar space group 3 − 3 6 [16] and a lattice parameter of 3.968 Å in the pseudocubic (pc) notation. This notation will be used throughout this article. In the case of BiFeO<sup>3</sup> thin films, epitaxial tensile or compressive strain results in a further distortion of the in-plane and out-of-plane lattice parameters. As consequence, different crystal structures including tetragonal (4), monoclinic ( ), monoclinic ( ), rhombohedral (3) and orthorhombic ( 2) are possible in BiFeO<sup>3</sup> thin films [17,18]. In the BiFeO<sup>3</sup> single crystals, the displacement of the A-site Bi 3+ ions and the lone pair selectrons cause the rhombohedral distortion which leads to the spontaneous ferroelectric polarization of about 100 μC cm−2 along the direction of the body diagonal [111]pc. This spontaneous ferroelectric polarization persists up to high temperatures (TC~1123 K) [19,20]. Below 643 K BiFeO<sup>3</sup> possesses a complex noncollinear magnetic structure arising from the Fe3+ ions along with nearly zero average magnetization [21]. The Fe3+ magnetic moments of the nearest neighbors of the adjacent (111)pc planes are ordered in a predominantly G-type antiferromagnetic spin structure. The Dzyaloshinskii-Moriya interaction, which is caused by both, the spin-orbit coupling and a broken inversion symmetry, tends to favor a particular spin rotation in BiFeO<sup>3</sup> which leads to a cycloidal spin order with a propagation length of about λ = 63 nm. In bulk BiFeO<sup>3</sup> the spin cycloid propagates along one of three crystallographic directions: [1�10]pc, [101�]pc or [011�]pc where the rotation axis of the spins is perpendicular to the plane defined by the propagation vector and the direction of the electric polarization along [111]pc [20-25]. Through the existence of s spin cycloid in BiFeO3 thin films was long time under debate, in 2010 Ke et al. found a type-I spin cycloid (~64 nm ) propagating along the [11�0]pc direction in a partially relaxed BiFeO<sup>3</sup> thin film with a thickness of about 800 nm deposited on a (100)-oriented SrTiO<sup>3</sup> substrate [26]. In 2011, Ratcliff et al. discovered the presence of a spin cycloid (~62 nm) in a 1 μm thin BiFeO<sup>3</sup> film deposited on a (110)-SrTiO<sup>3</sup> substrate [27], propagating along a unique [112�]pc direction, which is different from the direction [11�0]pc in the bulk single crystals [24,28]. In a neutron scattering study, Bertinshaw et al. discovered a spin cycloid in a BiFeO<sup>3</sup> thin film of just 100 nm when grown on a (110)-oriented SrTiO<sup>3</sup> substrate with a thin SrRuO<sup>3</sup> intermediate layer [29]. This report also showed that the length of the spin cycloid extends and diverges to infinity at the magnetic phase transition temperature of 650 ± 10 K. Finally, the report of Burns et al. revealed a dependence of the length of the spin cycloid on the film thickness. They observed that the length of spin cycloid increases for decreasing film thicknesses down to 20 nm [30]. Various factors can influence the spin cycloid in BiFeO3 thin films such as temperature, external electric and magnetic fields, film thickness, epitaxial strain, an intermediate layer, doping, etc. [8,10,26-34]. One of the key experimental strategies for understanding the fundamental physical interactions and domain states in magnetic materials is applying an external magnetic field. Several experimental techniques, including magnetization measurements and electric polarization measurements [7,14,15,35-38], electron spin resonance experiments (ESR) [39] and Raman light scattering experiments [40] have been performed to investigate the influence of an applied magnetic field (pulsed and static) on BiFeO3. They showed that applying a high magnetic field can suppress spin cycloid in BiFeO<sup>3</sup> single crystals or polycrystalline ceramics at a magnetic field of 16 T to 20 T [7,14,15,35-38,41,42]. A theoretical study by Gareeva et al. predicted the lowering of the required critical magnetic field for strained BiFeO<sup>3</sup> thin films as compared to BiFeO<sup>3</sup> single crystals [43,44]. Moreover, the Raman experiments by Agbelele et al. on BiFeO<sup>3</sup> thin films [40] reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T−6 T for BiFeO3 thin films grown on different substrates. Neutron scattering experiments would unambiguously reveal the nature of magnetic phase transition and would determine the precise magnetic spin structure in BiFeO3 thin films. Therefore, we have performed neutron diffraction experiments on a 100 nm thick BiFeO<sup>3</sup> film deposited by the PLD technique on a (110)-oriented SrTiO3 substrate. The experiment was performed using a triple-axis spectrometer at high magnetic fields of up to 10 T, in order to investigate the overall effect of a magnetic field on the spin structure in BiFeO3 thin films. ## 2. EXPERIMENTAL DETAILS The BiFeO<sup>3</sup> thin film sample with a film thickness of ~100 nm was grown by the Pulsed Laser Deposition on a (110)-oriented single-sided polished, 10 mm × 10 mm × 0.5 mm SrTiO<sup>3</sup> substrate (SHINKOSHA CO., LTD.). As an SrRuO3 intermediate layers can become magnetic at low temperature, the sample was grown without an intermediate layer. The lattice mismatch between SrTiO3 (apc = 3.905 Å [45]) and BiFeO3 (apc= 3.968 Å [16]) leads to an out-of-plane elongated lattice parameter, resulting in a monoclinically distorted structure of the BiFeO<sup>3</sup> films [18,27,29,30,46,47]. The triple-axis spectrometer instrument TAIPAN [48], located at the Australian Centre for Neutron Scattering (ACNS) at the Australian Nuclear Science and Technology Organisation (ANSTO) in Sydney, Australia, has demonstrated to be the ideal choice for the measurement of magnetic Bragg peaks of transition metal oxide thin films due to its enhanced resolution and the improved signal-to-background ratio as compared to conventional neutron diffraction instruments [29,30,49]. In order to apply a high magnetic field, a 12 T superconducting magnet was used. The magnetic field was applied in the direction perpendicular to the scattering plane, i.e. along the [11�0] in-plane direction of the film. In order to suppress contaminations from second order reflections from the SrTiO3 substrate at the � 1 2 1 2 1 2 � Bragg peak position, two pyrolytic-graphite, PG (002), filters with a total thickness of 60 mm were placed behind the sample. An incident energy of 14.86 eV (λ = 2.3462 Å) was used in the standard elastic scattering mode. This was provided by a PG-monochromator and a PG-analyzer with vertical focusing but horizontally flat configuration. In order to enhance the resolution and to suppress the background, 40′′ collimators were placed in the neutron beam before and after the sample (see also Ref. [29,30,49]). A linescan was performed around the � 1 2 1 2 1 2 � Bragg peak of the second order contamination of the SrTiO3 substrate without the two PG (002) filters to determine the instrumental resolution of a Full Width at Half Maximum of 0.0032 reciprocal lattice units (r.l.u.). The substrate SrTiO3 possesses a crystallographic phase transition at 105 K. In order to avoid corresponding effects on the BiFeO<sup>3</sup> thin film, the neutron scattering measurements were performed at the temperature of 150 K [49], which also provides a stronger magnetic signal from the BiFeO3thin film compared to room temperature [29]. ## 3. RESULTS AND DISCUSSION Figure 1 shows the neutron diffraction data of the 100 nm thick BiFeO3 film taken at a temperature of 150 K and zero magnetic field. In order to access the magnetic Bragg peaks arising from the spin cycloid, the film was grown on a (110)-oriented SrTiO3 substrate and mounted in the [110] x [001] scattering plane. Figure 1(a) shows the neutron scattering reciprocal space map (RSM) around the � 1 2 1 2 1 2 �pc Bragg reflection. The peak at precisely � 1 2 1 2 1 2 �pc arises from the second order contamination of the (111) structural Bragg reflection of the SrTiO3 substrate and is still visible despite the use of two PG (002) filters. The two Bragg peaks at around QHK = 0.485 can be attributed to magnetic scattering from the spin cycloid of the BiFeO3 thin film. The reduced QHK value is a consequence of the epitaxial strain, which leads to an increase of the out-of-plane lattice parameter [29,30]. The distance between the peak maxima indicates the length of the spin cycloid (λ = 2π/δ) and the direction of their separation, as shown by the black line, corresponds to the propagation direction of the spin cycloid along [112�]pc. This propagation direction was also observed by previous neutron diffraction experiments on BiFeO3 thin films [26,27,29,30]. Theoretical studies have predicted that a spin cycloid can possibly propagate either along the [112�]pc, [12�1]pc or [2�11]pc directions in a BiFeO3 thin film deposited on a (110)-oriented SrTiO<sup>3</sup> substrate as compared to the [1�10]pc, [101�]pc or [011�]pc directions in bulk BiFeO3 [27,50]. The change in the propagation direction is a consequence of the in-plane strain of the BiFeO3 film. Figure 1(b) shows the RSM around the � 1 2 1 2 1 2 ̅ �pc. Only one magnetic Bragg peak appears. Due to the orientation of the instrumental resolution ellipsoid the two magnetic Bragg peaks of the spin cycloid are overlapping in this measurement geometry (see Suppl. Mater of Ref. [30]). Figures 1(c) and 1(d) show the corresponding line scans along the black lines in Figs. 1(a) and (b), respectively. The red lines are Gaussian lineshapes fitted to the experimental data. For the fit the linewidth was kept constant for both magnetic Bragg peaks. The separation of the two magnetic Bragg peaks in Fig. 1(c) is = Δ = Δ = Δ = 0.00415, which corresponds to a length of the spin cycloid of 65.7 ± 1.0 nm. This is in good agreement to the value of about 63 nm obtained for bulk BiFeO3 [20-25] and for BiFeO3 thin films [27,29,30]. Figure 1(d) shows the corresponding linescan over the � 1 2 1 2 1 2 ̅ �pc RSM as indicated by the black line in Fig. 1(b). Note, to avoid the presence of the λ/2 contamination from the SrTiO3 substrate, the linescan was not taken precisely through the peak maximum but slightly shifted as indicated by the black line in Fig. 1(b). Only one sharp peak is observed. Its integrated intensity corresponds roughly to the integrated intensity of the two magnetic Bragg peaks in Fig. 1(b) [27,29,30]. The magnetic field dependent data are shown in Fig. 2. The data were taken around the � 1 2 1 2 1 2 �pc Bragg reflection following the black line of the RSM as shown in Fig. 1(a), i.e. along the [112�]pc direction. The magnetic field was increased in 1 T steps to a strength of 10 T (Fig. 2(a)) and then decreased to zero field again (Fig. 2(b)). The obtained data were analyzed by fitting two Gaussian lineshapes to the experimental data, where the linewidth and peak intensity of each of the two Gaussian peaks were kept constant using the values obtained for the 0 T measurement. With increasing magnetic field the distance between the two magnetic Bragg peaks decreases, which converge into one single peak at a magnetic field of 10 T (see Fig. 2(a)). The intensity of the peak maximum increases with increasing magnetic field to a value of about twice the maximum intensity at 0 T. However, the integrated intensity over both peaks remains almost constant. The merging into one single Bragg peak is an indication that the length of the spin cycloid increases continuously and finally diverges to infinity at a magnetic field of 10 T. The resulting spin structure is G-type antiferromagnetic, i.e. the spin cycloid collapses into a G-type antiferromagnetic spin arrangement. Upon decreasing magnetic field the peak splits again into two well separated peaks and the overall height of the peak maximum decreases with decreasing magnetic field see Fig. 2(b)). The initial lineshape, both in peak intensity and peak splitting has recovered when reaching 0 T again, i.e. the length of the spin cycloid reaches the same value of 65.7 ± 1.0 nm after the magnetic field cycle. In order to confirm that only one magnetic Bragg peak remains at 10 T, RSMs were taken around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc positions at 10 T and (see Figs. 3(a) and 3(b), respectively). The corresponding linescans along the [112�]pc direction through the peak maxima are shown in Figs. 3(c) and 3(d). At zero magnetic field two magnetic Bragg peaks were present (see Figs. 1(a) and 1(c)) which indicates the presence of a spin cycloid in the BiFeO3 thin film. At a magnetic field of 10 T the two magnetic Bragg peaks have converged into one single peak. The intensity of its peak maximum is almost twice the intensity of an initial single Bragg peak. Note, due to the orientation of the elliptical resolution function for the � 1 2 1 2 1 2 ̅ �pc Bragg reflections, in Figs (b) and (d) only one single peak is expected even in the presence of a peak splitting. Figure 4 shows the overall result of the magnetic field dependence of the neutron diffraction signal of the 100 nm thick BiFeO3 thin film. The magnetic field dependence of the peak maximum is shown in Fig. 4(a) for an increasing (blue symbols) and a decreasing (red symbols) magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. In Fig. 4(b) the magnetic field dependence of the length of the spin cycloid as extracted from the splitting of the two magnetic Bragg peaks, is presented. The distance between both Bragg peaks decreases continuously and converges towards zero at 10 T. This indicates that the length of the spin cycloid increases systematically and diverges to infinity at a magnetic field of about 10 T. The spin cycloid expands and finally collapses into a commensurate G-type antiferromagnetic spin structure. Since only one set of magnetic Bragg peaks was measured in this experiment, no conclusion can be drawn if the final antiferromagnetic structure is canted. It should be noted that both, the peak intensity and the length of the spin cycloid recover with decreasing magnetic field and exhibited a slight hysteresis behavior between 4 T and 10 T. For bulk BiFeO3 an abrupt phase transition was observed between about 16 T and 20 T. Based on measurements of the electric polarization, magnetization, and ESR in combination with theoretical modelling, the resulting phase was attributed to a canted antiferromagnetic structure [7,14,37-39]. In addition, an intermediate phase was observed by ESR and neutron diffraction experiments between about 10 T and 20 T [39,42]. The magnetic structure of this phase was described as conical. It is important to note that the neutron diffraction experiments on BiFeO3 single crystals under high magnetic field did not indicate any change in the incommensurable splitting in both phases, i.e. the length of the spin cycloid did not change with magnetic field [41,42]. This is in contrast to our neutron diffraction experiments on a 100 nm BiFeO3 thin film where the length of the spin cycloid increases systematically and diverges at 10 T. The present experiment on a BiFeO3 thin film does not provide any information about the conical intermediate phase. With a maximum of 10 T this phase was perhaps not reached. In addition, a conical spin structure would cause two magnetic Bragg peaks in the direction perpendicular to the scattering plane. Due to the finite out-of-plane instrumental resolution, they would appear as a single magnetic Bragg peak in the centre. Since the incommensurate splitting of the two magnetic Bragg peaks did converge to zero, we would consider this scenario as unlikely. An external magnetic field aligns the spins along the direction of the field. As the spin cycloid is partly anisotropic on a large scale, the applied magnetic field direction and magnetization direction are not same [15]. With increasing the magnetic field the Zeeman interaction strength can become comparable to the DM interaction. At a critical magnetic field this can lead to a suppression of the spin cycloidal. This can be expressed through the Landau-Ginzburg formalism of its free energy [25,38,39]. The Lifshitz invariant (a term in the Landau-Ginzburg theory that describes a coupling between the magnetization and its spatial variation due to the cycloidal spin structure) is increased by applying a magnetic field. For BiFeO<sup>3</sup> this can lead to the transformation of spin cycloid order into a simpler collinear antiferromagnetic state as observed in bulk BiFeO3 [7,14,37-39,41.42]. Theoretical studies have suggested that a lower critical magnetic field is required to suppress the cycloidal spin order in epitaxially grown BiFeO<sup>3</sup> thin films [43,44]. This was confirmed by the Raman experiments of Agbelele et al. in BiFeO<sup>3</sup> thin films [40], where they reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T − 6 T (depending on the chosen substrate) for BiFeO<sup>3</sup> thin films with a thickness of 70 nm grown on different (001)-oriented substrates, i.e. DyScO3, GdScO3, and SmScO3. In comparison to our BiFeO3 thin film grown on (110)-oriented SrTiO3 substrate they had a significantly smaller epitaxial strain (see Refs. [30] and [40], respectively). Therefore, our critical value of 10 T for the collapse of the spin cycloid in a BeFeO3 thin film is in accordance with the results of Agbelele et al. on differently strained BiFeO3 thin films [40]. ## 4. CONCLUSION In summary, our neutron diffraction experiment using a triple-axis spectrometer revealed a magnetic field-dependent change of the spin cycloid in a 100 nm BiFeO<sup>3</sup> film deposited by PLD on a (110)-oriented SrTiO<sup>3</sup> substrate. The experimental result show a continuous change of the spin cycloidal upon applying a magnetic field. The length of the spin cycloid expands systematically and diverges to infinity at an applied magnetic field of 10 T where the spin cycloidal transforms into a commensurate G-type antiferromagnetic spin structure. A transition to a canted antiferromagnetic spin structure was observed before for polycrystalline BiFeO<sup>3</sup> and single crystals but at much higher magnetic fields in the range of 16 T to 20 T. In contrast, for bulk BiFeO3 the incommensurable splitting of the magnetic Bragg peaks did not change with the magnetic field. Raman light scattering experiments showed that BiFeO3 thin films with lower epitaxial strain when grown on different (001)-oriented substrates had a transition from the cycloidal phase to a collinear spin structure at a lower magnetic field of 4-6 T, depending on the substrate. Our neutron diffraction experiments directly proof that the length of the cycloid systematically extends and transitions continuously into a G-type antiferromagnetic structure. This provides important information about the electronic correlations and microscopic physical effects behind the origin of the spin cycloid in BiFeO3. Furthermore, our results demonstrate that the application of a magnetic field offers a further route for the manipulation of the spin cycloid in multiferroics. Therefore, our experimental result will help to develop strategies to improve the linear magnetoelectric coupling in BiFeO<sup>3</sup> thin films with larger electric polarization, which is crucial for future technological applications. ## ACKNOWLEDGEMENT We would like to thank ANSTO for providing the neutron scattering beamtime at the instrument TAIPAN through the proposals P7934 and P9961, and Guochu Deng for his support as beamline scientist. ## REFERENCES - [1] M. Fiebig, T. [Lottermoser,](https://www.nature.com/articles/natrevmats201646#auth-Thomas-Lottermoser-Aff1) [D. Meier,](https://www.nature.com/articles/natrevmats201646#auth-Dennis-Meier-Aff1) and [M. Trassin,](javascript:;) Nature Rev. Mater. 1, 16046 (2016), doi: 10.1038/natrevmats.2016.46 - [8] W. Ratcliff, J. W. Lynn, V. [Kiryukhin,](https://www.nature.com/articles/npjquantmats20163#auth-Valery-Kiryukhin-Aff2) P. Jain, and [M. R. 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Figure 1: (a) and (c) Neutron scattering reciprocal space maps (RSMs) of the [110]-[001] plane of a 100 nm thick BiFeO<sup>3</sup> thin film grown on a (110)-oriented SrTiO3 substrate. The data were taken at 150 K and 0 T. The Bragg reflection at � 1 2 1 2 1 2 �pc corresponds to second order contamination from the SrTiO3 substrate. (a) RSM around the � 1 2 1 2 1 2 �pc Bragg reflection. Two additional Bragg peaks are observed. They originate from the spin cycloid of BiFeO3. Their spacing, as highlighted by the black line, indicates the propagation direction, i.e. [112�]pc and the length of the spin cycloid. (b) RSM around the � 1 2 1 2 1 2 ̅ �pc. Due to the elliptical instrumental resolution only one additional Bragg peak is visible. (c) and (d): corresponding line scans along the black lines in Figs. 1(a) and (b). The red lines are Gaussian lineshapes fitted to the experimental data with fixed linewidth and intensities for both magnetic Bragg reflections.	# Continuous Collapse of the Spin Cycloid in BiFeO3 Thin Films under an Applied Magnetic Field probed by Neutron Scattering <sup>1</sup> School of Physics, The University of New South Wales, NSW 2052, Australia 2 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA. Ningbo Institute of Materials Technology & Engineering, Chinese Academy of Sciences, Ningbo 315201, China 4 Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA 5 School of Engineering, Brown University, Providence, RI, 02912, USA. 6 Department of Physics, University of California, Berkeley, CA, 94720, USA. Md. Firoz Pervez, ORCID: [0000-0002-1453-8704](https://orcid.org/0000-0002-1453-8704) Clemens Ulrich, ORCID: 0000-0002-6829-9374 Hongrui Zhang, ORCID: [0000-0001-7896-019X](https://orcid.org/0000-0001-7896-019X) Yen-Lin Huang, ORCID: 0000-0002-6129-8547 Lucas Caretta, ORCID: 0000-0001-7229-7980 Ramamoorthy Ramesh, ORCID: 0000-0003-0524-1332 ## ABSTRACT Bismuth ferrite (BiFeO3) is one of the rare materials that exhibits multiferroic properties already at room-temperature. Therefore, it offers tremendous potential for future technological applications, such as memory and logic. However, a weak magnetoelectric coupling together with the presence of a noncollinear cycloidal spin order restricts various practical applications of BiFeO3. Therefore, there is a large interest in the search for suitable methods for the modulation of the spin cycloid in BiFeO3. By performing neutron diffraction experiments using a triple-axis instrument we have determined that the spin cycloid can be systematically suppressed by applying a high magnetic field of 10 T in a BiFeO3 thin film of about 100 nm grown on a (110)-oriented SrTiO3 substrate. As predicted by previous theoretical calculations, we observed that the required critical magnetic field to suppress the spin cycloid in a BiFeO3 thin film was lower as compared to the previously reported critical magnetic field for bulk BiFeO3 single crystals. Our experiment reveals that the spin cycloid continuously expands with increasing magnetic field before the complete transformation into a G-type antiferromagnetic spin order. Such tuning of the length of the spin cycloid up to a complete suppression offers new functionalities for future technological applications as in spintronics or magnonics. ## 1. INTROCUCTION Multiferroics are a fascinating class of materials that manifest multiple, simultaneous ferroic orders such as ferroelectric and magnetic polarizations. Magnetoelectric coupling between the ferroelectric and magnetic orders, including ferromagnetic, ferrimagnetic, antiferromagnetic or more complex noncollinear spin structures has been observed in single or multiphase multiferroic materials. Often this magnetoelectric coupling emerges from the simultaneous breaking of spatial inversion symmetry and time-reversal symmetry [1-2]. A magnetoelectric coupling in multiferroics allows for an efficient method of the electric control of the magnetic order as well as the magnetic control of electric polarization. Therefore, the magnetoelectric coupling offers an enormous potential for the application in next-generation spintronics and magnonic devices (e.g., memory devices, magnetic switches, magnetic sensors, high-frequency magnetic devices, spin valve devices, etc.) at much lower power consumption and faster operation as compared to the present conventional electronic devices [1-13]. Bismuth ferrite (BiFeO3) is one of the rare room temperature type-I, single-phase multiferroic oxides where primarily independent magnetic and ferroelectric order emerge [1,9]. However, a weak direct coupling between magnetic and ferroelectric order is present in BiFeO<sup>3</sup> [7,14,15]. Bulk BiFeO<sup>3</sup> is a rhombohedrally distorted cubic perovskite with the polar space group 3 − 3 6 [16] and a lattice parameter of 3.968 Å in the pseudocubic (pc) notation. This notation will be used throughout this article. In the case of BiFeO<sup>3</sup> thin films, epitaxial tensile or compressive strain results in a further distortion of the in-plane and out-of-plane lattice parameters. As consequence, different crystal structures including tetragonal (4), monoclinic ( ), monoclinic ( ), rhombohedral (3) and orthorhombic ( 2) are possible in BiFeO<sup>3</sup> thin films [17,18]. In the BiFeO<sup>3</sup> single crystals, the displacement of the A-site Bi 3+ ions and the lone pair selectrons cause the rhombohedral distortion which leads to the spontaneous ferroelectric polarization of about 100 μC cm−2 along the direction of the body diagonal [111]pc. This spontaneous ferroelectric polarization persists up to high temperatures (TC~1123 K) [19,20]. Below 643 K BiFeO<sup>3</sup> possesses a complex noncollinear magnetic structure arising from the Fe3+ ions along with nearly zero average magnetization [21]. The Fe3+ magnetic moments of the nearest neighbors of the adjacent (111)pc planes are ordered in a predominantly G-type antiferromagnetic spin structure. The Dzyaloshinskii-Moriya interaction, which is caused by both, the spin-orbit coupling and a broken inversion symmetry, tends to favor a particular spin rotation in BiFeO<sup>3</sup> which leads to a cycloidal spin order with a propagation length of about λ = 63 nm. In bulk BiFeO<sup>3</sup> the spin cycloid propagates along one of three crystallographic directions: [1�10]pc, [101�]pc or [011�]pc where the rotation axis of the spins is perpendicular to the plane defined by the propagation vector and the direction of the electric polarization along [111]pc [20-25]. Through the existence of s spin cycloid in BiFeO3 thin films was long time under debate, in 2010 Ke et al. found a type-I spin cycloid (~64 nm ) propagating along the [11�0]pc direction in a partially relaxed BiFeO<sup>3</sup> thin film with a thickness of about 800 nm deposited on a (100)-oriented SrTiO<sup>3</sup> substrate [26]. In 2011, Ratcliff et al. discovered the presence of a spin cycloid (~62 nm) in a 1 μm thin BiFeO<sup>3</sup> film deposited on a (110)-SrTiO<sup>3</sup> substrate [27], propagating along a unique [112�]pc direction, which is different from the direction [11�0]pc in the bulk single crystals [24,28]. In a neutron scattering study, Bertinshaw et al. discovered a spin cycloid in a BiFeO<sup>3</sup> thin film of just 100 nm when grown on a (110)-oriented SrTiO<sup>3</sup> substrate with a thin SrRuO<sup>3</sup> intermediate layer [29]. This report also showed that the length of the spin cycloid extends and diverges to infinity at the magnetic phase transition temperature of 650 ± 10 K. Finally, the report of Burns et al. revealed a dependence of the length of the spin cycloid on the film thickness. They observed that the length of spin cycloid increases for decreasing film thicknesses down to 20 nm [30]. Various factors can influence the spin cycloid in BiFeO3 thin films such as temperature, external electric and magnetic fields, film thickness, epitaxial strain, an intermediate layer, doping, etc. [8,10,26-34]. One of the key experimental strategies for understanding the fundamental physical interactions and domain states in magnetic materials is applying an external magnetic field. Several experimental techniques, including magnetization measurements and electric polarization measurements [7,14,15,35-38], electron spin resonance experiments (ESR) [39] and Raman light scattering experiments [40] have been performed to investigate the influence of an applied magnetic field (pulsed and static) on BiFeO3. They showed that applying a high magnetic field can suppress spin cycloid in BiFeO<sup>3</sup> single crystals or polycrystalline ceramics at a magnetic field of 16 T to 20 T [7,14,15,35-38,41,42]. A theoretical study by Gareeva et al. predicted the lowering of the required critical magnetic field for strained BiFeO<sup>3</sup> thin films as compared to BiFeO<sup>3</sup> single crystals [43,44]. Moreover, the Raman experiments by Agbelele et al. on BiFeO<sup>3</sup> thin films [40] reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T−6 T for BiFeO3 thin films grown on different substrates. Neutron scattering experiments would unambiguously reveal the nature of magnetic phase transition and would determine the precise magnetic spin structure in BiFeO3 thin films. Therefore, we have performed neutron diffraction experiments on a 100 nm thick BiFeO<sup>3</sup> film deposited by the PLD technique on a (110)-oriented SrTiO3 substrate. The experiment was performed using a triple-axis spectrometer at high magnetic fields of up to 10 T, in order to investigate the overall effect of a magnetic field on the spin structure in BiFeO3 thin films. ## 2. EXPERIMENTAL DETAILS The BiFeO<sup>3</sup> thin film sample with a film thickness of ~100 nm was grown by the Pulsed Laser Deposition on a (110)-oriented single-sided polished, 10 mm × 10 mm × 0.5 mm SrTiO<sup>3</sup> substrate (SHINKOSHA CO., LTD.). As an SrRuO3 intermediate layers can become magnetic at low temperature, the sample was grown without an intermediate layer. The lattice mismatch between SrTiO3 (apc = 3.905 Å [45]) and BiFeO3 (apc= 3.968 Å [16]) leads to an out-of-plane elongated lattice parameter, resulting in a monoclinically distorted structure of the BiFeO<sup>3</sup> films [18,27,29,30,46,47]. The triple-axis spectrometer instrument TAIPAN [48], located at the Australian Centre for Neutron Scattering (ACNS) at the Australian Nuclear Science and Technology Organisation (ANSTO) in Sydney, Australia, has demonstrated to be the ideal choice for the measurement of magnetic Bragg peaks of transition metal oxide thin films due to its enhanced resolution and the improved signal-to-background ratio as compared to conventional neutron diffraction instruments [29,30,49]. In order to apply a high magnetic field, a 12 T superconducting magnet was used. The magnetic field was applied in the direction perpendicular to the scattering plane, i.e. along the [11�0] in-plane direction of the film. In order to suppress contaminations from second order reflections from the SrTiO3 substrate at the � 1 2 1 2 1 2 � Bragg peak position, two pyrolytic-graphite, PG (002), filters with a total thickness of 60 mm were placed behind the sample. An incident energy of 14.86 eV (λ = 2.3462 Å) was used in the standard elastic scattering mode. This was provided by a PG-monochromator and a PG-analyzer with vertical focusing but horizontally flat configuration. In order to enhance the resolution and to suppress the background, 40′′ collimators were placed in the neutron beam before and after the sample (see also Ref. [29,30,49]). A linescan was performed around the � 1 2 1 2 1 2 � Bragg peak of the second order contamination of the SrTiO3 substrate without the two PG (002) filters to determine the instrumental resolution of a Full Width at Half Maximum of 0.0032 reciprocal lattice units (r.l.u.). The substrate SrTiO3 possesses a crystallographic phase transition at 105 K. In order to avoid corresponding effects on the BiFeO<sup>3</sup> thin film, the neutron scattering measurements were performed at the temperature of 150 K [49], which also provides a stronger magnetic signal from the BiFeO3thin film compared to room temperature [29]. ## 3. RESULTS AND DISCUSSION Figure 1 shows the neutron diffraction data of the 100 nm thick BiFeO3 film taken at a temperature of 150 K and zero magnetic field. In order to access the magnetic Bragg peaks arising from the spin cycloid, the film was grown on a (110)-oriented SrTiO3 substrate and mounted in the [110] x [001] scattering plane. Figure 1(a) shows the neutron scattering reciprocal space map (RSM) around the � 1 2 1 2 1 2 �pc Bragg reflection. The peak at precisely � 1 2 1 2 1 2 �pc arises from the second order contamination of the (111) structural Bragg reflection of the SrTiO3 substrate and is still visible despite the use of two PG (002) filters. The two Bragg peaks at around QHK = 0.485 can be attributed to magnetic scattering from the spin cycloid of the BiFeO3 thin film. The reduced QHK value is a consequence of the epitaxial strain, which leads to an increase of the out-of-plane lattice parameter [29,30]. The distance between the peak maxima indicates the length of the spin cycloid (λ = 2π/δ) and the direction of their separation, as shown by the black line, corresponds to the propagation direction of the spin cycloid along [112�]pc. This propagation direction was also observed by previous neutron diffraction experiments on BiFeO3 thin films [26,27,29,30]. Theoretical studies have predicted that a spin cycloid can possibly propagate either along the [112�]pc, [12�1]pc or [2�11]pc directions in a BiFeO3 thin film deposited on a (110)-oriented SrTiO<sup>3</sup> substrate as compared to the [1�10]pc, [101�]pc or [011�]pc directions in bulk BiFeO3 [27,50]. The change in the propagation direction is a consequence of the in-plane strain of the BiFeO3 film. Figure 1(b) shows the RSM around the � 1 2 1 2 1 2 ̅ �pc. Only one magnetic Bragg peak appears. Due to the orientation of the instrumental resolution ellipsoid the two magnetic Bragg peaks of the spin cycloid are overlapping in this measurement geometry (see Suppl. Mater of Ref. [30]). Figures 1(c) and 1(d) show the corresponding line scans along the black lines in Figs. 1(a) and (b), respectively. The red lines are Gaussian lineshapes fitted to the experimental data. For the fit the linewidth was kept constant for both magnetic Bragg peaks. The separation of the two magnetic Bragg peaks in Fig. 1(c) is = Δ = Δ = Δ = 0.00415, which corresponds to a length of the spin cycloid of 65.7 ± 1.0 nm. This is in good agreement to the value of about 63 nm obtained for bulk BiFeO3 [20-25] and for BiFeO3 thin films [27,29,30]. Figure 1(d) shows the corresponding linescan over the � 1 2 1 2 1 2 ̅ �pc RSM as indicated by the black line in Fig. 1(b). Note, to avoid the presence of the λ/2 contamination from the SrTiO3 substrate, the linescan was not taken precisely through the peak maximum but slightly shifted as indicated by the black line in Fig. 1(b). Only one sharp peak is observed. Its integrated intensity corresponds roughly to the integrated intensity of the two magnetic Bragg peaks in Fig. 1(b) [27,29,30]. The magnetic field dependent data are shown in Fig. 2. The data were taken around the � 1 2 1 2 1 2 �pc Bragg reflection following the black line of the RSM as shown in Fig. 1(a), i.e. along the [112�]pc direction. The magnetic field was increased in 1 T steps to a strength of 10 T (Fig. 2(a)) and then decreased to zero field again (Fig. 2(b)). The obtained data were analyzed by fitting two Gaussian lineshapes to the experimental data, where the linewidth and peak intensity of each of the two Gaussian peaks were kept constant using the values obtained for the 0 T measurement. With increasing magnetic field the distance between the two magnetic Bragg peaks decreases, which converge into one single peak at a magnetic field of 10 T (see Fig. 2(a)). The intensity of the peak maximum increases with increasing magnetic field to a value of about twice the maximum intensity at 0 T. However, the integrated intensity over both peaks remains almost constant. The merging into one single Bragg peak is an indication that the length of the spin cycloid increases continuously and finally diverges to infinity at a magnetic field of 10 T. The resulting spin structure is G-type antiferromagnetic, i.e. the spin cycloid collapses into a G-type antiferromagnetic spin arrangement. Upon decreasing magnetic field the peak splits again into two well separated peaks and the overall height of the peak maximum decreases with decreasing magnetic field see Fig. 2(b)). The initial lineshape, both in peak intensity and peak splitting has recovered when reaching 0 T again, i.e. the length of the spin cycloid reaches the same value of 65.7 ± 1.0 nm after the magnetic field cycle. In order to confirm that only one magnetic Bragg peak remains at 10 T, RSMs were taken around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc positions at 10 T and (see Figs. 3(a) and 3(b), respectively). The corresponding linescans along the [112�]pc direction through the peak maxima are shown in Figs. 3(c) and 3(d). At zero magnetic field two magnetic Bragg peaks were present (see Figs. 1(a) and 1(c)) which indicates the presence of a spin cycloid in the BiFeO3 thin film. At a magnetic field of 10 T the two magnetic Bragg peaks have converged into one single peak. The intensity of its peak maximum is almost twice the intensity of an initial single Bragg peak. Note, due to the orientation of the elliptical resolution function for the � 1 2 1 2 1 2 ̅ �pc Bragg reflections, in Figs (b) and (d) only one single peak is expected even in the presence of a peak splitting. Figure 4 shows the overall result of the magnetic field dependence of the neutron diffraction signal of the 100 nm thick BiFeO3 thin film. The magnetic field dependence of the peak maximum is shown in Fig. 4(a) for an increasing (blue symbols) and a decreasing (red symbols) magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. In Fig. 4(b) the magnetic field dependence of the length of the spin cycloid as extracted from the splitting of the two magnetic Bragg peaks, is presented. The distance between both Bragg peaks decreases continuously and converges towards zero at 10 T. This indicates that the length of the spin cycloid increases systematically and diverges to infinity at a magnetic field of about 10 T. The spin cycloid expands and finally collapses into a commensurate G-type antiferromagnetic spin structure. Since only one set of magnetic Bragg peaks was measured in this experiment, no conclusion can be drawn if the final antiferromagnetic structure is canted. It should be noted that both, the peak intensity and the length of the spin cycloid recover with decreasing magnetic field and exhibited a slight hysteresis behavior between 4 T and 10 T. For bulk BiFeO3 an abrupt phase transition was observed between about 16 T and 20 T. Based on measurements of the electric polarization, magnetization, and ESR in combination with theoretical modelling, the resulting phase was attributed to a canted antiferromagnetic structure [7,14,37-39]. In addition, an intermediate phase was observed by ESR and neutron diffraction experiments between about 10 T and 20 T [39,42]. The magnetic structure of this phase was described as conical. It is important to note that the neutron diffraction experiments on BiFeO3 single crystals under high magnetic field did not indicate any change in the incommensurable splitting in both phases, i.e. the length of the spin cycloid did not change with magnetic field [41,42]. This is in contrast to our neutron diffraction experiments on a 100 nm BiFeO3 thin film where the length of the spin cycloid increases systematically and diverges at 10 T. The present experiment on a BiFeO3 thin film does not provide any information about the conical intermediate phase. With a maximum of 10 T this phase was perhaps not reached. In addition, a conical spin structure would cause two magnetic Bragg peaks in the direction perpendicular to the scattering plane. Due to the finite out-of-plane instrumental resolution, they would appear as a single magnetic Bragg peak in the centre. Since the incommensurate splitting of the two magnetic Bragg peaks did converge to zero, we would consider this scenario as unlikely. An external magnetic field aligns the spins along the direction of the field. As the spin cycloid is partly anisotropic on a large scale, the applied magnetic field direction and magnetization direction are not same [15]. With increasing the magnetic field the Zeeman interaction strength can become comparable to the DM interaction. At a critical magnetic field this can lead to a suppression of the spin cycloidal. This can be expressed through the Landau-Ginzburg formalism of its free energy [25,38,39]. The Lifshitz invariant (a term in the Landau-Ginzburg theory that describes a coupling between the magnetization and its spatial variation due to the cycloidal spin structure) is increased by applying a magnetic field. For BiFeO<sup>3</sup> this can lead to the transformation of spin cycloid order into a simpler collinear antiferromagnetic state as observed in bulk BiFeO3 [7,14,37-39,41.42]. Theoretical studies have suggested that a lower critical magnetic field is required to suppress the cycloidal spin order in epitaxially grown BiFeO<sup>3</sup> thin films [43,44]. This was confirmed by the Raman experiments of Agbelele et al. in BiFeO<sup>3</sup> thin films [40], where they reported a magnetic phase transition from a cycloidal spin structure to a G-type antiferromagnetic spin order in the range of 4 T − 6 T (depending on the chosen substrate) for BiFeO<sup>3</sup> thin films with a thickness of 70 nm grown on different (001)-oriented substrates, i.e. DyScO3, GdScO3, and SmScO3. In comparison to our BiFeO3 thin film grown on (110)-oriented SrTiO3 substrate they had a significantly smaller epitaxial strain (see Refs. [30] and [40], respectively). Therefore, our critical value of 10 T for the collapse of the spin cycloid in a BeFeO3 thin film is in accordance with the results of Agbelele et al. on differently strained BiFeO3 thin films [40]. ## 4. CONCLUSION In summary, our neutron diffraction experiment using a triple-axis spectrometer revealed a magnetic field-dependent change of the spin cycloid in a 100 nm BiFeO<sup>3</sup> film deposited by PLD on a (110)-oriented SrTiO<sup>3</sup> substrate. The experimental result show a continuous change of the spin cycloidal upon applying a magnetic field. The length of the spin cycloid expands systematically and diverges to infinity at an applied magnetic field of 10 T where the spin cycloidal transforms into a commensurate G-type antiferromagnetic spin structure. A transition to a canted antiferromagnetic spin structure was observed before for polycrystalline BiFeO<sup>3</sup> and single crystals but at much higher magnetic fields in the range of 16 T to 20 T. In contrast, for bulk BiFeO3 the incommensurable splitting of the magnetic Bragg peaks did not change with the magnetic field. Raman light scattering experiments showed that BiFeO3 thin films with lower epitaxial strain when grown on different (001)-oriented substrates had a transition from the cycloidal phase to a collinear spin structure at a lower magnetic field of 4-6 T, depending on the substrate. Our neutron diffraction experiments directly proof that the length of the cycloid systematically extends and transitions continuously into a G-type antiferromagnetic structure. This provides important information about the electronic correlations and microscopic physical effects behind the origin of the spin cycloid in BiFeO3. Furthermore, our results demonstrate that the application of a magnetic field offers a further route for the manipulation of the spin cycloid in multiferroics. Therefore, our experimental result will help to develop strategies to improve the linear magnetoelectric coupling in BiFeO<sup>3</sup> thin films with larger electric polarization, which is crucial for future technological applications. ## ACKNOWLEDGEMENT We would like to thank ANSTO for providing the neutron scattering beamtime at the instrument TAIPAN through the proposals P7934 and P9961, and Guochu Deng for his support as beamline scientist. ## REFERENCES - [1] M. Fiebig, T. [Lottermoser,](https://www.nature.com/articles/natrevmats201646#auth-Thomas-Lottermoser-Aff1) [D. Meier,](https://www.nature.com/articles/natrevmats201646#auth-Dennis-Meier-Aff1) and [M. Trassin,](javascript:;) Nature Rev. Mater. 1, 16046 (2016), doi: 10.1038/natrevmats.2016.46 - [8] W. Ratcliff, J. W. Lynn, V. [Kiryukhin,](https://www.nature.com/articles/npjquantmats20163#auth-Valery-Kiryukhin-Aff2) P. Jain, and [M. R. 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Series. 340, 012003 (2012), doi: 10.1088/1742- 6596/340/1/012003 ![Figure 2: Neutron scattering data of the magnetic field dependence of the two incommensurate magnetic Bragg peaks in a 100 nm thick BiFeO3 thin films. The linescans follow the black line shown in Fig. 1(a) and were measured at a temperature of 150 K in the magnetic field range from 0 T to 10 T, i.e. in (a) for an increasing magnetic field and in (b) for a decreasing magnetic field back to 0 T. The dashed and solid lines correspond to the result of a fit of two Gaussian lineshapes to the experimental data. For the fit the linewidth was kept constant at the value determined at zero magnetic field and the intensity of both Bragg peaks was kept constant at each magnetic field.](path) ![Figure 3: Neutron scattering data taken in the [110] x [001] scattering plane of the 100 nm thick BiFeO3 thin film taken at a magnetic field of 10 T and a temperature of 150 K. Figure (a) and (b) show the RSMs around the � 1 2 1 2 1 2 �pc and � 1 2 1 2 1 2 ̅ �pc Bragg reflections, respectively. The black lines represent the corresponding linescans in [112�]pc direction as shown in Figs. (c) and (d), respectively. Gaussian lineshapes were fitted to the experimental data, as indicated by the dashed and solid lines.](path) ![Figure 4: (a) Intensity of the pek maximum of the neuron scattering data as a function of magnetic field as shown in Fig. 2. The blue symbols indicate the data obtained for an increasing magnetic field and the red symbols present the data of a decreasing magnetic field. Note that the integrated intensity over both magnetic Bragg peaks remains almost unchanged. (b) length of the spin cycloid extracted from the splitting of the two magnetic Bragg peaks. The length of the spin cycloid is 65.7 ± 1.0 nm at zero magnetic field, expands systematically with increasing magnetic field and diverges towards infinity at a magnetic field of 10 T. The inset shows the zoomed-in data of the cycloidal length and the dashed horizontal line indicates the spin cycloid length of 63 nm obtained previously on bulk and thin film BiFeO3 samples. The solid lines serve as a guides to the eye.](path)
Fig. 1 - The schematic unit cell of (a) Mo2GaB<sup>2</sup> and (b) Mo2GaB compound.	# Exploration of new 212 MAB phases: M2AB2 (M=Mo, Ta; A=Ga, Ge) via DFT calculations #### Abstract The recently developed MAB phases, an extension of the MAX phase, have sparked interest in research among scientists because of their better thermo-mechanical properties. In this paper, we have explored four new MAB phases M2AB2 (M=Mo, Ta and A=Ga, Ge) and studied the elastic, electronic, thermal, and optical properties to predict the possible applications. The stability of the new phases has been confirmed by calculating formation energy (Ef), formation enthalpy (∆H), phonon dispersion curve (PDC), and elastic constant (Cij). The study reveals that M2AB<sup>2</sup> (M=Mo, Ta and A=Ga, Ge) exhibit significantly higher elastic constants, elastic moduli, and Vickers hardness values than their counterpart 211 borides. Higher Vickers hardness values of Ta2AB2 (A=Ga, Ge) than Mo2AB2 (A=Ga, Ge) have been explained based on the values of the bond overlap population. The analysis of the density of states and electronic band structure revealed the metallic nature of the borides under examination. The thermodynamic characteristics of M2AB2 (M=Mo, Ta and A=Ga, Ge) under high temperatures (0–1000 K) are investigated using the quasi-harmonic Debye model. Critical thermal properties such as melting temperature (Tm), Grüneisen parameter (γ), minimum thermal conductivity (Kmin), Debye temperature (ΘD), and others are also computed. Compared with 211 MAX phases, the 212 phases exhibit higher values of (ΘD) and Tm, along with a lower value of Kmin. These findings suggest that the studied compounds exhibit superior thermal properties that are suitable for practical applications. The optical characteristics have been examined, and the reflectance spectrum indicates that the materials have the potential to mitigate solar heating across various energy regions. #First two authors contributed equally. ### 1. Introduction The MAX phase has garnered significant attention in the present era due to its outstanding mechanical and thermal characteristics at high temperatures, showcasing attributes shared by both metals and ceramics. The increased interest in MAX phase materials can be traced back to Barsoum's noteworthy contributions [1], [2]. The term MAX phase represents a family of multilayer solids where M is an earlier transition metal, A is an element from the IIIA or IVA group of the periodic table, and X is an atom of C/N/B [3]. MAX phase materials showcase metallic behavior due to alternate metallic A-layers and ceramic behavior attributed to the MX layers [4]. Like most metals and alloys, MAX phase materials have excellent thermal shock resistance, superior machinability, and enhanced thermal and electrical conductivities. On the other hand, they have high melting or decomposition temperatures and strong elastic stiffness, similar to many ceramics[5]. The unique combination of metallic and ceramic properties makes them versatile, with applications ranging from high-temperature coatings to nuclear accidenttolerant fuel (ATF), concentrated solar power (CSP), catalysis, and as precursors for MXenes [6], [7], [8], [9]. The diversity of MAX phases was confined to C and N as X elements for years (from 1960 to 2014). However, recent advancements have overcome this limitation by successfully synthesizing MAX phases that contain B [10]. The physical and chemical properties of B and Bcontaining compounds highlight the potential of MAX phase borides, which replace C/N with boron [11]. Due to the presence of B in the composition, they are also called the MAB phase. Furthermore, scientific communities are actively working to broaden the diversity of MAX phases by introducing structural changes, as seen in examples like Cr3AlB<sup>4</sup> (space group Immm), Cr4AlB<sup>6</sup> (space group Cmmm), Cr4AlB<sup>4</sup> [12], [13], [14]. Khazaei et al. reported on the first investigation of the theoretical MAX phase borides M2AlB (where M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta). They evaluated these compounds' electronic structure, mechanical characteristics, and dynamical stability in their investigation [15]. The diversity of MAX phases is a subject of research interest due to their structure variations and the number of atoms in the compounds, resulting in changes to their characteristics. In examples like the 211 MAX phase [16], 312 MAX phase [8][17], 212 MAX phase [3], and i-MAX phase [18], the recent additions to the MAX family include the 314 MAX phase and 212 MAX phase, where the element B has been incorporated as an X element [3] [19]. This introduces a novel aspect to the MAX phase family, contributing to its diversity. The unique structural features of MAB phases include both orthorhombic and hexagonal symmetry observed in crystals. These distinctive symmetries set MAB phases apart from the typical MAX phases. Yinqiao Liu's synthesis of the orthorhombic phase of M2AlB2, particularly the 212-MAB phases with M = Sc, Ti, Zr, Hf, V, Nb, Cr, Mo, W, Mn, Tc, Fe, Co, and Ni, has revealed unique structural stability and notable electrical and mechanical properties [20]. The MAB phase structure slightly differs from the standard MAX phases, which typically crystallize in the hexagonal system with a space group -6<sup>2</sup> (No. 187). Ali et al [3]. investigated the diverse physical properties of Zr2AB<sup>2</sup> (A = In, Tl). Martin Ade et al [13] synthesized ternary borides, namely Cr2AlB2, Cr3AlB4, and Cr4AlB6, and subsequently compared their mechanical properties. Qureshi et al. [19] investigated the 314 Zr3CdB<sup>4</sup> MAX phase boride, calculating its mechanical, thermodynamic, and optical properties. The 314 MAX phase Hf3PB<sup>4</sup> has been thoroughly studied using Density Functional Theory (DFT) and revealed that it was the hardest MAX phase compound discovered until that date [21]. The structure of 212 phases exhibits a slight deviation from 211 MAX phases. In the case of 212 phases, a 2D layer of B is situated between M layers, featuring an additional B atom at the X position, unlike in 211 phases [22] [23], [24]. The B-B bonding in 212 MAX phase borides has improved mechanical and thermal properties. To date, the physical properties of Zr2AB<sup>2</sup> (A = In, Tl) [25] and Hf2AB<sup>2</sup> (A = In, Sn) [26], M2AB (M= Ti, Zr, Hf; A=Al, Ga, In) [27] MAX phases, as well as Nb2AC (A = Ga, Ge, Tl, Zn, P, In, and Cd) MAX phases, have been investigated using density functional theory (DFT). In each instance, the mechanical properties of B-containing compounds show significant improvement compared to their traditional C/N containing 211 MAX phases. The Debye temperature and melting temperature are higher for boron-containing 212 phases than for 211 carbides/nitrides, while the minimum thermal conductivity is lower. The thermal expansion coefficient of borides remains well-suited for use as coating materials. Consequently, the superior thermomechanical properties of B-containing 212 MAX phases demonstrate their suitability for high-temperature technological applications, surpassing the commonly used 211 MAX phase carbides. It should be noted that the 212 MAB phase with a hexagonal structure has already been synthesized [24]. In addition, Ga and Ge-based MAX phases have also been synthesized previously [28]. Thus, the reports on the synthesis of 212 phase and Ga and Gebased MAX phases motivated us to select the 212 MAB phases: M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) for our present study, and we have performed an in-depth investigation of their physical properties through DFT method. Therefore, in this paper, the first-time prediction of the stability and mechanical, electronic, thermal, and optical properties of M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) phases has been presented. The results revealed that M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) compounds are stable and suitable for thermal barrier coating (TBC) and reflection coating applications. Additionally, to provide a comparison, the properties determined for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) are compared with those of other 212 and 211 compounds MAX phase borides. ### 2. Methods of calculations First principles density-functional theory (DFT) computations are conducted utilizing the Cambridge Serial Total Energy Package (CASTEP) module integrated within Materials Studio 2017 [29], [30]. The exchange-correlation function is estimated using the Generalized Gradient Approximation (GGA) method, originally suggested by Perdew, Burke, and Ernzerhof [31]. Pseudo-atomic simulations accounted for electronic orbitals corresponding to B (2s 2 2p 1 ), Ga (3d <sup>10</sup> 4s 2 4p 1 ), Ge (4s 2 4p 2 ), Mo (4s 2 4p 6 4d 5 5s 1 ), and Ta (5d 3 6s 2 ). The energy cutoff and k-point grids were established at 650 eV and 11 × 11 × 4, respectively. The structural relaxation was performed utilizing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique [32], while the electronic structure was computed employing density mixing. The parameters for relaxed structures incorporate the following tolerance thresholds: the self-consistent convergence of the total energy is set at 5 × 10-6 eV/atom, the maximum force exerted on the atom is limited to 0.01 eV/Å, the maximum ionic displacement is constrained to 5 × 10-4 Å, and a maximum stress threshold of 0.02 GPa is imposed. The finite strain method [33], grounded in density functional theory (DFT), is utilized to compute the elastic properties within this framework. All necessary equations for determining various properties are provided in the supplementary document. #### 3 Results and discussion #### 3.1 Structural properties The M2AB<sup>2</sup> compounds (where M=Mo or Ta; A=Ga or Ge) belong to the P6m<sup>2</sup> (No. 187)[23] space group and crystallize in the hexagonal system. Unlike conventional MAX phases, which typically belong to the P63/mmc (194) space group, the 212 MAX phases exhibit distinct characteristics. In Fig. 1, the unit cell structure of Mo2GaB<sup>2</sup> is depicted as a representative of M2AB<sup>2</sup> alongside Mo2GaB, facilitating a comparison to discern their differences easily. The atomic positions are as follows: M (Mo or Ta) at (0.3333, 0.6667, 0.6935), A at (0.6667, 0.3333, 0.0), and two B atoms positioned at (0.6667, 0.3333, 0.5) and (0.0, 0.0, 0.5). The B components are arranged at the corners of the unit cell in typical 211 MAX phases, but in 212 boride MAX phases, they form a 2D layer between the M layers. This structural arrangement results in B-B covalent bonds in the 2D layer, enhancing stability compared to conventional 211 MAX phases. The calculated lattice parameters of M2AB<sup>2</sup> are presented in Table 1 alongside those of other 211 and 212 MAX phases for comparison, demonstrating consistency with prior results [16], [34], [35] and affirming the accuracy of the computational methodology used. The primary distinction between 212- and 211-unit cell structures arises from differences in the lattice parameter c, where the c value for 211 exceeds that of 212. Moreover, the volumes of Mo- and Ta-based 211 MAX phase borides surpass those of 212 MAX phase borides. #### 3.2 Stability Examining a compound's stability is significant for multiple purposes, as it yields valuable information regarding the compound's synthesis parameters and aids in assessing the material's resilience across diverse environments, including thermal, compressive, and mechanical pressures. In this section, we delve into a comprehensive theoretical analysis concerning the chemical, dynamic, and mechanical stability of M2AB<sup>2</sup> compounds. The compound's chemical stability is determined by computing its formation energy by the following equation [36]: 2<sup>2</sup> = 22−( + + ) ++ . In the context provided, 22 represents the total energy of the compound after optimizing the unit cell. , , and denote the energies of the individual elements M, A, and B, respectively. The variables x, y, and z correspond to the number of atoms in the unit cell for M, Ga, and B, respectively. For Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, the calculated formation energy (Ef) are -1.8079 eV/atom, -1.8859 eV/atom, -2.1252 eV/atom, and -2.1933 eV/atom, respectively. The negative values signify the chemical stability of all compounds. The order of chemical stability can be expressed as Ta2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Mo2GaB2, indicating that Ta2GeB<sup>2</sup> is the most stable. Additionally, it's observed that MAX phase borides containing Ta are more stable than those containing Mo. Negative formation energy alone may not fully explain the chemical stability of M2AB2 (M=Mo, Ta and A=Ga, Ge). We calculated its formation enthalpy by examining potential pathways to evaluate its thermodynamic stability. For this analysis, we used the experimentally identified stable phases of MoB[37] and TaB[38], Ga4Mo[39], and B2Mo[40]. The potential decomposition pathways for our compounds, as determined from the Open Quantum Materials Database (OQMD), are outlined below. ![We have calculated the reaction energy as follows [41]:](path) Where, M=Mo, Ta and A=Ga, Ge. The following formula can calculate the decomposition energy associated with the reaction energy. = . Where n is the number of participating atoms. The calculated decomposition energies for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> are -0.16, -0.17, -0.26, and -0.28 meV/atom, respectively. So, we can state that the M2AB2 (M=Mo, Ta and A=Ga, Ge) system exhibits thermodynamic stability. Phonon dispersion curves (PDCs) have been calculated at the ground state utilizing the density functional perturbation theory (DFPT) linear-response approach to evaluate the dynamic stability of the MAX phase borides under investigation [42]. The PDCs, depicting the phonon dispersion along the high symmetry directions of the crystal Brillouin zone (BZ), along with the total phonon density of states (PHDOS) of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, are illustrated in Fig. 2(a, b, c and d). Analysis of the PDCs reveals no negative phonon frequencies for any of the compounds, indicating their dynamic stability. The PHDOS of the M2AB<sup>2</sup> compounds are derived from the PDCs and are presented alongside the PDCs in Fig. 2(a, b, c, and d), facilitating band identification through comparison of corresponding peaks. From Fig. 2, it is observed that in Mo2GaB2, the flatness of the bands for the Transverse Optical (TO) modes results in a prominent peak in the PHDOS, whereas non-flat bands for the Longitudinal Optical (LO) modes lead to weaker peaks in the PHDOS. Similar trends are observed in Mo2GeB2, Ta2GaB2, and Ta2GeB2. Notably, a distinct discrepancy arises between the optical and acoustic branches, with the top of the LO and bottom of the TO modes situated at the G point, with separations of 7.49 THz, 6.39 THz, 9.39 THz, and 8.51 THz for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, respectively. ![Fig. 2 – Phonon DOS and Phonon dispersion curves of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) Materials are subjected to various forces and loads in practical applications, necessitating understanding their mechanical stability. The mechanical stability of a compound can be assessed using stiffness constants. For a hexagonal system, the conditions for mechanical stability are as follows [43]: C<sup>11</sup> > 0, C<sup>11</sup> > C12, C<sup>44</sup> > 0, and (C<sup>11</sup> + C12)C<sup>33</sup> - 2(C13)² > 0. As indicated in Table 2, the Cij values of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) satisfy these conditions, thus confirming the mechanical stability of herein predicted phases: M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge). #### 3.3 Electronic properties Analyzing the electronic band structure (EBS) is crucial for gaining insights into the electronic behavior of a compound. The EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) MAX phases are depicted in Fig. 3(a, b, c, and d), with the Fermi energy (EF) level set at 0 eV, represented by a horizontal line. Observing the EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge), it is evident that the conduction band overlaps with the valence band, indicating the absence of a band gap. This observation confirms that the M2AB<sup>2</sup> compounds exhibit metallic behavior, which aligns with conventional MAX phases. The red lines illustrate the overlapping band at the Fermi level. Fig. 3 (a, b) shows that for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> compounds, the maximum band overlap occurs along the A-H path. Conversely, in Fig. 3(c, d), the maximum band overlap is observed along the G-M paths. Utilizing the band structure, we can analyze the electrical anisotropy of M2AB<sup>2</sup> MAX phase compounds. The anisotropic nature can be understood by analyzing the energy dispersion in the basal plane and along the c-axis. The paths G-A, H-K, and M-L show energy dispersion along the c-direction, and A-H, K-G, G-M, and L-H show energy dispersion in the basal plane. In comparison to the paths A-H, K-G, G-M, and L-H (basal plane), there is less energy dispersion along the lines G-A, H–K, and M-L (c-direction), as shown by Fig. 2(a, b, c and d). Lower energy dispersion in the c-direction results from a higher effective mass [44], indicating the strong electronic anisotropy of the M2AB<sup>2</sup> MAX phase compound. Consequently, conductivity along the c-axis is expected to be lower than in the basal planes. These findings are consistent with prior studies [34], [45]. ![Fig. 3 – Band structure of (a) Mo2GaB2, (b) Mo2GeB2, (c) Ta2GaB2, and (d) Ta2GeB<sup>2</sup> compounds.](path) To investigate the bonding nature and electronic conductivity, the total density of states (TDOS) and partial density of states (PDOS) of M2AB<sup>2</sup> compounds were computed. Figure 4 (a, b, c, and d) illustrates the TDOS and PDOS of these compounds, with the Fermi energy (EF) set at zero energy level, indicated by a straight line. These profiles exhibit typical characteristics of MAX phase materials. The Mo or Ta-d electronic states predominantly contribute to the Fermi level, with a minor contribution from the B-p and Ga or Ge-p electronic states. To ascertain the hybridization characteristics of various electronic states within the valence band, the energy spectrum of the valence band has been partitioned into two distinct segments. The first segment encompasses the lower valence band region spanning from -7 eV to -3.5 eV, originating from the hybridization of Mo-p, Mo-d, and B-s orbitals in the case of the Mo2GaB<sup>2</sup> compound. Conversely, for the Mo2GeB2 compound, the lower valence band region arises from the hybridization of Mo-d, Ge-p, and B-s orbitals. Notably, for both the Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compounds, the dominance of the B-p state characterizes the lower valence band region. The second segment pertains to the upper valence band region from -3.5 eV to 0 eV. In the case of the Mo2GaB<sup>2</sup> compound, this region arises from the hybridization of Mo-p and Mo-d orbitals. However, for the Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, the upper valence band region results from the hybridization of Mo-d orbitals and (Ga/Ge)-p orbitals. Notably, the Fermi level of M2AB<sup>2</sup> resides near the pseudogap in the TDOS profile, indicating a high level of electronic stability. Similar trends are observed in other compounds like Zr2AlN, V2AlN, Sc2AlB, Sc2GaB, Ta2GaB, and Hf2GaB<sup>2</sup> [15], [35], [45]. ![Fig. 4 – TDOS and PDOS of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) The electron charge density mapping is helpful in understanding the distribution of electron densities linked to chemical bonds. It delineates areas of positive and negative charge densities, signifying the development and exhaustion of electrical charges, respectively. As depicted in the map, covalent bonds become apparent by accumulating charges between two atoms. Furthermore, the presence of ionic bonds can be inferred from a balance between negative and positive charges at specific atom positions[46]. The valence electronic CDM, denoted in units of eÅ-3 , for M2AB<sup>2</sup> (where M = Mo, Ta; A = Ga, Ge) is showcased in Fig. 5(a, b, c, and d) along the (110) crystallographic plane. The accompanying scale illustrates the intensity of electronic charge density, with red and blue colors indicating low and high electronic charge density, respectively. As depicted in Fig. 5(a, b, c, and d), it is evident that charges accumulate in the regions between the B sites. Consequently, it is anticipated that strong covalent B‒B bonding occurs through the formation of two center-two electron (2c‒2e) bonds in the M2AB<sup>2</sup> (where M=Mo, Ta; A=Ga, Ge) compound, similar to other 212 MAX phase borides like Ti2PB2, Zr2PbB2, Nb2SB2, Zr2GaB<sup>2</sup> and Hf2GaB<sup>2</sup> [34], [45]. Mulliken analysis has corroborated the charge transfer from Mo/Ta atoms to B atoms. The charge received from Mo/Ta atoms is distributed among the B atoms positioned at the transitions and those located at the edges, facilitating the formation of a two center-two electron (2c‒2e) bond between B atoms within the 2D layer of B, as illustrated in Fig. 1(a). The hardness of each bond value presented in Table 5 also aligns with the results obtained from charge density mapping (CDM) and our analysis using moduli and elastic stiffness constants. ![Fig. 5 – The charge density mapping of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) ## 3.4.1 Mechanical properties The mechanical properties of materials play a pivotal role in determining their potential applications, serving as crucial indicators of their behavior and suitability in materials engineering endeavors. These properties are equally applicable to MAX phase materials. Initially, to assess the mechanical properties of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phases, we employed the stress-strain method within the CASTEP code to compute the elastic constants (Cij) [22], [47], [48]. These calculated elastic constants (Cij) are presented in Table 2, alongside those of other 212 and 211 MAX phases. Due to the hexagonal crystal structure of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phase borides, five stiffness constants emerge: C11, C12, C13, C33, and C<sup>44</sup> [49]. The mechanical stability was evaluated using these stiffness constants in the preceding section. For instance, C<sup>11</sup> and C<sup>33</sup> determine stiffness when stress is exerted along the (100) and (001) directions, respectively, whereas C<sup>44</sup> evaluates resistance to shear deformation on the (100) and (001) planes. The stiffness constants C<sup>11</sup> and C<sup>12</sup> directly reflect the strength of atomic bonds along the a- and c-axes. When C<sup>11</sup> exceeds C<sup>33</sup> (or vice versa), it signifies stronger atomic bonding along the a-axis (or c-axis). In Table 2, for Mo2GaB2, Mo2GeB2, and Ta2GeB<sup>2</sup> compounds, C<sup>11</sup> surpasses C33, indicating superior atomic bonding along the a-axis compared to the c-axis. This robust bonding along the a-axis suggests heightened resistance to a-axial deformation. Conversely, in Ta2GeB2, where C<sup>11</sup> < C33, stronger atomic bonding along the c-axis translates to increased resistance against c-axial deformation. Analysis of Table-2 reveals that the values of C<sup>11</sup> and C<sup>22</sup> are notably higher for 212 MAX phase borides than 211 MAX phase borides. Consequently, it can be inferred that 212 MAX phases exhibit stronger resistance to axial deformation when juxtaposed with 211 MAX phases. C<sup>44</sup> is commonly utilized to gauge shear deformation tolerance among the elastic constants. Notably, Ta2GaB exhibits superior shear deformation resistance owing to its highest C<sup>44</sup> value among the studied compounds. Furthermore, the individual elastic constants C12, C13, and C<sup>44</sup> denote shear deformation response under external stress. The observation that C<sup>11</sup> and C<sup>33</sup> possess larger magnitudes than C<sup>44</sup> implies that shear deformation is more facile than axial strain. Another crucial parameter derived from the stiffness constants is the Cauchy pressure (CP), calculated as C<sup>12</sup> - C44, which provides vital insights relevant to the practical applications of solids [51]. Pettifor [51] emphasized the significance of Cauchy pressure (CP) in discerning the chemical bonding and ductile/brittle properties of solids. A negative CP value indicates covalently bonded brittle solids, whereas a positive value signifies isotropic ionic ductile solids. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> fall into covalently bonded brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit metallic ductile characteristics. Similarly, like Ga-containing 211 MAX phase borides, Ga-containing 212 MAX phases also demonstrate negative CP values and brittle behavior [52][53]. The elastic constants obtained are used to produce several bulk elastic parameters that are used to characterize polycrystalline materials, such as Young's modulus (Y), bulk modulus (B), and shear modulus (G). In Table 2, the bulk modulus (B) and shear modulus (G) calculated using Hill's approximation [54] are also presented. Hill's values represent the average of the upper limit (Voigt [55]) and lower limit (Reuss [56]) of B. The necessary equations for these calculations are provided in the supplementary document (S1). Young's modulus (Y) is a crucial indicator of material stiffness. A higher Y value indicates a stiffer material. It can be observed from the table that Mo2GaB<sup>2</sup> exhibits the highest Y value, signifying its greater stiffness compared to others. According to the sequence of Y values, stiffness can be ranked as follows: Mo2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Ta2GeB2. Therefore, Mo2GaB2, with its higher Young's modulus, is anticipated to demonstrate superior mechanical stability and deformation resistance compared to the other compounds under investigation. This quality is essential for aircraft parts or high-performance machinery applications where critical dimensional stability and structural integrity are required [57], [58]. Table 2 shows that 212 MAX phase materials exhibit larger Young's modulus (Y) values than 211 MAX phases. This suggests that 212 MAX phase borides are stiffer than their 211 MAX phase counterparts. Young's modulus (E) also correlates well with thermal shock resistance (R): R ∝ 1/E [59]. Lower Young's modulus values correspond to higher thermal shock resistance. Therefore, materials with higher thermal shock resistance (i.e., lower Young's modulus) are more suitable for use as Thermal Barrier Coating (TBC) materials. Given that Ta2GaB<sup>2</sup> possesses the lowest Y value among the materials studied, it should be considered a superior candidate for TBC material due to its higher thermal shock resistance. The material's ability to withstand shape distortion is elucidated by the shear modulus (G). On the other hand, the bulk modulus (B) indicates the strength of a material's chemical bonds and its ability to withstand uniform compression or volume change. We computed Young's modulus (Y), bulk modulus (B), and shear modulus for our analysis. Based on the bulk modulus values presented in Table 2, the sequence of a material that resists compression when pressure is applied can be outlined as follows: Mo2GeB<sup>2</sup> > Ta2GeB<sup>2</sup> > Mo2GaB<sup>2</sup> > Ta2GeB2. Mo2GeB2, boasting a higher bulk modulus, may exhibit reduced plastic deformation and superior resistance to stress-induced deformation compared to other compounds examined. The ductile and brittle characteristics of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds can also be assessed through Poisson's ratio (ν) and Pugh's ratio (G/B). A compound demonstrates ductile (brittle) behavior if the ν value surpasses (falls below) 0.26 [60]. Furthermore, if the G/B value exceeds (is less than) 0.571, then the compound exhibits brittle (ductile) behavior [19]. According to both criteria, Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are classified as brittle compounds, whereas Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> are categorized as ductile compounds. The ductile nature of Ge-based MAX phases has been reported previously [61]. Fracture toughness (KIC) is a vital property that gauges a material's ability to resist crack propagation. In the case of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, Equation (S2) was employed to determine KIC. The KIC values are documented in Table 2, and notably, these values surpass those of 211 MAX phases [16], [35]. Another parameter, the "f-index," characterizes the isotropic nature and strength of atom-atom bonds within a single hexagonal crystalline lattice along the a- and c-directions. If the f-index is less than 1, chemical bonds exhibit greater rigidity along the c-axis; conversely, if f exceeds 1, bonds are more rigid in the ab-plane. When the value of f is set to 1, atomic bonds exhibit similar strength and uniformity in all directions [15]. The fvalue is computed using Equation (S3) and displayed in Table 2. Table 2 shows that the f-values of Mo2GeB2, Ta2GaB2, and Ta2GeB2, which are close to one, indicate a slight anisotropic bonding strength. Nevertheless, substances with robust bonds in the horizontal plane (ab plane) (f > 1), such as Mo2GaB2, are deemed optimal candidates for the exfoliation process. In engineering applications, the hardness of a solid material serves as a valuable criterion for designing various devices. The elastic properties of polycrystalline materials can be utilized to calculate hardness values, as the ability to resist indentation is closely linked to a material's hardness. Both micro-hardness (Hmicro) and macro-hardness (Hmacro) were computed using Equation (S4) and are presented in Table 2. Based on the values of Hmicro and Hmacro, Mo2GaB<sup>2</sup> emerges as the toughest among the studied phases. The order of hardness is as follows: Mo2GaB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GeB2. Interestingly, Ga-containing MAX phase borides exhibit superior hardness compared to Ge-containing MAX phase borides. In Table 2, we compare our findings with previously reported MAX phases and observe that our data align perfectly with the earlier results. #### 3.4.2 Elastic anisotropy Additionally, anisotropy is linked to other crucial events like anisotropic plastic deformation and the development and spread of micro-cracks within mechanical stress. By providing directiondependent elastic constants, the understanding of anisotropy also offers a framework for improving the mechanical stability of materials in extreme circumstances. The following formulae are used for hexagonal structures to calculate the various anisotropic variables from elastic constants Cij [62]. $A\_{1} = \frac{\frac{1}{6}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{44}};$ $A\_{2} = \frac{2\mathcal{C}\_{44}}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ and $A\_{3} = A\_{1}.$ $A\_{2} = \frac{\frac{1}{3}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ Table 3 lists every anisotropy parameter. Since the value of A<sup>i</sup> should be 1 to be isotropic, the computed value of A<sup>i</sup> (i = 1-3) indicates that all of the compounds under research exhibit anisotropic behavior [63]. An additional method for estimating elastic anisotropy is to use the percentage anisotropy to compressibility and shear (A<sup>B</sup> & AG). This gives polycrystalline materials a helpful way to measure elastic anisotropy. They have been described as [45]; Zero values for AB, AG, and the universal anisotropy factor (AU) show elastic isotropy; the maximum amount of anisotropy is represented by a value of 1. Table 3 shows that, compared to other compounds, the values of AB, AG, and A U for Mo2GeB<sup>2</sup> and Ta2GaB<sup>2</sup> are incredibly close to zero, suggesting that these compounds have nearly isotropic characteristics. #### 3.4.3 Mulliken Populations The Mulliken charge assigned to an atomic species quantifies the effective valence by calculating the absolute difference between the formal ionic charge and the Mulliken charge. Equations (S5) and (S6) are employed to ascertain the Mulliken charge for each atom (α). Table-4 provides the Mulliken atomic population and effective valence charge. Transition metals Mo and Ta in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have pure valence states of 4d<sup>5</sup> and 5d<sup>3</sup> , respectively. The d- orbital electrons of transition metals have been found to influence their effective valence charge significantly. A non-zero positive value indicates a combination of covalent and ionic attributes within chemical bonds. As this value decreases towards zero, it signifies a rise in ionicity. A zero value suggests an ideal ionic character in the bond. Conversely, a progression from zero with a positive value indicates an elevation in the covalency level of the bonds. Based on their effective valence, M atoms move from the left to the right in the periodic table, increasing the covalency of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). Table-4 reveals that the Mulliken atomic charge ascribed to the B atoms is solely negative. Conversely, positive Mulliken atomic charges are associated with transition metals (M) and A. This suggests a charge transfer from M and A to B for each compound within M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge), thereby fostering ionic chemical bonds among these atoms. Bond population serves as another indicator of bond covalency within a crystal, as a high value of bond population essentially signifies a heightened degree of covalency within the chemical bond. The bonding and anti-bonding states influence the populations with positive and negative bond overlap. As demonstrated in Table-5, the B-B bond exhibits greater covalency compared to any other bond in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The presence of an antibonding state between two relevant atoms, which subsequently decreases their chemical bonding, is indicated by a hostile bond overlap population. In Mo2GeB<sup>2</sup> and Ta2GeB2, a hostile bond overlap population is observed in the Ge-Mo and Ge-Ta bonds, indicating the presence of an antibonding state. Therefore, the existence of ionic bonding is guaranteed by electronic charge transfer. On the other hand, the high positive value of the bond overlap population (BOP) denotes the presence of covalent bonding, a feature shared by materials in the MAX phase. ## 3.4.4 Theoretical Vickers Hardness Vickers hardness, derived from the atomic bonds found in solids, indicates how resistant a material is to deformation in extreme circumstances. Several variables influence this feature, such as the crystal flaws, solid structure, atomic arrangement, and bond strength. The Vickers hardness of the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) MAX phases is determined using the Mulliken bond population method, as described by Gou et al. [66] using the formula (S7-S10). This method is particularly suitable for partial metallic systems such as MAX phases. Table-5 lists the computed Vickers hardness values for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The calculated values are 3.35 GPa, 4.76 GPa, 8.21 GPa, and 8.74 GPa for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, respectively. We observed that Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> have much higher hardness values than Mo2GaB<sup>2</sup> and Mo2GeB2. These values are also higher than that of other 212 phases, like, Ti2InB<sup>2</sup> (4.05 GPa) [50], Hf2InB<sup>2</sup> (3.94 GPa), and Hf2SnB<sup>2</sup> (4.41 GPa) [26], Zr2InB<sup>2</sup> (2.92 GPa) and Zr2TlB<sup>2</sup> (2.19 GPa) [25], Zr2GaB<sup>2</sup> (2.53GPa), Zr2GeB<sup>2</sup> (3.31GPa), Hf2GaB<sup>2</sup> (4.73GPa) and Hf2GeB2(4.83GPa) [67]The Hv calculated by the geometrical average of the individual bonding, where the bonding strength mainly determined by the BOP values. In the case of Ta2GaB<sup>2</sup> and Ta2GeB2, the BOP of M-B bonding is much higher compared to Mo2GaB<sup>2</sup> and Mo2GeB2. Even though the BOP of M-B for Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> is higher than that of the other 212 phases mentioned earlier. Therefore, higher values of Hv are expected for Ta2GaB<sup>2</sup> and Ta2GeB2. Additionally, we looked at the Vickers hardness value between 211 and 212 compounds and discovered that the 212 MAX phase compounds had a higher H<sup>V</sup> value. This is because a 2D layer of B atoms is positioned between the M atoms. The B atoms share two center-two electrons to form an extremely strong B-B bond [3]. Table 5 Calculated data for Mulliken bond number (n μ ), bond length (d μ ), bond overlap populations BOP, (P μ ), metallic populations (P μˊ), Vickers hardness (HV) M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. # 3.5 Thermal Properties MAX phases are ideal for high-temperature applications due to their exceptional mechanical qualities at elevated temperatures. As a result, researching the fundamental parameters necessary for predicting their application is of great interest, and these can be obtained from the vibrations of atoms or phonons. The Debye temperature (ΘD) of a solid is directly connected to its bonding strength, melting temperature, thermal expansion, and conductivity. Using the sound velocity and Anderson's technique [68], the Θ<sup>D</sup> of the phases under study has been computed using the formula (S11). Equation (S12) can be used to get the average sound velocity (Vm) from the longitudinal and transverse sound velocities. Equations (S13–14) were used to determine v<sup>l</sup> and vt. The calculated values of Debye's temperature are shown in Table 6, where Mo2GaB<sup>2</sup> has the highest Θ<sup>D</sup> and Ta2GeB<sup>2</sup> has the lowest. If we rank them, it is as follows: Mo2GaB<sup>2</sup> < Mo2GeB<sup>2</sup> < Ta2GaB2 < Ta2GeB2. Hadi et al. recently reported a MAX phase (V2SnC) as a TBC material with a Θ<sup>D</sup> value of 472 K [69]. Thus, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) exhibit encouraging potential as TBC materials, as shown in Table 6. Table 6-Data for density (ρ), longitudinal, transverse, and average sound velocities (vl, vt, and vm), Debye temperature (ΘD), minimum thermal conductivity (Kmin), Grüneisen parameter (γ), thermal expansion coefficient (TEC) at 300K and melting temperature (Tm) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The constant thermal conductivity value at high temperatures is the minimum thermal conductivity (Kmin). As the name implies, this conductivity is minimal because, at high temperatures, phonon coupling breaks. The formula (S15) for the minimum thermal conductivity of solids was derived using the Clarke model [70]. Table 6 lists the calculated value of kmin, with Ta2GaB<sup>2</sup> having the lowest value and Mo2GaB<sup>2</sup> having the highest. When selecting suitable materials for TBC applications, a minimum thermal conductivity of 1.25 W/mK is used as a screening criterion [71]. Our compounds exhibit lower minimum thermal conductivity values, holding promising potential as TBC materials. Gd2Zr2O<sup>7</sup> and Y2SiO5, two recently developed thermal barrier coating (TBC) materials, have minimal thermal conductivities (Kmin) of 1.22 W/m.K. and 1.3 W/m.K. [72], respectively, as confirmed by experiment. These numbers roughly match the values we computed for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The Grüneisen parameter (γ) is a crucial thermal parameter that helps explain the anharmonic effects of lattice dynamics; solids utilized at high temperatures are expected to have lower anharmonic effects. The Grüneisen parameter (γ) can be determined with the help of Poisson's ratio using equation (S16) [73]. The computed γ values, as shown in Table 6, suggest that the compounds under investigation exhibit a weak anharmonic effect. Additionally, for solids with a Poisson's ratio between 0.05 and 0.46, the values similarly fall within the range of 0.85 and 3.53 [74]. The melting temperature (Tm) of the compounds under investigation has been calculated using the following formula (S17). The strength of atomic bonding is the primary factor determining the melting temperature of solids; the higher the Tm, the stronger the atomic bonding. The order of T<sup>m</sup> for the titled phases is found to mirror the Y-based (Young's modulus) order, indicating a close link between T<sup>m</sup> and Y [75]. As observed in Table 6, our compounds roughly follow the Ybased ranking with Mo2GeB<sup>2</sup> < Mo2GaB<sup>2</sup> < Ta2GeB<sup>2</sup> < Ta2GaB2. The T<sup>m</sup> value for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) is also comparable with the TBC material Y4Al2O<sup>9</sup> (2000 K)[75]. ![Fig. 6: Temperature dependence of the calculated thermodynamic parameters. (a) Specific heat at constant pressure, C<sup>p</sup> (b) Specific heat at constant volume, C<sup>v</sup> (c) Free energy, F (d) Entropy, S (e) Internal energy, E and (f) Thermal expansion coefficient, TEC of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge).](path) The temperature dependence of specific heats, Cv, Cp, for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds was obtained using the formulas (S19-S20), as shown in Fig.6(a,b). Assuming that the quasi-harmonic model is accurate and that phase transitions are not anticipated for the compounds under study, these properties are approximated across a temperature range of 0 to 1000 K. Because phonon thermal softening occurs at higher temperatures, the heat capacity rises with temperature. Heat capacities increase quickly and follow the Debye-T 3 power law at lower temperatures. At higher temperature regimes, where C<sup>v</sup> and C<sup>p</sup> do not greatly depend on temperature, they approach the Dulong-Petit (3) limit.[76]. Using the quasi-harmonic approximation, various temperature-dependent thermodynamic potential functions for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have been estimated at zero pressure and displayed in Fig.6 [77]. These functions include the Helmholtz free energy (F), internal energy (E), and entropy (S) within 0-1000 K temperature using equation (S21-S23). The free energy progressively decreases as the temperature rises, as shown in Fig.6(c). Free energy typically declines, and this trend becomes increasingly negative as a natural process proceeds. As demonstrated in Fig. 6(d), the internal energy (E) shows a rising trend with temperature, in contrast to the free energy. Since thermal agitation creates disorder, a system's entropy rises as temperature rises. This is illustrated in Fig.6(d). A material's thermal expansion coefficient (TEC) is derived from the anharmonicity in the lattice dynamics and can be found using equation (S18). The measure of a material's capacity to expand or contract with heat or cold is called the Thermal Expansion Coefficient, or TEC. As observed in Fig. 5(f), the Thermal Expansion Coefficient (TEC) increases rapidly up to 365 K. Then it approaches a constant value, which indicates lower saturations in the materials with temperature changes. The materials under study have a very low TEC value, a crucial characteristic of materials intended for application in high-temperature technology. To be effective as thermal barrier coating (TBC) materials, compounds must exhibit a low thermal conductivity (Kmin) to impede heat transfer, a high melting temperature to withstand extreme heat, and a low thermal expansion coefficient (TEC) to maintain dimensional stability under thermal stress. The compounds M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge) possess these properties, making them suitable candidates for use as TBC materials #### 3.6 Optical Properties Different materials exhibit unique behaviors when exposed to electromagnetic radiation. The optical constants determine the overall response of the sample to the incident radiation. The complex dielectric function, defined as ε(ω)=ε1(ω)+iε2(ω), is one of the main optical characteristics of solids. The following formula determines the imaginary part of the dielectric function ε2(ω) from the momentum matrix element between the occupied and unoccupied electronic states. In this formula, e stands for an electronic charge, ω for light angular frequency, u for the polarization vector of the incident electric field, and and for the conduction and valence band wave functions, respectively, at k. The Kramers-Kronig equation can estimate the real part of the dielectric function, ε1(ω). In contrast, ε2(ω) and ε1(ω) are utilized to evaluate all other optical parameters, such as the absorption coefficient, photoconductivity, reflectivity, and loss function [78]. In this part, several energy-dependent optical properties of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) 212 MAX phases are calculated and analyzed in detail for the photon energy range of 0 to 30 eV, for [100] plane enabling the first assessment of the compounds' practical applicability. Given the metallic conductivity of the MAX compounds' electronic structure, additional parameters were chosen to analyze the optical properties. These include a plasma frequency of 3 eV, damping of 0.05 eV, and Gaussian smearing of 0.5 eV [79]. Figure 7(a) shows the real component of the dielectric function, ε1, which exhibits metallic behavior. In metallic systems, ε<sup>1</sup> has a considerably high negative value in the low-energy range, with the real component reaching negative, which aligns with the band structure finding. Fig. 7(b) depicts the imaginary part of the dielectric function, ε2(ω), representing dielectric losses about frequency. Mo2GeB<sup>2</sup> demonstrates the highest peak in the low-energy region, with all compounds approaching zero from above at around 17 eV. This observation confirms Drude's behavior. ![Fig. 7- (a) Real part of dielectric constant, ε<sup>1</sup> (b) Imaginary part of dielectric constant, ε<sup>2</sup> (c) Reflectivity, R and (d) Loss function, (e) The coefficient of absorption, (α) (f) Refractive index, (n) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) for [100] electric field directions.](path) The reflectivity of a material measures the percentage of incident light energy reflected off it. Equation (S24) was employed to compute reflectivity using the dielectric function, as shown in Fig.7(c). The spectra's visible and infrared (IR) portions consistently exhibit values exceeding 55%. Consequently, the materials under examination are expected to appear metallic gray. In the infrared (IR) region, the maximum reflectivity for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> is 65% and 74%, respectively, occurring at 0.33 eV. Conversely, for Ta2GaB2, the maximum reflectivity is 59% at 14.02 eV, while for Ta2GeB2, it is 65% at 14.78 eV in the UV region. After 18 eV, the reflectance decreases significantly. According to reports, substances with an average reflectivity value above 44% in the visible light region can effectively reduce solar heating by reflecting a significant amount of the solar spectrum. Among all the compounds we examined, Ta2GaB<sup>2</sup> has a reflectivity in the visible range of more than 44% [78]. Therefore, Ta2GaB<sup>2</sup> can be utilized as a coating material and should be able to mitigate solar heating. plasma frequencies (ωp). The plasma frequency for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is observed at 23 eV, 25 eV, 21 eV, and 23 eV, respectively. At this specific frequency, the absorption coefficient rapidly decreases, ε<sup>1</sup> crosses zero from the negative side, and the reflectance R(ω) displays a falling tail. Above this distinctive frequency, the materials are transparent to the incident electromagnetic radiation. Fig. 7(e) illustrates the absorbance coefficient (α) of compounds M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) determined using equation (S26). As α begins to rise from zero photon energy, the metallic nature of the substances under study is again indicated. The visible light region experiences a sharp increase in absorption, peaking in the UV region at around 14 eV and then progressively declining. The IR region exhibits negligible absorption. However, the materials mentioned above appear to have a significant absorption band primarily located in the visible and ultraviolet spectrum. This suggests that the materials can be used in UV surface-disinfection devices, medical sterilization equipment, and other optoelectronic device designs. Its refractive index is crucial to a material's potential application in optical devices like waveguides and photonic crystals. Equations (S27) and (S28) were used to derive the refractive index (n) and extinction coefficient (k) for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), as depicted in Fig.7(c) and Fig.S-1(a). The variations of n and k in MAX phase carbides with incident photon energy closely resemble ε1(ω) and ε2(ω). The static refractive index n(0) for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is 9.4,13.07,7 and 9.2, respectively, and decreases gradually with the increase in photon energy. The extinction coefficient for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2 gradually increases in the IR region, reaching their maximum values of 2.56, 4.27, 2.42, and 3.28, respectively. Following this, they slowly decrease in the visible and UV regions. Figure S-1(b) illustrates the photoconductivity (σ) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) across various photon energies. The photoconductivity (σ) parameter quantifies the impact of photon irradiation on a material's electrical conductivity. Similar to the absorbance coefficient (α) spectrum, the σ spectrum aligns with the metallic nature of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). #### 4 Conclusions In summary, we employed DFT calculations to explore four 212 MAB phases, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), and investigated the structural, electronic, mechanical, lattice dynamical, and optical properties to predict their possible applications. The phonon dispersion curves, formation energy, and elastic constants collectively suggest that the M2AB<sup>2</sup> boride maintains dynamic, mechanical, chemical, and thermodynamic stability. The electronic band structure and density of states (DOS) offer evidence supporting the metallic nature of the compounds under investigation. Concurrently, the charge density mapping and atomic Mulliken population both confirm the presence of a strong B-B covalent bond. The stiffness constants, elastic moduli, findex, fracture toughness (KIC), Pugh's ratio (G/B), hardness parameters, and Cauchy Pressure (CP) of M2AB<sup>2</sup> were computed and compared with those of their 211 equivalents. We found that the values of the 212-phase borides are higher than those of the 211-phase carbides or borides. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are identified as brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit ductile characteristics, as indicated by Poisson's ratio (ν) and Pugh's ratio (B/G or G/B). The elastic characteristics display anisotropy due to the distinct atomic configurations along the aand c-directions. Vickers hardness calculations are considered reliable indicators of material hardness. The results suggest that Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> possess more pliable characteristics than Ta2GaB<sup>2</sup> and Ta2GeB2. The high hardness values of Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compared to the Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> due to higher BOP values of M-B bonding in Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> than in Mo2GaB<sup>2</sup> and Mo2GeB2. The value of minimum thermal conductivity (Kmin), thermal expansion coefficient, and melting temperature (Tm) collectively suggest the potential suitability of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) as a material for thermal barrier coating (TBC) applications in high-temperature devices. The optical conductivity and absorption coefficient corroborate the findings of the electronic band structure. Reflectivity is notably high in infrared (IR) regions and remains nearly constant in the visible and moderate ultraviolet (UV) regions, with an average value exceeding 44% for Ta2GaB2. This suggests that Ta2GaB<sup>2</sup> can be effectively utilized as a coating material to reduce solar heating. We expect that the comprehensive analysis of the diverse physical characteristics of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) presented in this study will establish a robust foundation for future theoretical and experimental explorations of these fascinating MAB phases. ## CRediT author statement A. K. M Naim Ishtiaq and Md Nasir Uddin: Data curation, Writing- Original draft preparation. Md. Rasel Rana, Shariful Islam and Noor Afsary: Reviewing and Editing. Md. 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Fig. 7- (a) Real part of dielectric constant, ε<sup>1</sup> (b) Imaginary part of dielectric constant, ε<sup>2</sup> (c) Reflectivity, R and (d) Loss function, (e) The coefficient of absorption, (α) (f) Refractive index, (n) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) for [100] electric field directions.	# Exploration of new 212 MAB phases: M2AB2 (M=Mo, Ta; A=Ga, Ge) via DFT calculations #### Abstract The recently developed MAB phases, an extension of the MAX phase, have sparked interest in research among scientists because of their better thermo-mechanical properties. In this paper, we have explored four new MAB phases M2AB2 (M=Mo, Ta and A=Ga, Ge) and studied the elastic, electronic, thermal, and optical properties to predict the possible applications. The stability of the new phases has been confirmed by calculating formation energy (Ef), formation enthalpy (∆H), phonon dispersion curve (PDC), and elastic constant (Cij). The study reveals that M2AB<sup>2</sup> (M=Mo, Ta and A=Ga, Ge) exhibit significantly higher elastic constants, elastic moduli, and Vickers hardness values than their counterpart 211 borides. Higher Vickers hardness values of Ta2AB2 (A=Ga, Ge) than Mo2AB2 (A=Ga, Ge) have been explained based on the values of the bond overlap population. The analysis of the density of states and electronic band structure revealed the metallic nature of the borides under examination. The thermodynamic characteristics of M2AB2 (M=Mo, Ta and A=Ga, Ge) under high temperatures (0–1000 K) are investigated using the quasi-harmonic Debye model. Critical thermal properties such as melting temperature (Tm), Grüneisen parameter (γ), minimum thermal conductivity (Kmin), Debye temperature (ΘD), and others are also computed. Compared with 211 MAX phases, the 212 phases exhibit higher values of (ΘD) and Tm, along with a lower value of Kmin. These findings suggest that the studied compounds exhibit superior thermal properties that are suitable for practical applications. The optical characteristics have been examined, and the reflectance spectrum indicates that the materials have the potential to mitigate solar heating across various energy regions. #First two authors contributed equally. ### 1. Introduction The MAX phase has garnered significant attention in the present era due to its outstanding mechanical and thermal characteristics at high temperatures, showcasing attributes shared by both metals and ceramics. The increased interest in MAX phase materials can be traced back to Barsoum's noteworthy contributions [1], [2]. The term MAX phase represents a family of multilayer solids where M is an earlier transition metal, A is an element from the IIIA or IVA group of the periodic table, and X is an atom of C/N/B [3]. MAX phase materials showcase metallic behavior due to alternate metallic A-layers and ceramic behavior attributed to the MX layers [4]. Like most metals and alloys, MAX phase materials have excellent thermal shock resistance, superior machinability, and enhanced thermal and electrical conductivities. On the other hand, they have high melting or decomposition temperatures and strong elastic stiffness, similar to many ceramics[5]. The unique combination of metallic and ceramic properties makes them versatile, with applications ranging from high-temperature coatings to nuclear accidenttolerant fuel (ATF), concentrated solar power (CSP), catalysis, and as precursors for MXenes [6], [7], [8], [9]. The diversity of MAX phases was confined to C and N as X elements for years (from 1960 to 2014). However, recent advancements have overcome this limitation by successfully synthesizing MAX phases that contain B [10]. The physical and chemical properties of B and Bcontaining compounds highlight the potential of MAX phase borides, which replace C/N with boron [11]. Due to the presence of B in the composition, they are also called the MAB phase. Furthermore, scientific communities are actively working to broaden the diversity of MAX phases by introducing structural changes, as seen in examples like Cr3AlB<sup>4</sup> (space group Immm), Cr4AlB<sup>6</sup> (space group Cmmm), Cr4AlB<sup>4</sup> [12], [13], [14]. Khazaei et al. reported on the first investigation of the theoretical MAX phase borides M2AlB (where M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta). They evaluated these compounds' electronic structure, mechanical characteristics, and dynamical stability in their investigation [15]. The diversity of MAX phases is a subject of research interest due to their structure variations and the number of atoms in the compounds, resulting in changes to their characteristics. In examples like the 211 MAX phase [16], 312 MAX phase [8][17], 212 MAX phase [3], and i-MAX phase [18], the recent additions to the MAX family include the 314 MAX phase and 212 MAX phase, where the element B has been incorporated as an X element [3] [19]. This introduces a novel aspect to the MAX phase family, contributing to its diversity. The unique structural features of MAB phases include both orthorhombic and hexagonal symmetry observed in crystals. These distinctive symmetries set MAB phases apart from the typical MAX phases. Yinqiao Liu's synthesis of the orthorhombic phase of M2AlB2, particularly the 212-MAB phases with M = Sc, Ti, Zr, Hf, V, Nb, Cr, Mo, W, Mn, Tc, Fe, Co, and Ni, has revealed unique structural stability and notable electrical and mechanical properties [20]. The MAB phase structure slightly differs from the standard MAX phases, which typically crystallize in the hexagonal system with a space group -6<sup>2</sup> (No. 187). Ali et al [3]. investigated the diverse physical properties of Zr2AB<sup>2</sup> (A = In, Tl). Martin Ade et al [13] synthesized ternary borides, namely Cr2AlB2, Cr3AlB4, and Cr4AlB6, and subsequently compared their mechanical properties. Qureshi et al. [19] investigated the 314 Zr3CdB<sup>4</sup> MAX phase boride, calculating its mechanical, thermodynamic, and optical properties. The 314 MAX phase Hf3PB<sup>4</sup> has been thoroughly studied using Density Functional Theory (DFT) and revealed that it was the hardest MAX phase compound discovered until that date [21]. The structure of 212 phases exhibits a slight deviation from 211 MAX phases. In the case of 212 phases, a 2D layer of B is situated between M layers, featuring an additional B atom at the X position, unlike in 211 phases [22] [23], [24]. The B-B bonding in 212 MAX phase borides has improved mechanical and thermal properties. To date, the physical properties of Zr2AB<sup>2</sup> (A = In, Tl) [25] and Hf2AB<sup>2</sup> (A = In, Sn) [26], M2AB (M= Ti, Zr, Hf; A=Al, Ga, In) [27] MAX phases, as well as Nb2AC (A = Ga, Ge, Tl, Zn, P, In, and Cd) MAX phases, have been investigated using density functional theory (DFT). In each instance, the mechanical properties of B-containing compounds show significant improvement compared to their traditional C/N containing 211 MAX phases. The Debye temperature and melting temperature are higher for boron-containing 212 phases than for 211 carbides/nitrides, while the minimum thermal conductivity is lower. The thermal expansion coefficient of borides remains well-suited for use as coating materials. Consequently, the superior thermomechanical properties of B-containing 212 MAX phases demonstrate their suitability for high-temperature technological applications, surpassing the commonly used 211 MAX phase carbides. It should be noted that the 212 MAB phase with a hexagonal structure has already been synthesized [24]. In addition, Ga and Ge-based MAX phases have also been synthesized previously [28]. Thus, the reports on the synthesis of 212 phase and Ga and Gebased MAX phases motivated us to select the 212 MAB phases: M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) for our present study, and we have performed an in-depth investigation of their physical properties through DFT method. Therefore, in this paper, the first-time prediction of the stability and mechanical, electronic, thermal, and optical properties of M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) phases has been presented. The results revealed that M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) compounds are stable and suitable for thermal barrier coating (TBC) and reflection coating applications. Additionally, to provide a comparison, the properties determined for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) are compared with those of other 212 and 211 compounds MAX phase borides. ### 2. Methods of calculations First principles density-functional theory (DFT) computations are conducted utilizing the Cambridge Serial Total Energy Package (CASTEP) module integrated within Materials Studio 2017 [29], [30]. The exchange-correlation function is estimated using the Generalized Gradient Approximation (GGA) method, originally suggested by Perdew, Burke, and Ernzerhof [31]. Pseudo-atomic simulations accounted for electronic orbitals corresponding to B (2s 2 2p 1 ), Ga (3d <sup>10</sup> 4s 2 4p 1 ), Ge (4s 2 4p 2 ), Mo (4s 2 4p 6 4d 5 5s 1 ), and Ta (5d 3 6s 2 ). The energy cutoff and k-point grids were established at 650 eV and 11 × 11 × 4, respectively. The structural relaxation was performed utilizing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique [32], while the electronic structure was computed employing density mixing. The parameters for relaxed structures incorporate the following tolerance thresholds: the self-consistent convergence of the total energy is set at 5 × 10-6 eV/atom, the maximum force exerted on the atom is limited to 0.01 eV/Å, the maximum ionic displacement is constrained to 5 × 10-4 Å, and a maximum stress threshold of 0.02 GPa is imposed. The finite strain method [33], grounded in density functional theory (DFT), is utilized to compute the elastic properties within this framework. All necessary equations for determining various properties are provided in the supplementary document. #### 3 Results and discussion #### 3.1 Structural properties The M2AB<sup>2</sup> compounds (where M=Mo or Ta; A=Ga or Ge) belong to the P6m<sup>2</sup> (No. 187)[23] space group and crystallize in the hexagonal system. Unlike conventional MAX phases, which typically belong to the P63/mmc (194) space group, the 212 MAX phases exhibit distinct characteristics. In Fig. 1, the unit cell structure of Mo2GaB<sup>2</sup> is depicted as a representative of M2AB<sup>2</sup> alongside Mo2GaB, facilitating a comparison to discern their differences easily. The atomic positions are as follows: M (Mo or Ta) at (0.3333, 0.6667, 0.6935), A at (0.6667, 0.3333, 0.0), and two B atoms positioned at (0.6667, 0.3333, 0.5) and (0.0, 0.0, 0.5). The B components are arranged at the corners of the unit cell in typical 211 MAX phases, but in 212 boride MAX phases, they form a 2D layer between the M layers. This structural arrangement results in B-B covalent bonds in the 2D layer, enhancing stability compared to conventional 211 MAX phases. The reflectivity of a material measures the percentage of incident light energy reflected off it. Equation (S24) was employed to compute reflectivity using the dielectric function, as shown in Fig.7(c). The spectra's visible and infrared (IR) portions consistently exhibit values exceeding 55%. Consequently, the materials under examination are expected to appear metallic gray. In the infrared (IR) region, the maximum reflectivity for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> is 65% and 74%, respectively, occurring at 0.33 eV. Conversely, for Ta2GaB2, the maximum reflectivity is 59% at 14.02 eV, while for Ta2GeB2, it is 65% at 14.78 eV in the UV region. After 18 eV, the reflectance decreases significantly. According to reports, substances with an average reflectivity value above 44% in the visible light region can effectively reduce solar heating by reflecting a significant amount of the solar spectrum. Among all the compounds we examined, Ta2GaB<sup>2</sup> has a reflectivity in the visible range of more than 44% [78]. Therefore, Ta2GaB<sup>2</sup> can be utilized as a coating material and should be able to mitigate solar heating. plasma frequencies (ωp). The plasma frequency for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is observed at 23 eV, 25 eV, 21 eV, and 23 eV, respectively. At this specific frequency, the absorption coefficient rapidly decreases, ε<sup>1</sup> crosses zero from the negative side, and the reflectance R(ω) displays a falling tail. Above this distinctive frequency, the materials are transparent to the incident electromagnetic radiation. Fig. 7(e) illustrates the absorbance coefficient (α) of compounds M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) determined using equation (S26). As α begins to rise from zero photon energy, the metallic nature of the substances under study is again indicated. The visible light region experiences a sharp increase in absorption, peaking in the UV region at around 14 eV and then progressively declining. The IR region exhibits negligible absorption. However, the materials mentioned above appear to have a significant absorption band primarily located in the visible and ultraviolet spectrum. This suggests that the materials can be used in UV surface-disinfection devices, medical sterilization equipment, and other optoelectronic device designs. Its refractive index is crucial to a material's potential application in optical devices like waveguides and photonic crystals. Equations (S27) and (S28) were used to derive the refractive index (n) and extinction coefficient (k) for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), as depicted in Fig.7(c) and Fig.S-1(a). The variations of n and k in MAX phase carbides with incident photon energy closely resemble ε1(ω) and ε2(ω). The static refractive index n(0) for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is 9.4,13.07,7 and 9.2, respectively, and decreases gradually with the increase in photon energy. The extinction coefficient for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2 gradually increases in the IR region, reaching their maximum values of 2.56, 4.27, 2.42, and 3.28, respectively. Following this, they slowly decrease in the visible and UV regions. Figure S-1(b) illustrates the photoconductivity (σ) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) across various photon energies. The photoconductivity (σ) parameter quantifies the impact of photon irradiation on a material's electrical conductivity. Similar to the absorbance coefficient (α) spectrum, the σ spectrum aligns with the metallic nature of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). #### 4 Conclusions In summary, we employed DFT calculations to explore four 212 MAB phases, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), and investigated the structural, electronic, mechanical, lattice dynamical, and optical properties to predict their possible applications. The phonon dispersion curves, formation energy, and elastic constants collectively suggest that the M2AB<sup>2</sup> boride maintains dynamic, mechanical, chemical, and thermodynamic stability. The electronic band structure and density of states (DOS) offer evidence supporting the metallic nature of the compounds under investigation. Concurrently, the charge density mapping and atomic Mulliken population both confirm the presence of a strong B-B covalent bond. The stiffness constants, elastic moduli, findex, fracture toughness (KIC), Pugh's ratio (G/B), hardness parameters, and Cauchy Pressure (CP) of M2AB<sup>2</sup> were computed and compared with those of their 211 equivalents. We found that the values of the 212-phase borides are higher than those of the 211-phase carbides or borides. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are identified as brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit ductile characteristics, as indicated by Poisson's ratio (ν) and Pugh's ratio (B/G or G/B). The elastic characteristics display anisotropy due to the distinct atomic configurations along the aand c-directions. Vickers hardness calculations are considered reliable indicators of material hardness. The results suggest that Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> possess more pliable characteristics than Ta2GaB<sup>2</sup> and Ta2GeB2. The high hardness values of Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compared to the Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> due to higher BOP values of M-B bonding in Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> than in Mo2GaB<sup>2</sup> and Mo2GeB2. The value of minimum thermal conductivity (Kmin), thermal expansion coefficient, and melting temperature (Tm) collectively suggest the potential suitability of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) as a material for thermal barrier coating (TBC) applications in high-temperature devices. The optical conductivity and absorption coefficient corroborate the findings of the electronic band structure. Reflectivity is notably high in infrared (IR) regions and remains nearly constant in the visible and moderate ultraviolet (UV) regions, with an average value exceeding 44% for Ta2GaB2. This suggests that Ta2GaB<sup>2</sup> can be effectively utilized as a coating material to reduce solar heating. We expect that the comprehensive analysis of the diverse physical characteristics of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) presented in this study will establish a robust foundation for future theoretical and experimental explorations of these fascinating MAB phases. ## CRediT author statement A. K. M Naim Ishtiaq and Md Nasir Uddin: Data curation, Writing- Original draft preparation. Md. Rasel Rana, Shariful Islam and Noor Afsary: Reviewing and Editing. Md. 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Fig. 3 – Band structure of (a) Mo2GaB2, (b) Mo2GeB2, (c) Ta2GaB2, and (d) Ta2GeB<sup>2</sup> compounds.	# Exploration of new 212 MAB phases: M2AB2 (M=Mo, Ta; A=Ga, Ge) via DFT calculations #### Abstract The recently developed MAB phases, an extension of the MAX phase, have sparked interest in research among scientists because of their better thermo-mechanical properties. In this paper, we have explored four new MAB phases M2AB2 (M=Mo, Ta and A=Ga, Ge) and studied the elastic, electronic, thermal, and optical properties to predict the possible applications. The stability of the new phases has been confirmed by calculating formation energy (Ef), formation enthalpy (∆H), phonon dispersion curve (PDC), and elastic constant (Cij). The study reveals that M2AB<sup>2</sup> (M=Mo, Ta and A=Ga, Ge) exhibit significantly higher elastic constants, elastic moduli, and Vickers hardness values than their counterpart 211 borides. Higher Vickers hardness values of Ta2AB2 (A=Ga, Ge) than Mo2AB2 (A=Ga, Ge) have been explained based on the values of the bond overlap population. The analysis of the density of states and electronic band structure revealed the metallic nature of the borides under examination. The thermodynamic characteristics of M2AB2 (M=Mo, Ta and A=Ga, Ge) under high temperatures (0–1000 K) are investigated using the quasi-harmonic Debye model. Critical thermal properties such as melting temperature (Tm), Grüneisen parameter (γ), minimum thermal conductivity (Kmin), Debye temperature (ΘD), and others are also computed. Compared with 211 MAX phases, the 212 phases exhibit higher values of (ΘD) and Tm, along with a lower value of Kmin. These findings suggest that the studied compounds exhibit superior thermal properties that are suitable for practical applications. The optical characteristics have been examined, and the reflectance spectrum indicates that the materials have the potential to mitigate solar heating across various energy regions. #First two authors contributed equally. ### 1. Introduction The MAX phase has garnered significant attention in the present era due to its outstanding mechanical and thermal characteristics at high temperatures, showcasing attributes shared by both metals and ceramics. The increased interest in MAX phase materials can be traced back to Barsoum's noteworthy contributions [1], [2]. The term MAX phase represents a family of multilayer solids where M is an earlier transition metal, A is an element from the IIIA or IVA group of the periodic table, and X is an atom of C/N/B [3]. MAX phase materials showcase metallic behavior due to alternate metallic A-layers and ceramic behavior attributed to the MX layers [4]. Like most metals and alloys, MAX phase materials have excellent thermal shock resistance, superior machinability, and enhanced thermal and electrical conductivities. On the other hand, they have high melting or decomposition temperatures and strong elastic stiffness, similar to many ceramics[5]. The unique combination of metallic and ceramic properties makes them versatile, with applications ranging from high-temperature coatings to nuclear accidenttolerant fuel (ATF), concentrated solar power (CSP), catalysis, and as precursors for MXenes [6], [7], [8], [9]. The diversity of MAX phases was confined to C and N as X elements for years (from 1960 to 2014). However, recent advancements have overcome this limitation by successfully synthesizing MAX phases that contain B [10]. The physical and chemical properties of B and Bcontaining compounds highlight the potential of MAX phase borides, which replace C/N with boron [11]. Due to the presence of B in the composition, they are also called the MAB phase. Furthermore, scientific communities are actively working to broaden the diversity of MAX phases by introducing structural changes, as seen in examples like Cr3AlB<sup>4</sup> (space group Immm), Cr4AlB<sup>6</sup> (space group Cmmm), Cr4AlB<sup>4</sup> [12], [13], [14]. Khazaei et al. reported on the first investigation of the theoretical MAX phase borides M2AlB (where M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta). They evaluated these compounds' electronic structure, mechanical characteristics, and dynamical stability in their investigation [15]. The diversity of MAX phases is a subject of research interest due to their structure variations and the number of atoms in the compounds, resulting in changes to their characteristics. In examples like the 211 MAX phase [16], 312 MAX phase [8][17], 212 MAX phase [3], and i-MAX phase [18], the recent additions to the MAX family include the 314 MAX phase and 212 MAX phase, where the element B has been incorporated as an X element [3] [19]. This introduces a novel aspect to the MAX phase family, contributing to its diversity. The unique structural features of MAB phases include both orthorhombic and hexagonal symmetry observed in crystals. These distinctive symmetries set MAB phases apart from the typical MAX phases. Yinqiao Liu's synthesis of the orthorhombic phase of M2AlB2, particularly the 212-MAB phases with M = Sc, Ti, Zr, Hf, V, Nb, Cr, Mo, W, Mn, Tc, Fe, Co, and Ni, has revealed unique structural stability and notable electrical and mechanical properties [20]. The MAB phase structure slightly differs from the standard MAX phases, which typically crystallize in the hexagonal system with a space group -6<sup>2</sup> (No. 187). Ali et al [3]. investigated the diverse physical properties of Zr2AB<sup>2</sup> (A = In, Tl). Martin Ade et al [13] synthesized ternary borides, namely Cr2AlB2, Cr3AlB4, and Cr4AlB6, and subsequently compared their mechanical properties. Qureshi et al. [19] investigated the 314 Zr3CdB<sup>4</sup> MAX phase boride, calculating its mechanical, thermodynamic, and optical properties. The 314 MAX phase Hf3PB<sup>4</sup> has been thoroughly studied using Density Functional Theory (DFT) and revealed that it was the hardest MAX phase compound discovered until that date [21]. The structure of 212 phases exhibits a slight deviation from 211 MAX phases. In the case of 212 phases, a 2D layer of B is situated between M layers, featuring an additional B atom at the X position, unlike in 211 phases [22] [23], [24]. The B-B bonding in 212 MAX phase borides has improved mechanical and thermal properties. To date, the physical properties of Zr2AB<sup>2</sup> (A = In, Tl) [25] and Hf2AB<sup>2</sup> (A = In, Sn) [26], M2AB (M= Ti, Zr, Hf; A=Al, Ga, In) [27] MAX phases, as well as Nb2AC (A = Ga, Ge, Tl, Zn, P, In, and Cd) MAX phases, have been investigated using density functional theory (DFT). In each instance, the mechanical properties of B-containing compounds show significant improvement compared to their traditional C/N containing 211 MAX phases. The Debye temperature and melting temperature are higher for boron-containing 212 phases than for 211 carbides/nitrides, while the minimum thermal conductivity is lower. The thermal expansion coefficient of borides remains well-suited for use as coating materials. Consequently, the superior thermomechanical properties of B-containing 212 MAX phases demonstrate their suitability for high-temperature technological applications, surpassing the commonly used 211 MAX phase carbides. It should be noted that the 212 MAB phase with a hexagonal structure has already been synthesized [24]. In addition, Ga and Ge-based MAX phases have also been synthesized previously [28]. Thus, the reports on the synthesis of 212 phase and Ga and Gebased MAX phases motivated us to select the 212 MAB phases: M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) for our present study, and we have performed an in-depth investigation of their physical properties through DFT method. Therefore, in this paper, the first-time prediction of the stability and mechanical, electronic, thermal, and optical properties of M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) phases has been presented. The results revealed that M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) compounds are stable and suitable for thermal barrier coating (TBC) and reflection coating applications. Additionally, to provide a comparison, the properties determined for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) are compared with those of other 212 and 211 compounds MAX phase borides. ### 2. Methods of calculations First principles density-functional theory (DFT) computations are conducted utilizing the Cambridge Serial Total Energy Package (CASTEP) module integrated within Materials Studio 2017 [29], [30]. The exchange-correlation function is estimated using the Generalized Gradient Approximation (GGA) method, originally suggested by Perdew, Burke, and Ernzerhof [31]. Pseudo-atomic simulations accounted for electronic orbitals corresponding to B (2s 2 2p 1 ), Ga (3d <sup>10</sup> 4s 2 4p 1 ), Ge (4s 2 4p 2 ), Mo (4s 2 4p 6 4d 5 5s 1 ), and Ta (5d 3 6s 2 ). The energy cutoff and k-point grids were established at 650 eV and 11 × 11 × 4, respectively. The structural relaxation was performed utilizing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique [32], while the electronic structure was computed employing density mixing. The parameters for relaxed structures incorporate the following tolerance thresholds: the self-consistent convergence of the total energy is set at 5 × 10-6 eV/atom, the maximum force exerted on the atom is limited to 0.01 eV/Å, the maximum ionic displacement is constrained to 5 × 10-4 Å, and a maximum stress threshold of 0.02 GPa is imposed. The finite strain method [33], grounded in density functional theory (DFT), is utilized to compute the elastic properties within this framework. All necessary equations for determining various properties are provided in the supplementary document. #### 3 Results and discussion #### 3.1 Structural properties The M2AB<sup>2</sup> compounds (where M=Mo or Ta; A=Ga or Ge) belong to the P6m<sup>2</sup> (No. 187)[23] space group and crystallize in the hexagonal system. Unlike conventional MAX phases, which typically belong to the P63/mmc (194) space group, the 212 MAX phases exhibit distinct characteristics. In Fig. 1, the unit cell structure of Mo2GaB<sup>2</sup> is depicted as a representative of M2AB<sup>2</sup> alongside Mo2GaB, facilitating a comparison to discern their differences easily. The atomic positions are as follows: M (Mo or Ta) at (0.3333, 0.6667, 0.6935), A at (0.6667, 0.3333, 0.0), and two B atoms positioned at (0.6667, 0.3333, 0.5) and (0.0, 0.0, 0.5). The B components are arranged at the corners of the unit cell in typical 211 MAX phases, but in 212 boride MAX phases, they form a 2D layer between the M layers. This structural arrangement results in B-B covalent bonds in the 2D layer, enhancing stability compared to conventional 211 MAX phases. ![Fig. 1 - The schematic unit cell of (a) Mo2GaB<sup>2</sup> and (b) Mo2GaB compound.](path) The calculated lattice parameters of M2AB<sup>2</sup> are presented in Table 1 alongside those of other 211 and 212 MAX phases for comparison, demonstrating consistency with prior results [16], [34], [35] and affirming the accuracy of the computational methodology used. The primary distinction between 212- and 211-unit cell structures arises from differences in the lattice parameter c, where the c value for 211 exceeds that of 212. Moreover, the volumes of Mo- and Ta-based 211 MAX phase borides surpass those of 212 MAX phase borides. #### 3.2 Stability Examining a compound's stability is significant for multiple purposes, as it yields valuable information regarding the compound's synthesis parameters and aids in assessing the material's resilience across diverse environments, including thermal, compressive, and mechanical pressures. In this section, we delve into a comprehensive theoretical analysis concerning the chemical, dynamic, and mechanical stability of M2AB<sup>2</sup> compounds. The compound's chemical stability is determined by computing its formation energy by the following equation [36]: 2<sup>2</sup> = 22−( + + ) ++ . In the context provided, 22 represents the total energy of the compound after optimizing the unit cell. , , and denote the energies of the individual elements M, A, and B, respectively. The variables x, y, and z correspond to the number of atoms in the unit cell for M, Ga, and B, respectively. For Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, the calculated formation energy (Ef) are -1.8079 eV/atom, -1.8859 eV/atom, -2.1252 eV/atom, and -2.1933 eV/atom, respectively. The negative values signify the chemical stability of all compounds. The order of chemical stability can be expressed as Ta2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Mo2GaB2, indicating that Ta2GeB<sup>2</sup> is the most stable. Additionally, it's observed that MAX phase borides containing Ta are more stable than those containing Mo. Negative formation energy alone may not fully explain the chemical stability of M2AB2 (M=Mo, Ta and A=Ga, Ge). We calculated its formation enthalpy by examining potential pathways to evaluate its thermodynamic stability. For this analysis, we used the experimentally identified stable phases of MoB[37] and TaB[38], Ga4Mo[39], and B2Mo[40]. The potential decomposition pathways for our compounds, as determined from the Open Quantum Materials Database (OQMD), are outlined below. ![We have calculated the reaction energy as follows [41]:](path) Where, M=Mo, Ta and A=Ga, Ge. The following formula can calculate the decomposition energy associated with the reaction energy. = . Where n is the number of participating atoms. The calculated decomposition energies for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> are -0.16, -0.17, -0.26, and -0.28 meV/atom, respectively. So, we can state that the M2AB2 (M=Mo, Ta and A=Ga, Ge) system exhibits thermodynamic stability. Phonon dispersion curves (PDCs) have been calculated at the ground state utilizing the density functional perturbation theory (DFPT) linear-response approach to evaluate the dynamic stability of the MAX phase borides under investigation [42]. The PDCs, depicting the phonon dispersion along the high symmetry directions of the crystal Brillouin zone (BZ), along with the total phonon density of states (PHDOS) of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, are illustrated in Fig. 2(a, b, c and d). Analysis of the PDCs reveals no negative phonon frequencies for any of the compounds, indicating their dynamic stability. The PHDOS of the M2AB<sup>2</sup> compounds are derived from the PDCs and are presented alongside the PDCs in Fig. 2(a, b, c, and d), facilitating band identification through comparison of corresponding peaks. From Fig. 2, it is observed that in Mo2GaB2, the flatness of the bands for the Transverse Optical (TO) modes results in a prominent peak in the PHDOS, whereas non-flat bands for the Longitudinal Optical (LO) modes lead to weaker peaks in the PHDOS. Similar trends are observed in Mo2GeB2, Ta2GaB2, and Ta2GeB2. Notably, a distinct discrepancy arises between the optical and acoustic branches, with the top of the LO and bottom of the TO modes situated at the G point, with separations of 7.49 THz, 6.39 THz, 9.39 THz, and 8.51 THz for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, respectively. ![Fig. 2 – Phonon DOS and Phonon dispersion curves of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) Materials are subjected to various forces and loads in practical applications, necessitating understanding their mechanical stability. The mechanical stability of a compound can be assessed using stiffness constants. For a hexagonal system, the conditions for mechanical stability are as follows [43]: C<sup>11</sup> > 0, C<sup>11</sup> > C12, C<sup>44</sup> > 0, and (C<sup>11</sup> + C12)C<sup>33</sup> - 2(C13)² > 0. As indicated in Table 2, the Cij values of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) satisfy these conditions, thus confirming the mechanical stability of herein predicted phases: M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge). #### 3.3 Electronic properties Analyzing the electronic band structure (EBS) is crucial for gaining insights into the electronic behavior of a compound. The EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) MAX phases are depicted in Fig. 3(a, b, c, and d), with the Fermi energy (EF) level set at 0 eV, represented by a horizontal line. Observing the EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge), it is evident that the conduction band overlaps with the valence band, indicating the absence of a band gap. This observation confirms that the M2AB<sup>2</sup> compounds exhibit metallic behavior, which aligns with conventional MAX phases. The red lines illustrate the overlapping band at the Fermi level. Fig. 3 (a, b) shows that for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> compounds, the maximum band overlap occurs along the A-H path. Conversely, in Fig. 3(c, d), the maximum band overlap is observed along the G-M paths. Utilizing the band structure, we can analyze the electrical anisotropy of M2AB<sup>2</sup> MAX phase compounds. The anisotropic nature can be understood by analyzing the energy dispersion in the basal plane and along the c-axis. The paths G-A, H-K, and M-L show energy dispersion along the c-direction, and A-H, K-G, G-M, and L-H show energy dispersion in the basal plane. In comparison to the paths A-H, K-G, G-M, and L-H (basal plane), there is less energy dispersion along the lines G-A, H–K, and M-L (c-direction), as shown by Fig. 2(a, b, c and d). Lower energy dispersion in the c-direction results from a higher effective mass [44], indicating the strong electronic anisotropy of the M2AB<sup>2</sup> MAX phase compound. Consequently, conductivity along the c-axis is expected to be lower than in the basal planes. These findings are consistent with prior studies [34], [45]. ![Fig. 3 – Band structure of (a) Mo2GaB2, (b) Mo2GeB2, (c) Ta2GaB2, and (d) Ta2GeB<sup>2</sup> compounds.](path) To investigate the bonding nature and electronic conductivity, the total density of states (TDOS) and partial density of states (PDOS) of M2AB<sup>2</sup> compounds were computed. Figure 4 (a, b, c, and d) illustrates the TDOS and PDOS of these compounds, with the Fermi energy (EF) set at zero energy level, indicated by a straight line. These profiles exhibit typical characteristics of MAX phase materials. The Mo or Ta-d electronic states predominantly contribute to the Fermi level, with a minor contribution from the B-p and Ga or Ge-p electronic states. To ascertain the hybridization characteristics of various electronic states within the valence band, the energy spectrum of the valence band has been partitioned into two distinct segments. The first segment encompasses the lower valence band region spanning from -7 eV to -3.5 eV, originating from the hybridization of Mo-p, Mo-d, and B-s orbitals in the case of the Mo2GaB<sup>2</sup> compound. Conversely, for the Mo2GeB2 compound, the lower valence band region arises from the hybridization of Mo-d, Ge-p, and B-s orbitals. Notably, for both the Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compounds, the dominance of the B-p state characterizes the lower valence band region. The second segment pertains to the upper valence band region from -3.5 eV to 0 eV. In the case of the Mo2GaB<sup>2</sup> compound, this region arises from the hybridization of Mo-p and Mo-d orbitals. However, for the Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, the upper valence band region results from the hybridization of Mo-d orbitals and (Ga/Ge)-p orbitals. Notably, the Fermi level of M2AB<sup>2</sup> resides near the pseudogap in the TDOS profile, indicating a high level of electronic stability. Similar trends are observed in other compounds like Zr2AlN, V2AlN, Sc2AlB, Sc2GaB, Ta2GaB, and Hf2GaB<sup>2</sup> [15], [35], [45]. ![Fig. 4 – TDOS and PDOS of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) The electron charge density mapping is helpful in understanding the distribution of electron densities linked to chemical bonds. It delineates areas of positive and negative charge densities, signifying the development and exhaustion of electrical charges, respectively. As depicted in the map, covalent bonds become apparent by accumulating charges between two atoms. Furthermore, the presence of ionic bonds can be inferred from a balance between negative and positive charges at specific atom positions[46]. The valence electronic CDM, denoted in units of eÅ-3 , for M2AB<sup>2</sup> (where M = Mo, Ta; A = Ga, Ge) is showcased in Fig. 5(a, b, c, and d) along the (110) crystallographic plane. The accompanying scale illustrates the intensity of electronic charge density, with red and blue colors indicating low and high electronic charge density, respectively. As depicted in Fig. 5(a, b, c, and d), it is evident that charges accumulate in the regions between the B sites. Consequently, it is anticipated that strong covalent B‒B bonding occurs through the formation of two center-two electron (2c‒2e) bonds in the M2AB<sup>2</sup> (where M=Mo, Ta; A=Ga, Ge) compound, similar to other 212 MAX phase borides like Ti2PB2, Zr2PbB2, Nb2SB2, Zr2GaB<sup>2</sup> and Hf2GaB<sup>2</sup> [34], [45]. Mulliken analysis has corroborated the charge transfer from Mo/Ta atoms to B atoms. The charge received from Mo/Ta atoms is distributed among the B atoms positioned at the transitions and those located at the edges, facilitating the formation of a two center-two electron (2c‒2e) bond between B atoms within the 2D layer of B, as illustrated in Fig. 1(a). The hardness of each bond value presented in Table 5 also aligns with the results obtained from charge density mapping (CDM) and our analysis using moduli and elastic stiffness constants. ![Fig. 5 – The charge density mapping of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) ## 3.4.1 Mechanical properties The mechanical properties of materials play a pivotal role in determining their potential applications, serving as crucial indicators of their behavior and suitability in materials engineering endeavors. These properties are equally applicable to MAX phase materials. Initially, to assess the mechanical properties of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phases, we employed the stress-strain method within the CASTEP code to compute the elastic constants (Cij) [22], [47], [48]. These calculated elastic constants (Cij) are presented in Table 2, alongside those of other 212 and 211 MAX phases. Due to the hexagonal crystal structure of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phase borides, five stiffness constants emerge: C11, C12, C13, C33, and C<sup>44</sup> [49]. The mechanical stability was evaluated using these stiffness constants in the preceding section. For instance, C<sup>11</sup> and C<sup>33</sup> determine stiffness when stress is exerted along the (100) and (001) directions, respectively, whereas C<sup>44</sup> evaluates resistance to shear deformation on the (100) and (001) planes. The stiffness constants C<sup>11</sup> and C<sup>12</sup> directly reflect the strength of atomic bonds along the a- and c-axes. When C<sup>11</sup> exceeds C<sup>33</sup> (or vice versa), it signifies stronger atomic bonding along the a-axis (or c-axis). In Table 2, for Mo2GaB2, Mo2GeB2, and Ta2GeB<sup>2</sup> compounds, C<sup>11</sup> surpasses C33, indicating superior atomic bonding along the a-axis compared to the c-axis. This robust bonding along the a-axis suggests heightened resistance to a-axial deformation. Conversely, in Ta2GeB2, where C<sup>11</sup> < C33, stronger atomic bonding along the c-axis translates to increased resistance against c-axial deformation. Analysis of Table-2 reveals that the values of C<sup>11</sup> and C<sup>22</sup> are notably higher for 212 MAX phase borides than 211 MAX phase borides. Consequently, it can be inferred that 212 MAX phases exhibit stronger resistance to axial deformation when juxtaposed with 211 MAX phases. C<sup>44</sup> is commonly utilized to gauge shear deformation tolerance among the elastic constants. Notably, Ta2GaB exhibits superior shear deformation resistance owing to its highest C<sup>44</sup> value among the studied compounds. Furthermore, the individual elastic constants C12, C13, and C<sup>44</sup> denote shear deformation response under external stress. The observation that C<sup>11</sup> and C<sup>33</sup> possess larger magnitudes than C<sup>44</sup> implies that shear deformation is more facile than axial strain. Another crucial parameter derived from the stiffness constants is the Cauchy pressure (CP), calculated as C<sup>12</sup> - C44, which provides vital insights relevant to the practical applications of solids [51]. Pettifor [51] emphasized the significance of Cauchy pressure (CP) in discerning the chemical bonding and ductile/brittle properties of solids. A negative CP value indicates covalently bonded brittle solids, whereas a positive value signifies isotropic ionic ductile solids. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> fall into covalently bonded brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit metallic ductile characteristics. Similarly, like Ga-containing 211 MAX phase borides, Ga-containing 212 MAX phases also demonstrate negative CP values and brittle behavior [52][53]. The elastic constants obtained are used to produce several bulk elastic parameters that are used to characterize polycrystalline materials, such as Young's modulus (Y), bulk modulus (B), and shear modulus (G). In Table 2, the bulk modulus (B) and shear modulus (G) calculated using Hill's approximation [54] are also presented. Hill's values represent the average of the upper limit (Voigt [55]) and lower limit (Reuss [56]) of B. The necessary equations for these calculations are provided in the supplementary document (S1). Young's modulus (Y) is a crucial indicator of material stiffness. A higher Y value indicates a stiffer material. It can be observed from the table that Mo2GaB<sup>2</sup> exhibits the highest Y value, signifying its greater stiffness compared to others. According to the sequence of Y values, stiffness can be ranked as follows: Mo2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Ta2GeB2. Therefore, Mo2GaB2, with its higher Young's modulus, is anticipated to demonstrate superior mechanical stability and deformation resistance compared to the other compounds under investigation. This quality is essential for aircraft parts or high-performance machinery applications where critical dimensional stability and structural integrity are required [57], [58]. Table 2 shows that 212 MAX phase materials exhibit larger Young's modulus (Y) values than 211 MAX phases. This suggests that 212 MAX phase borides are stiffer than their 211 MAX phase counterparts. Young's modulus (E) also correlates well with thermal shock resistance (R): R ∝ 1/E [59]. Lower Young's modulus values correspond to higher thermal shock resistance. Therefore, materials with higher thermal shock resistance (i.e., lower Young's modulus) are more suitable for use as Thermal Barrier Coating (TBC) materials. Given that Ta2GaB<sup>2</sup> possesses the lowest Y value among the materials studied, it should be considered a superior candidate for TBC material due to its higher thermal shock resistance. The material's ability to withstand shape distortion is elucidated by the shear modulus (G). On the other hand, the bulk modulus (B) indicates the strength of a material's chemical bonds and its ability to withstand uniform compression or volume change. We computed Young's modulus (Y), bulk modulus (B), and shear modulus for our analysis. Based on the bulk modulus values presented in Table 2, the sequence of a material that resists compression when pressure is applied can be outlined as follows: Mo2GeB<sup>2</sup> > Ta2GeB<sup>2</sup> > Mo2GaB<sup>2</sup> > Ta2GeB2. Mo2GeB2, boasting a higher bulk modulus, may exhibit reduced plastic deformation and superior resistance to stress-induced deformation compared to other compounds examined. The ductile and brittle characteristics of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds can also be assessed through Poisson's ratio (ν) and Pugh's ratio (G/B). A compound demonstrates ductile (brittle) behavior if the ν value surpasses (falls below) 0.26 [60]. Furthermore, if the G/B value exceeds (is less than) 0.571, then the compound exhibits brittle (ductile) behavior [19]. According to both criteria, Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are classified as brittle compounds, whereas Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> are categorized as ductile compounds. The ductile nature of Ge-based MAX phases has been reported previously [61]. Fracture toughness (KIC) is a vital property that gauges a material's ability to resist crack propagation. In the case of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, Equation (S2) was employed to determine KIC. The KIC values are documented in Table 2, and notably, these values surpass those of 211 MAX phases [16], [35]. Another parameter, the "f-index," characterizes the isotropic nature and strength of atom-atom bonds within a single hexagonal crystalline lattice along the a- and c-directions. If the f-index is less than 1, chemical bonds exhibit greater rigidity along the c-axis; conversely, if f exceeds 1, bonds are more rigid in the ab-plane. When the value of f is set to 1, atomic bonds exhibit similar strength and uniformity in all directions [15]. The fvalue is computed using Equation (S3) and displayed in Table 2. Table 2 shows that the f-values of Mo2GeB2, Ta2GaB2, and Ta2GeB2, which are close to one, indicate a slight anisotropic bonding strength. Nevertheless, substances with robust bonds in the horizontal plane (ab plane) (f > 1), such as Mo2GaB2, are deemed optimal candidates for the exfoliation process. In engineering applications, the hardness of a solid material serves as a valuable criterion for designing various devices. The elastic properties of polycrystalline materials can be utilized to calculate hardness values, as the ability to resist indentation is closely linked to a material's hardness. Both micro-hardness (Hmicro) and macro-hardness (Hmacro) were computed using Equation (S4) and are presented in Table 2. Based on the values of Hmicro and Hmacro, Mo2GaB<sup>2</sup> emerges as the toughest among the studied phases. The order of hardness is as follows: Mo2GaB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GeB2. Interestingly, Ga-containing MAX phase borides exhibit superior hardness compared to Ge-containing MAX phase borides. In Table 2, we compare our findings with previously reported MAX phases and observe that our data align perfectly with the earlier results. #### 3.4.2 Elastic anisotropy Additionally, anisotropy is linked to other crucial events like anisotropic plastic deformation and the development and spread of micro-cracks within mechanical stress. By providing directiondependent elastic constants, the understanding of anisotropy also offers a framework for improving the mechanical stability of materials in extreme circumstances. The following formulae are used for hexagonal structures to calculate the various anisotropic variables from elastic constants Cij [62]. $A\_{1} = \frac{\frac{1}{6}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{44}};$ $A\_{2} = \frac{2\mathcal{C}\_{44}}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ and $A\_{3} = A\_{1}.$ $A\_{2} = \frac{\frac{1}{3}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ Table 3 lists every anisotropy parameter. Since the value of A<sup>i</sup> should be 1 to be isotropic, the computed value of A<sup>i</sup> (i = 1-3) indicates that all of the compounds under research exhibit anisotropic behavior [63]. An additional method for estimating elastic anisotropy is to use the percentage anisotropy to compressibility and shear (A<sup>B</sup> & AG). This gives polycrystalline materials a helpful way to measure elastic anisotropy. They have been described as [45]; Zero values for AB, AG, and the universal anisotropy factor (AU) show elastic isotropy; the maximum amount of anisotropy is represented by a value of 1. Table 3 shows that, compared to other compounds, the values of AB, AG, and A U for Mo2GeB<sup>2</sup> and Ta2GaB<sup>2</sup> are incredibly close to zero, suggesting that these compounds have nearly isotropic characteristics. #### 3.4.3 Mulliken Populations The Mulliken charge assigned to an atomic species quantifies the effective valence by calculating the absolute difference between the formal ionic charge and the Mulliken charge. Equations (S5) and (S6) are employed to ascertain the Mulliken charge for each atom (α). Table-4 provides the Mulliken atomic population and effective valence charge. Transition metals Mo and Ta in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have pure valence states of 4d<sup>5</sup> and 5d<sup>3</sup> , respectively. The d- orbital electrons of transition metals have been found to influence their effective valence charge significantly. A non-zero positive value indicates a combination of covalent and ionic attributes within chemical bonds. As this value decreases towards zero, it signifies a rise in ionicity. A zero value suggests an ideal ionic character in the bond. Conversely, a progression from zero with a positive value indicates an elevation in the covalency level of the bonds. Based on their effective valence, M atoms move from the left to the right in the periodic table, increasing the covalency of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). Table-4 reveals that the Mulliken atomic charge ascribed to the B atoms is solely negative. Conversely, positive Mulliken atomic charges are associated with transition metals (M) and A. This suggests a charge transfer from M and A to B for each compound within M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge), thereby fostering ionic chemical bonds among these atoms. Bond population serves as another indicator of bond covalency within a crystal, as a high value of bond population essentially signifies a heightened degree of covalency within the chemical bond. The bonding and anti-bonding states influence the populations with positive and negative bond overlap. As demonstrated in Table-5, the B-B bond exhibits greater covalency compared to any other bond in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The presence of an antibonding state between two relevant atoms, which subsequently decreases their chemical bonding, is indicated by a hostile bond overlap population. In Mo2GeB<sup>2</sup> and Ta2GeB2, a hostile bond overlap population is observed in the Ge-Mo and Ge-Ta bonds, indicating the presence of an antibonding state. Therefore, the existence of ionic bonding is guaranteed by electronic charge transfer. On the other hand, the high positive value of the bond overlap population (BOP) denotes the presence of covalent bonding, a feature shared by materials in the MAX phase. ## 3.4.4 Theoretical Vickers Hardness Vickers hardness, derived from the atomic bonds found in solids, indicates how resistant a material is to deformation in extreme circumstances. Several variables influence this feature, such as the crystal flaws, solid structure, atomic arrangement, and bond strength. The Vickers hardness of the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) MAX phases is determined using the Mulliken bond population method, as described by Gou et al. [66] using the formula (S7-S10). This method is particularly suitable for partial metallic systems such as MAX phases. Table-5 lists the computed Vickers hardness values for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The calculated values are 3.35 GPa, 4.76 GPa, 8.21 GPa, and 8.74 GPa for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, respectively. We observed that Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> have much higher hardness values than Mo2GaB<sup>2</sup> and Mo2GeB2. These values are also higher than that of other 212 phases, like, Ti2InB<sup>2</sup> (4.05 GPa) [50], Hf2InB<sup>2</sup> (3.94 GPa), and Hf2SnB<sup>2</sup> (4.41 GPa) [26], Zr2InB<sup>2</sup> (2.92 GPa) and Zr2TlB<sup>2</sup> (2.19 GPa) [25], Zr2GaB<sup>2</sup> (2.53GPa), Zr2GeB<sup>2</sup> (3.31GPa), Hf2GaB<sup>2</sup> (4.73GPa) and Hf2GeB2(4.83GPa) [67]The Hv calculated by the geometrical average of the individual bonding, where the bonding strength mainly determined by the BOP values. In the case of Ta2GaB<sup>2</sup> and Ta2GeB2, the BOP of M-B bonding is much higher compared to Mo2GaB<sup>2</sup> and Mo2GeB2. Even though the BOP of M-B for Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> is higher than that of the other 212 phases mentioned earlier. Therefore, higher values of Hv are expected for Ta2GaB<sup>2</sup> and Ta2GeB2. Additionally, we looked at the Vickers hardness value between 211 and 212 compounds and discovered that the 212 MAX phase compounds had a higher H<sup>V</sup> value. This is because a 2D layer of B atoms is positioned between the M atoms. The B atoms share two center-two electrons to form an extremely strong B-B bond [3]. Table 5 Calculated data for Mulliken bond number (n μ ), bond length (d μ ), bond overlap populations BOP, (P μ ), metallic populations (P μˊ), Vickers hardness (HV) M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. # 3.5 Thermal Properties MAX phases are ideal for high-temperature applications due to their exceptional mechanical qualities at elevated temperatures. As a result, researching the fundamental parameters necessary for predicting their application is of great interest, and these can be obtained from the vibrations of atoms or phonons. The Debye temperature (ΘD) of a solid is directly connected to its bonding strength, melting temperature, thermal expansion, and conductivity. Using the sound velocity and Anderson's technique [68], the Θ<sup>D</sup> of the phases under study has been computed using the formula (S11). Equation (S12) can be used to get the average sound velocity (Vm) from the longitudinal and transverse sound velocities. Equations (S13–14) were used to determine v<sup>l</sup> and vt. The calculated values of Debye's temperature are shown in Table 6, where Mo2GaB<sup>2</sup> has the highest Θ<sup>D</sup> and Ta2GeB<sup>2</sup> has the lowest. If we rank them, it is as follows: Mo2GaB<sup>2</sup> < Mo2GeB<sup>2</sup> < Ta2GaB2 < Ta2GeB2. Hadi et al. recently reported a MAX phase (V2SnC) as a TBC material with a Θ<sup>D</sup> value of 472 K [69]. Thus, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) exhibit encouraging potential as TBC materials, as shown in Table 6. Table 6-Data for density (ρ), longitudinal, transverse, and average sound velocities (vl, vt, and vm), Debye temperature (ΘD), minimum thermal conductivity (Kmin), Grüneisen parameter (γ), thermal expansion coefficient (TEC) at 300K and melting temperature (Tm) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The constant thermal conductivity value at high temperatures is the minimum thermal conductivity (Kmin). As the name implies, this conductivity is minimal because, at high temperatures, phonon coupling breaks. The formula (S15) for the minimum thermal conductivity of solids was derived using the Clarke model [70]. Table 6 lists the calculated value of kmin, with Ta2GaB<sup>2</sup> having the lowest value and Mo2GaB<sup>2</sup> having the highest. When selecting suitable materials for TBC applications, a minimum thermal conductivity of 1.25 W/mK is used as a screening criterion [71]. Our compounds exhibit lower minimum thermal conductivity values, holding promising potential as TBC materials. Gd2Zr2O<sup>7</sup> and Y2SiO5, two recently developed thermal barrier coating (TBC) materials, have minimal thermal conductivities (Kmin) of 1.22 W/m.K. and 1.3 W/m.K. [72], respectively, as confirmed by experiment. These numbers roughly match the values we computed for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The Grüneisen parameter (γ) is a crucial thermal parameter that helps explain the anharmonic effects of lattice dynamics; solids utilized at high temperatures are expected to have lower anharmonic effects. The Grüneisen parameter (γ) can be determined with the help of Poisson's ratio using equation (S16) [73]. The computed γ values, as shown in Table 6, suggest that the compounds under investigation exhibit a weak anharmonic effect. Additionally, for solids with a Poisson's ratio between 0.05 and 0.46, the values similarly fall within the range of 0.85 and 3.53 [74]. The melting temperature (Tm) of the compounds under investigation has been calculated using the following formula (S17). The strength of atomic bonding is the primary factor determining the melting temperature of solids; the higher the Tm, the stronger the atomic bonding. The order of T<sup>m</sup> for the titled phases is found to mirror the Y-based (Young's modulus) order, indicating a close link between T<sup>m</sup> and Y [75]. As observed in Table 6, our compounds roughly follow the Ybased ranking with Mo2GeB<sup>2</sup> < Mo2GaB<sup>2</sup> < Ta2GeB<sup>2</sup> < Ta2GaB2. The T<sup>m</sup> value for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) is also comparable with the TBC material Y4Al2O<sup>9</sup> (2000 K)[75]. ![Fig. 6: Temperature dependence of the calculated thermodynamic parameters. (a) Specific heat at constant pressure, C<sup>p</sup> (b) Specific heat at constant volume, C<sup>v</sup> (c) Free energy, F (d) Entropy, S (e) Internal energy, E and (f) Thermal expansion coefficient, TEC of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge).](path) The temperature dependence of specific heats, Cv, Cp, for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds was obtained using the formulas (S19-S20), as shown in Fig.6(a,b). Assuming that the quasi-harmonic model is accurate and that phase transitions are not anticipated for the compounds under study, these properties are approximated across a temperature range of 0 to 1000 K. Because phonon thermal softening occurs at higher temperatures, the heat capacity rises with temperature. Heat capacities increase quickly and follow the Debye-T 3 power law at lower temperatures. At higher temperature regimes, where C<sup>v</sup> and C<sup>p</sup> do not greatly depend on temperature, they approach the Dulong-Petit (3) limit.[76]. Using the quasi-harmonic approximation, various temperature-dependent thermodynamic potential functions for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have been estimated at zero pressure and displayed in Fig.6 [77]. These functions include the Helmholtz free energy (F), internal energy (E), and entropy (S) within 0-1000 K temperature using equation (S21-S23). The free energy progressively decreases as the temperature rises, as shown in Fig.6(c). Free energy typically declines, and this trend becomes increasingly negative as a natural process proceeds. As demonstrated in Fig. 6(d), the internal energy (E) shows a rising trend with temperature, in contrast to the free energy. Since thermal agitation creates disorder, a system's entropy rises as temperature rises. This is illustrated in Fig.6(d). A material's thermal expansion coefficient (TEC) is derived from the anharmonicity in the lattice dynamics and can be found using equation (S18). The measure of a material's capacity to expand or contract with heat or cold is called the Thermal Expansion Coefficient, or TEC. As observed in Fig. 5(f), the Thermal Expansion Coefficient (TEC) increases rapidly up to 365 K. Then it approaches a constant value, which indicates lower saturations in the materials with temperature changes. The materials under study have a very low TEC value, a crucial characteristic of materials intended for application in high-temperature technology. To be effective as thermal barrier coating (TBC) materials, compounds must exhibit a low thermal conductivity (Kmin) to impede heat transfer, a high melting temperature to withstand extreme heat, and a low thermal expansion coefficient (TEC) to maintain dimensional stability under thermal stress. The compounds M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge) possess these properties, making them suitable candidates for use as TBC materials #### 3.6 Optical Properties Different materials exhibit unique behaviors when exposed to electromagnetic radiation. The optical constants determine the overall response of the sample to the incident radiation. The complex dielectric function, defined as ε(ω)=ε1(ω)+iε2(ω), is one of the main optical characteristics of solids. The following formula determines the imaginary part of the dielectric function ε2(ω) from the momentum matrix element between the occupied and unoccupied electronic states. In this formula, e stands for an electronic charge, ω for light angular frequency, u for the polarization vector of the incident electric field, and and for the conduction and valence band wave functions, respectively, at k. The Kramers-Kronig equation can estimate the real part of the dielectric function, ε1(ω). In contrast, ε2(ω) and ε1(ω) are utilized to evaluate all other optical parameters, such as the absorption coefficient, photoconductivity, reflectivity, and loss function [78]. In this part, several energy-dependent optical properties of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) 212 MAX phases are calculated and analyzed in detail for the photon energy range of 0 to 30 eV, for [100] plane enabling the first assessment of the compounds' practical applicability. Given the metallic conductivity of the MAX compounds' electronic structure, additional parameters were chosen to analyze the optical properties. These include a plasma frequency of 3 eV, damping of 0.05 eV, and Gaussian smearing of 0.5 eV [79]. Figure 7(a) shows the real component of the dielectric function, ε1, which exhibits metallic behavior. In metallic systems, ε<sup>1</sup> has a considerably high negative value in the low-energy range, with the real component reaching negative, which aligns with the band structure finding. Fig. 7(b) depicts the imaginary part of the dielectric function, ε2(ω), representing dielectric losses about frequency. Mo2GeB<sup>2</sup> demonstrates the highest peak in the low-energy region, with all compounds approaching zero from above at around 17 eV. This observation confirms Drude's behavior. ![Fig. 7- (a) Real part of dielectric constant, ε<sup>1</sup> (b) Imaginary part of dielectric constant, ε<sup>2</sup> (c) Reflectivity, R and (d) Loss function, (e) The coefficient of absorption, (α) (f) Refractive index, (n) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) for [100] electric field directions.](path) The reflectivity of a material measures the percentage of incident light energy reflected off it. Equation (S24) was employed to compute reflectivity using the dielectric function, as shown in Fig.7(c). The spectra's visible and infrared (IR) portions consistently exhibit values exceeding 55%. Consequently, the materials under examination are expected to appear metallic gray. In the infrared (IR) region, the maximum reflectivity for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> is 65% and 74%, respectively, occurring at 0.33 eV. Conversely, for Ta2GaB2, the maximum reflectivity is 59% at 14.02 eV, while for Ta2GeB2, it is 65% at 14.78 eV in the UV region. After 18 eV, the reflectance decreases significantly. According to reports, substances with an average reflectivity value above 44% in the visible light region can effectively reduce solar heating by reflecting a significant amount of the solar spectrum. Among all the compounds we examined, Ta2GaB<sup>2</sup> has a reflectivity in the visible range of more than 44% [78]. Therefore, Ta2GaB<sup>2</sup> can be utilized as a coating material and should be able to mitigate solar heating. plasma frequencies (ωp). The plasma frequency for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is observed at 23 eV, 25 eV, 21 eV, and 23 eV, respectively. At this specific frequency, the absorption coefficient rapidly decreases, ε<sup>1</sup> crosses zero from the negative side, and the reflectance R(ω) displays a falling tail. Above this distinctive frequency, the materials are transparent to the incident electromagnetic radiation. Fig. 7(e) illustrates the absorbance coefficient (α) of compounds M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) determined using equation (S26). As α begins to rise from zero photon energy, the metallic nature of the substances under study is again indicated. The visible light region experiences a sharp increase in absorption, peaking in the UV region at around 14 eV and then progressively declining. The IR region exhibits negligible absorption. However, the materials mentioned above appear to have a significant absorption band primarily located in the visible and ultraviolet spectrum. This suggests that the materials can be used in UV surface-disinfection devices, medical sterilization equipment, and other optoelectronic device designs. Its refractive index is crucial to a material's potential application in optical devices like waveguides and photonic crystals. Equations (S27) and (S28) were used to derive the refractive index (n) and extinction coefficient (k) for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), as depicted in Fig.7(c) and Fig.S-1(a). The variations of n and k in MAX phase carbides with incident photon energy closely resemble ε1(ω) and ε2(ω). The static refractive index n(0) for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is 9.4,13.07,7 and 9.2, respectively, and decreases gradually with the increase in photon energy. The extinction coefficient for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2 gradually increases in the IR region, reaching their maximum values of 2.56, 4.27, 2.42, and 3.28, respectively. Following this, they slowly decrease in the visible and UV regions. Figure S-1(b) illustrates the photoconductivity (σ) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) across various photon energies. The photoconductivity (σ) parameter quantifies the impact of photon irradiation on a material's electrical conductivity. Similar to the absorbance coefficient (α) spectrum, the σ spectrum aligns with the metallic nature of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). #### 4 Conclusions In summary, we employed DFT calculations to explore four 212 MAB phases, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), and investigated the structural, electronic, mechanical, lattice dynamical, and optical properties to predict their possible applications. The phonon dispersion curves, formation energy, and elastic constants collectively suggest that the M2AB<sup>2</sup> boride maintains dynamic, mechanical, chemical, and thermodynamic stability. The electronic band structure and density of states (DOS) offer evidence supporting the metallic nature of the compounds under investigation. Concurrently, the charge density mapping and atomic Mulliken population both confirm the presence of a strong B-B covalent bond. The stiffness constants, elastic moduli, findex, fracture toughness (KIC), Pugh's ratio (G/B), hardness parameters, and Cauchy Pressure (CP) of M2AB<sup>2</sup> were computed and compared with those of their 211 equivalents. We found that the values of the 212-phase borides are higher than those of the 211-phase carbides or borides. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are identified as brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit ductile characteristics, as indicated by Poisson's ratio (ν) and Pugh's ratio (B/G or G/B). The elastic characteristics display anisotropy due to the distinct atomic configurations along the aand c-directions. Vickers hardness calculations are considered reliable indicators of material hardness. The results suggest that Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> possess more pliable characteristics than Ta2GaB<sup>2</sup> and Ta2GeB2. The high hardness values of Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compared to the Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> due to higher BOP values of M-B bonding in Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> than in Mo2GaB<sup>2</sup> and Mo2GeB2. The value of minimum thermal conductivity (Kmin), thermal expansion coefficient, and melting temperature (Tm) collectively suggest the potential suitability of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) as a material for thermal barrier coating (TBC) applications in high-temperature devices. The optical conductivity and absorption coefficient corroborate the findings of the electronic band structure. Reflectivity is notably high in infrared (IR) regions and remains nearly constant in the visible and moderate ultraviolet (UV) regions, with an average value exceeding 44% for Ta2GaB2. This suggests that Ta2GaB<sup>2</sup> can be effectively utilized as a coating material to reduce solar heating. We expect that the comprehensive analysis of the diverse physical characteristics of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) presented in this study will establish a robust foundation for future theoretical and experimental explorations of these fascinating MAB phases. ## CRediT author statement A. K. M Naim Ishtiaq and Md Nasir Uddin: Data curation, Writing- Original draft preparation. Md. Rasel Rana, Shariful Islam and Noor Afsary: Reviewing and Editing. Md. 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Fig. 5 – The charge density mapping of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.	# Exploration of new 212 MAB phases: M2AB2 (M=Mo, Ta; A=Ga, Ge) via DFT calculations #### Abstract The recently developed MAB phases, an extension of the MAX phase, have sparked interest in research among scientists because of their better thermo-mechanical properties. In this paper, we have explored four new MAB phases M2AB2 (M=Mo, Ta and A=Ga, Ge) and studied the elastic, electronic, thermal, and optical properties to predict the possible applications. The stability of the new phases has been confirmed by calculating formation energy (Ef), formation enthalpy (∆H), phonon dispersion curve (PDC), and elastic constant (Cij). The study reveals that M2AB<sup>2</sup> (M=Mo, Ta and A=Ga, Ge) exhibit significantly higher elastic constants, elastic moduli, and Vickers hardness values than their counterpart 211 borides. Higher Vickers hardness values of Ta2AB2 (A=Ga, Ge) than Mo2AB2 (A=Ga, Ge) have been explained based on the values of the bond overlap population. The analysis of the density of states and electronic band structure revealed the metallic nature of the borides under examination. The thermodynamic characteristics of M2AB2 (M=Mo, Ta and A=Ga, Ge) under high temperatures (0–1000 K) are investigated using the quasi-harmonic Debye model. Critical thermal properties such as melting temperature (Tm), Grüneisen parameter (γ), minimum thermal conductivity (Kmin), Debye temperature (ΘD), and others are also computed. Compared with 211 MAX phases, the 212 phases exhibit higher values of (ΘD) and Tm, along with a lower value of Kmin. These findings suggest that the studied compounds exhibit superior thermal properties that are suitable for practical applications. The optical characteristics have been examined, and the reflectance spectrum indicates that the materials have the potential to mitigate solar heating across various energy regions. #First two authors contributed equally. ### 1. Introduction The MAX phase has garnered significant attention in the present era due to its outstanding mechanical and thermal characteristics at high temperatures, showcasing attributes shared by both metals and ceramics. The increased interest in MAX phase materials can be traced back to Barsoum's noteworthy contributions [1], [2]. The term MAX phase represents a family of multilayer solids where M is an earlier transition metal, A is an element from the IIIA or IVA group of the periodic table, and X is an atom of C/N/B [3]. MAX phase materials showcase metallic behavior due to alternate metallic A-layers and ceramic behavior attributed to the MX layers [4]. Like most metals and alloys, MAX phase materials have excellent thermal shock resistance, superior machinability, and enhanced thermal and electrical conductivities. On the other hand, they have high melting or decomposition temperatures and strong elastic stiffness, similar to many ceramics[5]. The unique combination of metallic and ceramic properties makes them versatile, with applications ranging from high-temperature coatings to nuclear accidenttolerant fuel (ATF), concentrated solar power (CSP), catalysis, and as precursors for MXenes [6], [7], [8], [9]. The diversity of MAX phases was confined to C and N as X elements for years (from 1960 to 2014). However, recent advancements have overcome this limitation by successfully synthesizing MAX phases that contain B [10]. The physical and chemical properties of B and Bcontaining compounds highlight the potential of MAX phase borides, which replace C/N with boron [11]. Due to the presence of B in the composition, they are also called the MAB phase. Furthermore, scientific communities are actively working to broaden the diversity of MAX phases by introducing structural changes, as seen in examples like Cr3AlB<sup>4</sup> (space group Immm), Cr4AlB<sup>6</sup> (space group Cmmm), Cr4AlB<sup>4</sup> [12], [13], [14]. Khazaei et al. reported on the first investigation of the theoretical MAX phase borides M2AlB (where M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta). They evaluated these compounds' electronic structure, mechanical characteristics, and dynamical stability in their investigation [15]. The diversity of MAX phases is a subject of research interest due to their structure variations and the number of atoms in the compounds, resulting in changes to their characteristics. In examples like the 211 MAX phase [16], 312 MAX phase [8][17], 212 MAX phase [3], and i-MAX phase [18], the recent additions to the MAX family include the 314 MAX phase and 212 MAX phase, where the element B has been incorporated as an X element [3] [19]. This introduces a novel aspect to the MAX phase family, contributing to its diversity. The unique structural features of MAB phases include both orthorhombic and hexagonal symmetry observed in crystals. These distinctive symmetries set MAB phases apart from the typical MAX phases. Yinqiao Liu's synthesis of the orthorhombic phase of M2AlB2, particularly the 212-MAB phases with M = Sc, Ti, Zr, Hf, V, Nb, Cr, Mo, W, Mn, Tc, Fe, Co, and Ni, has revealed unique structural stability and notable electrical and mechanical properties [20]. The MAB phase structure slightly differs from the standard MAX phases, which typically crystallize in the hexagonal system with a space group -6<sup>2</sup> (No. 187). Ali et al [3]. investigated the diverse physical properties of Zr2AB<sup>2</sup> (A = In, Tl). Martin Ade et al [13] synthesized ternary borides, namely Cr2AlB2, Cr3AlB4, and Cr4AlB6, and subsequently compared their mechanical properties. Qureshi et al. [19] investigated the 314 Zr3CdB<sup>4</sup> MAX phase boride, calculating its mechanical, thermodynamic, and optical properties. The 314 MAX phase Hf3PB<sup>4</sup> has been thoroughly studied using Density Functional Theory (DFT) and revealed that it was the hardest MAX phase compound discovered until that date [21]. The structure of 212 phases exhibits a slight deviation from 211 MAX phases. In the case of 212 phases, a 2D layer of B is situated between M layers, featuring an additional B atom at the X position, unlike in 211 phases [22] [23], [24]. The B-B bonding in 212 MAX phase borides has improved mechanical and thermal properties. To date, the physical properties of Zr2AB<sup>2</sup> (A = In, Tl) [25] and Hf2AB<sup>2</sup> (A = In, Sn) [26], M2AB (M= Ti, Zr, Hf; A=Al, Ga, In) [27] MAX phases, as well as Nb2AC (A = Ga, Ge, Tl, Zn, P, In, and Cd) MAX phases, have been investigated using density functional theory (DFT). In each instance, the mechanical properties of B-containing compounds show significant improvement compared to their traditional C/N containing 211 MAX phases. The Debye temperature and melting temperature are higher for boron-containing 212 phases than for 211 carbides/nitrides, while the minimum thermal conductivity is lower. The thermal expansion coefficient of borides remains well-suited for use as coating materials. Consequently, the superior thermomechanical properties of B-containing 212 MAX phases demonstrate their suitability for high-temperature technological applications, surpassing the commonly used 211 MAX phase carbides. It should be noted that the 212 MAB phase with a hexagonal structure has already been synthesized [24]. In addition, Ga and Ge-based MAX phases have also been synthesized previously [28]. Thus, the reports on the synthesis of 212 phase and Ga and Gebased MAX phases motivated us to select the 212 MAB phases: M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) for our present study, and we have performed an in-depth investigation of their physical properties through DFT method. Therefore, in this paper, the first-time prediction of the stability and mechanical, electronic, thermal, and optical properties of M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) phases has been presented. The results revealed that M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) compounds are stable and suitable for thermal barrier coating (TBC) and reflection coating applications. Additionally, to provide a comparison, the properties determined for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) are compared with those of other 212 and 211 compounds MAX phase borides. ### 2. Methods of calculations First principles density-functional theory (DFT) computations are conducted utilizing the Cambridge Serial Total Energy Package (CASTEP) module integrated within Materials Studio 2017 [29], [30]. The exchange-correlation function is estimated using the Generalized Gradient Approximation (GGA) method, originally suggested by Perdew, Burke, and Ernzerhof [31]. Pseudo-atomic simulations accounted for electronic orbitals corresponding to B (2s 2 2p 1 ), Ga (3d <sup>10</sup> 4s 2 4p 1 ), Ge (4s 2 4p 2 ), Mo (4s 2 4p 6 4d 5 5s 1 ), and Ta (5d 3 6s 2 ). The energy cutoff and k-point grids were established at 650 eV and 11 × 11 × 4, respectively. The structural relaxation was performed utilizing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique [32], while the electronic structure was computed employing density mixing. The parameters for relaxed structures incorporate the following tolerance thresholds: the self-consistent convergence of the total energy is set at 5 × 10-6 eV/atom, the maximum force exerted on the atom is limited to 0.01 eV/Å, the maximum ionic displacement is constrained to 5 × 10-4 Å, and a maximum stress threshold of 0.02 GPa is imposed. The finite strain method [33], grounded in density functional theory (DFT), is utilized to compute the elastic properties within this framework. All necessary equations for determining various properties are provided in the supplementary document. #### 3 Results and discussion #### 3.1 Structural properties The M2AB<sup>2</sup> compounds (where M=Mo or Ta; A=Ga or Ge) belong to the P6m<sup>2</sup> (No. 187)[23] space group and crystallize in the hexagonal system. Unlike conventional MAX phases, which typically belong to the P63/mmc (194) space group, the 212 MAX phases exhibit distinct characteristics. In Fig. 1, the unit cell structure of Mo2GaB<sup>2</sup> is depicted as a representative of M2AB<sup>2</sup> alongside Mo2GaB, facilitating a comparison to discern their differences easily. The atomic positions are as follows: M (Mo or Ta) at (0.3333, 0.6667, 0.6935), A at (0.6667, 0.3333, 0.0), and two B atoms positioned at (0.6667, 0.3333, 0.5) and (0.0, 0.0, 0.5). The B components are arranged at the corners of the unit cell in typical 211 MAX phases, but in 212 boride MAX phases, they form a 2D layer between the M layers. This structural arrangement results in B-B covalent bonds in the 2D layer, enhancing stability compared to conventional 211 MAX phases. ## 3.4.1 Mechanical properties The mechanical properties of materials play a pivotal role in determining their potential applications, serving as crucial indicators of their behavior and suitability in materials engineering endeavors. These properties are equally applicable to MAX phase materials. Initially, to assess the mechanical properties of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phases, we employed the stress-strain method within the CASTEP code to compute the elastic constants (Cij) [22], [47], [48]. These calculated elastic constants (Cij) are presented in Table 2, alongside those of other 212 and 211 MAX phases. Due to the hexagonal crystal structure of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phase borides, five stiffness constants emerge: C11, C12, C13, C33, and C<sup>44</sup> [49]. The mechanical stability was evaluated using these stiffness constants in the preceding section. For instance, C<sup>11</sup> and C<sup>33</sup> determine stiffness when stress is exerted along the (100) and (001) directions, respectively, whereas C<sup>44</sup> evaluates resistance to shear deformation on the (100) and (001) planes. The stiffness constants C<sup>11</sup> and C<sup>12</sup> directly reflect the strength of atomic bonds along the a- and c-axes. When C<sup>11</sup> exceeds C<sup>33</sup> (or vice versa), it signifies stronger atomic bonding along the a-axis (or c-axis). In Table 2, for Mo2GaB2, Mo2GeB2, and Ta2GeB<sup>2</sup> compounds, C<sup>11</sup> surpasses C33, indicating superior atomic bonding along the a-axis compared to the c-axis. This robust bonding along the a-axis suggests heightened resistance to a-axial deformation. Conversely, in Ta2GeB2, where C<sup>11</sup> < C33, stronger atomic bonding along the c-axis translates to increased resistance against c-axial deformation. Analysis of Table-2 reveals that the values of C<sup>11</sup> and C<sup>22</sup> are notably higher for 212 MAX phase borides than 211 MAX phase borides. Consequently, it can be inferred that 212 MAX phases exhibit stronger resistance to axial deformation when juxtaposed with 211 MAX phases. C<sup>44</sup> is commonly utilized to gauge shear deformation tolerance among the elastic constants. Notably, Ta2GaB exhibits superior shear deformation resistance owing to its highest C<sup>44</sup> value among the studied compounds. Furthermore, the individual elastic constants C12, C13, and C<sup>44</sup> denote shear deformation response under external stress. The observation that C<sup>11</sup> and C<sup>33</sup> possess larger magnitudes than C<sup>44</sup> implies that shear deformation is more facile than axial strain. Another crucial parameter derived from the stiffness constants is the Cauchy pressure (CP), calculated as C<sup>12</sup> - C44, which provides vital insights relevant to the practical applications of solids [51]. Pettifor [51] emphasized the significance of Cauchy pressure (CP) in discerning the chemical bonding and ductile/brittle properties of solids. A negative CP value indicates covalently bonded brittle solids, whereas a positive value signifies isotropic ionic ductile solids. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> fall into covalently bonded brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit metallic ductile characteristics. Similarly, like Ga-containing 211 MAX phase borides, Ga-containing 212 MAX phases also demonstrate negative CP values and brittle behavior [52][53]. The elastic constants obtained are used to produce several bulk elastic parameters that are used to characterize polycrystalline materials, such as Young's modulus (Y), bulk modulus (B), and shear modulus (G). In Table 2, the bulk modulus (B) and shear modulus (G) calculated using Hill's approximation [54] are also presented. Hill's values represent the average of the upper limit (Voigt [55]) and lower limit (Reuss [56]) of B. The necessary equations for these calculations are provided in the supplementary document (S1). Young's modulus (Y) is a crucial indicator of material stiffness. A higher Y value indicates a stiffer material. It can be observed from the table that Mo2GaB<sup>2</sup> exhibits the highest Y value, signifying its greater stiffness compared to others. According to the sequence of Y values, stiffness can be ranked as follows: Mo2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Ta2GeB2. Therefore, Mo2GaB2, with its higher Young's modulus, is anticipated to demonstrate superior mechanical stability and deformation resistance compared to the other compounds under investigation. This quality is essential for aircraft parts or high-performance machinery applications where critical dimensional stability and structural integrity are required [57], [58]. Table 2 shows that 212 MAX phase materials exhibit larger Young's modulus (Y) values than 211 MAX phases. This suggests that 212 MAX phase borides are stiffer than their 211 MAX phase counterparts. Young's modulus (E) also correlates well with thermal shock resistance (R): R ∝ 1/E [59]. Lower Young's modulus values correspond to higher thermal shock resistance. Therefore, materials with higher thermal shock resistance (i.e., lower Young's modulus) are more suitable for use as Thermal Barrier Coating (TBC) materials. Given that Ta2GaB<sup>2</sup> possesses the lowest Y value among the materials studied, it should be considered a superior candidate for TBC material due to its higher thermal shock resistance. The material's ability to withstand shape distortion is elucidated by the shear modulus (G). On the other hand, the bulk modulus (B) indicates the strength of a material's chemical bonds and its ability to withstand uniform compression or volume change. We computed Young's modulus (Y), bulk modulus (B), and shear modulus for our analysis. Based on the bulk modulus values presented in Table 2, the sequence of a material that resists compression when pressure is applied can be outlined as follows: Mo2GeB<sup>2</sup> > Ta2GeB<sup>2</sup> > Mo2GaB<sup>2</sup> > Ta2GeB2. Mo2GeB2, boasting a higher bulk modulus, may exhibit reduced plastic deformation and superior resistance to stress-induced deformation compared to other compounds examined. The ductile and brittle characteristics of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds can also be assessed through Poisson's ratio (ν) and Pugh's ratio (G/B). A compound demonstrates ductile (brittle) behavior if the ν value surpasses (falls below) 0.26 [60]. Furthermore, if the G/B value exceeds (is less than) 0.571, then the compound exhibits brittle (ductile) behavior [19]. According to both criteria, Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are classified as brittle compounds, whereas Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> are categorized as ductile compounds. The ductile nature of Ge-based MAX phases has been reported previously [61]. Fracture toughness (KIC) is a vital property that gauges a material's ability to resist crack propagation. In the case of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, Equation (S2) was employed to determine KIC. The KIC values are documented in Table 2, and notably, these values surpass those of 211 MAX phases [16], [35]. Another parameter, the "f-index," characterizes the isotropic nature and strength of atom-atom bonds within a single hexagonal crystalline lattice along the a- and c-directions. If the f-index is less than 1, chemical bonds exhibit greater rigidity along the c-axis; conversely, if f exceeds 1, bonds are more rigid in the ab-plane. When the value of f is set to 1, atomic bonds exhibit similar strength and uniformity in all directions [15]. The fvalue is computed using Equation (S3) and displayed in Table 2. Table 2 shows that the f-values of Mo2GeB2, Ta2GaB2, and Ta2GeB2, which are close to one, indicate a slight anisotropic bonding strength. Nevertheless, substances with robust bonds in the horizontal plane (ab plane) (f > 1), such as Mo2GaB2, are deemed optimal candidates for the exfoliation process. In engineering applications, the hardness of a solid material serves as a valuable criterion for designing various devices. The elastic properties of polycrystalline materials can be utilized to calculate hardness values, as the ability to resist indentation is closely linked to a material's hardness. Both micro-hardness (Hmicro) and macro-hardness (Hmacro) were computed using Equation (S4) and are presented in Table 2. Based on the values of Hmicro and Hmacro, Mo2GaB<sup>2</sup> emerges as the toughest among the studied phases. The order of hardness is as follows: Mo2GaB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GeB2. Interestingly, Ga-containing MAX phase borides exhibit superior hardness compared to Ge-containing MAX phase borides. In Table 2, we compare our findings with previously reported MAX phases and observe that our data align perfectly with the earlier results. #### 3.4.2 Elastic anisotropy Additionally, anisotropy is linked to other crucial events like anisotropic plastic deformation and the development and spread of micro-cracks within mechanical stress. By providing directiondependent elastic constants, the understanding of anisotropy also offers a framework for improving the mechanical stability of materials in extreme circumstances. The following formulae are used for hexagonal structures to calculate the various anisotropic variables from elastic constants Cij [62]. $A\_{1} = \frac{\frac{1}{6}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{44}};$ $A\_{2} = \frac{2\mathcal{C}\_{44}}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ and $A\_{3} = A\_{1}.$ $A\_{2} = \frac{\frac{1}{3}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ Table 3 lists every anisotropy parameter. Since the value of A<sup>i</sup> should be 1 to be isotropic, the computed value of A<sup>i</sup> (i = 1-3) indicates that all of the compounds under research exhibit anisotropic behavior [63]. An additional method for estimating elastic anisotropy is to use the percentage anisotropy to compressibility and shear (A<sup>B</sup> & AG). This gives polycrystalline materials a helpful way to measure elastic anisotropy. They have been described as [45]; Zero values for AB, AG, and the universal anisotropy factor (AU) show elastic isotropy; the maximum amount of anisotropy is represented by a value of 1. Table 3 shows that, compared to other compounds, the values of AB, AG, and A U for Mo2GeB<sup>2</sup> and Ta2GaB<sup>2</sup> are incredibly close to zero, suggesting that these compounds have nearly isotropic characteristics. #### 3.4.3 Mulliken Populations The Mulliken charge assigned to an atomic species quantifies the effective valence by calculating the absolute difference between the formal ionic charge and the Mulliken charge. Equations (S5) and (S6) are employed to ascertain the Mulliken charge for each atom (α). Table-4 provides the Mulliken atomic population and effective valence charge. Transition metals Mo and Ta in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have pure valence states of 4d<sup>5</sup> and 5d<sup>3</sup> , respectively. The d- orbital electrons of transition metals have been found to influence their effective valence charge significantly. A non-zero positive value indicates a combination of covalent and ionic attributes within chemical bonds. As this value decreases towards zero, it signifies a rise in ionicity. A zero value suggests an ideal ionic character in the bond. Conversely, a progression from zero with a positive value indicates an elevation in the covalency level of the bonds. Based on their effective valence, M atoms move from the left to the right in the periodic table, increasing the covalency of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). Table-4 reveals that the Mulliken atomic charge ascribed to the B atoms is solely negative. Conversely, positive Mulliken atomic charges are associated with transition metals (M) and A. This suggests a charge transfer from M and A to B for each compound within M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge), thereby fostering ionic chemical bonds among these atoms. Bond population serves as another indicator of bond covalency within a crystal, as a high value of bond population essentially signifies a heightened degree of covalency within the chemical bond. The bonding and anti-bonding states influence the populations with positive and negative bond overlap. As demonstrated in Table-5, the B-B bond exhibits greater covalency compared to any other bond in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The presence of an antibonding state between two relevant atoms, which subsequently decreases their chemical bonding, is indicated by a hostile bond overlap population. In Mo2GeB<sup>2</sup> and Ta2GeB2, a hostile bond overlap population is observed in the Ge-Mo and Ge-Ta bonds, indicating the presence of an antibonding state. Therefore, the existence of ionic bonding is guaranteed by electronic charge transfer. On the other hand, the high positive value of the bond overlap population (BOP) denotes the presence of covalent bonding, a feature shared by materials in the MAX phase. ## 3.4.4 Theoretical Vickers Hardness Vickers hardness, derived from the atomic bonds found in solids, indicates how resistant a material is to deformation in extreme circumstances. Several variables influence this feature, such as the crystal flaws, solid structure, atomic arrangement, and bond strength. The Vickers hardness of the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) MAX phases is determined using the Mulliken bond population method, as described by Gou et al. [66] using the formula (S7-S10). This method is particularly suitable for partial metallic systems such as MAX phases. Table-5 lists the computed Vickers hardness values for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The calculated values are 3.35 GPa, 4.76 GPa, 8.21 GPa, and 8.74 GPa for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, respectively. We observed that Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> have much higher hardness values than Mo2GaB<sup>2</sup> and Mo2GeB2. These values are also higher than that of other 212 phases, like, Ti2InB<sup>2</sup> (4.05 GPa) [50], Hf2InB<sup>2</sup> (3.94 GPa), and Hf2SnB<sup>2</sup> (4.41 GPa) [26], Zr2InB<sup>2</sup> (2.92 GPa) and Zr2TlB<sup>2</sup> (2.19 GPa) [25], Zr2GaB<sup>2</sup> (2.53GPa), Zr2GeB<sup>2</sup> (3.31GPa), Hf2GaB<sup>2</sup> (4.73GPa) and Hf2GeB2(4.83GPa) [67]The Hv calculated by the geometrical average of the individual bonding, where the bonding strength mainly determined by the BOP values. In the case of Ta2GaB<sup>2</sup> and Ta2GeB2, the BOP of M-B bonding is much higher compared to Mo2GaB<sup>2</sup> and Mo2GeB2. Even though the BOP of M-B for Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> is higher than that of the other 212 phases mentioned earlier. Therefore, higher values of Hv are expected for Ta2GaB<sup>2</sup> and Ta2GeB2. Additionally, we looked at the Vickers hardness value between 211 and 212 compounds and discovered that the 212 MAX phase compounds had a higher H<sup>V</sup> value. This is because a 2D layer of B atoms is positioned between the M atoms. The B atoms share two center-two electrons to form an extremely strong B-B bond [3]. Table 5 Calculated data for Mulliken bond number (n μ ), bond length (d μ ), bond overlap populations BOP, (P μ ), metallic populations (P μˊ), Vickers hardness (HV) M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. # 3.5 Thermal Properties MAX phases are ideal for high-temperature applications due to their exceptional mechanical qualities at elevated temperatures. As a result, researching the fundamental parameters necessary for predicting their application is of great interest, and these can be obtained from the vibrations of atoms or phonons. The Debye temperature (ΘD) of a solid is directly connected to its bonding strength, melting temperature, thermal expansion, and conductivity. Using the sound velocity and Anderson's technique [68], the Θ<sup>D</sup> of the phases under study has been computed using the formula (S11). Equation (S12) can be used to get the average sound velocity (Vm) from the longitudinal and transverse sound velocities. Equations (S13–14) were used to determine v<sup>l</sup> and vt. The calculated values of Debye's temperature are shown in Table 6, where Mo2GaB<sup>2</sup> has the highest Θ<sup>D</sup> and Ta2GeB<sup>2</sup> has the lowest. If we rank them, it is as follows: Mo2GaB<sup>2</sup> < Mo2GeB<sup>2</sup> < Ta2GaB2 < Ta2GeB2. Hadi et al. recently reported a MAX phase (V2SnC) as a TBC material with a Θ<sup>D</sup> value of 472 K [69]. Thus, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) exhibit encouraging potential as TBC materials, as shown in Table 6. Table 6-Data for density (ρ), longitudinal, transverse, and average sound velocities (vl, vt, and vm), Debye temperature (ΘD), minimum thermal conductivity (Kmin), Grüneisen parameter (γ), thermal expansion coefficient (TEC) at 300K and melting temperature (Tm) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The constant thermal conductivity value at high temperatures is the minimum thermal conductivity (Kmin). As the name implies, this conductivity is minimal because, at high temperatures, phonon coupling breaks. The formula (S15) for the minimum thermal conductivity of solids was derived using the Clarke model [70]. Table 6 lists the calculated value of kmin, with Ta2GaB<sup>2</sup> having the lowest value and Mo2GaB<sup>2</sup> having the highest. When selecting suitable materials for TBC applications, a minimum thermal conductivity of 1.25 W/mK is used as a screening criterion [71]. Our compounds exhibit lower minimum thermal conductivity values, holding promising potential as TBC materials. Gd2Zr2O<sup>7</sup> and Y2SiO5, two recently developed thermal barrier coating (TBC) materials, have minimal thermal conductivities (Kmin) of 1.22 W/m.K. and 1.3 W/m.K. [72], respectively, as confirmed by experiment. These numbers roughly match the values we computed for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The Grüneisen parameter (γ) is a crucial thermal parameter that helps explain the anharmonic effects of lattice dynamics; solids utilized at high temperatures are expected to have lower anharmonic effects. The Grüneisen parameter (γ) can be determined with the help of Poisson's ratio using equation (S16) [73]. The computed γ values, as shown in Table 6, suggest that the compounds under investigation exhibit a weak anharmonic effect. Additionally, for solids with a Poisson's ratio between 0.05 and 0.46, the values similarly fall within the range of 0.85 and 3.53 [74]. The melting temperature (Tm) of the compounds under investigation has been calculated using the following formula (S17). The strength of atomic bonding is the primary factor determining the melting temperature of solids; the higher the Tm, the stronger the atomic bonding. The order of T<sup>m</sup> for the titled phases is found to mirror the Y-based (Young's modulus) order, indicating a close link between T<sup>m</sup> and Y [75]. As observed in Table 6, our compounds roughly follow the Ybased ranking with Mo2GeB<sup>2</sup> < Mo2GaB<sup>2</sup> < Ta2GeB<sup>2</sup> < Ta2GaB2. The T<sup>m</sup> value for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) is also comparable with the TBC material Y4Al2O<sup>9</sup> (2000 K)[75]. ![Fig. 6: Temperature dependence of the calculated thermodynamic parameters. (a) Specific heat at constant pressure, C<sup>p</sup> (b) Specific heat at constant volume, C<sup>v</sup> (c) Free energy, F (d) Entropy, S (e) Internal energy, E and (f) Thermal expansion coefficient, TEC of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge).](path) The temperature dependence of specific heats, Cv, Cp, for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds was obtained using the formulas (S19-S20), as shown in Fig.6(a,b). Assuming that the quasi-harmonic model is accurate and that phase transitions are not anticipated for the compounds under study, these properties are approximated across a temperature range of 0 to 1000 K. Because phonon thermal softening occurs at higher temperatures, the heat capacity rises with temperature. Heat capacities increase quickly and follow the Debye-T 3 power law at lower temperatures. At higher temperature regimes, where C<sup>v</sup> and C<sup>p</sup> do not greatly depend on temperature, they approach the Dulong-Petit (3) limit.[76]. Using the quasi-harmonic approximation, various temperature-dependent thermodynamic potential functions for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have been estimated at zero pressure and displayed in Fig.6 [77]. These functions include the Helmholtz free energy (F), internal energy (E), and entropy (S) within 0-1000 K temperature using equation (S21-S23). The free energy progressively decreases as the temperature rises, as shown in Fig.6(c). Free energy typically declines, and this trend becomes increasingly negative as a natural process proceeds. As demonstrated in Fig. 6(d), the internal energy (E) shows a rising trend with temperature, in contrast to the free energy. Since thermal agitation creates disorder, a system's entropy rises as temperature rises. This is illustrated in Fig.6(d). A material's thermal expansion coefficient (TEC) is derived from the anharmonicity in the lattice dynamics and can be found using equation (S18). The measure of a material's capacity to expand or contract with heat or cold is called the Thermal Expansion Coefficient, or TEC. As observed in Fig. 5(f), the Thermal Expansion Coefficient (TEC) increases rapidly up to 365 K. Then it approaches a constant value, which indicates lower saturations in the materials with temperature changes. The materials under study have a very low TEC value, a crucial characteristic of materials intended for application in high-temperature technology. To be effective as thermal barrier coating (TBC) materials, compounds must exhibit a low thermal conductivity (Kmin) to impede heat transfer, a high melting temperature to withstand extreme heat, and a low thermal expansion coefficient (TEC) to maintain dimensional stability under thermal stress. The compounds M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge) possess these properties, making them suitable candidates for use as TBC materials #### 3.6 Optical Properties Different materials exhibit unique behaviors when exposed to electromagnetic radiation. The optical constants determine the overall response of the sample to the incident radiation. The complex dielectric function, defined as ε(ω)=ε1(ω)+iε2(ω), is one of the main optical characteristics of solids. The following formula determines the imaginary part of the dielectric function ε2(ω) from the momentum matrix element between the occupied and unoccupied electronic states. In this formula, e stands for an electronic charge, ω for light angular frequency, u for the polarization vector of the incident electric field, and and for the conduction and valence band wave functions, respectively, at k. The Kramers-Kronig equation can estimate the real part of the dielectric function, ε1(ω). In contrast, ε2(ω) and ε1(ω) are utilized to evaluate all other optical parameters, such as the absorption coefficient, photoconductivity, reflectivity, and loss function [78]. In this part, several energy-dependent optical properties of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) 212 MAX phases are calculated and analyzed in detail for the photon energy range of 0 to 30 eV, for [100] plane enabling the first assessment of the compounds' practical applicability. Given the metallic conductivity of the MAX compounds' electronic structure, additional parameters were chosen to analyze the optical properties. These include a plasma frequency of 3 eV, damping of 0.05 eV, and Gaussian smearing of 0.5 eV [79]. Figure 7(a) shows the real component of the dielectric function, ε1, which exhibits metallic behavior. In metallic systems, ε<sup>1</sup> has a considerably high negative value in the low-energy range, with the real component reaching negative, which aligns with the band structure finding. Fig. 7(b) depicts the imaginary part of the dielectric function, ε2(ω), representing dielectric losses about frequency. Mo2GeB<sup>2</sup> demonstrates the highest peak in the low-energy region, with all compounds approaching zero from above at around 17 eV. This observation confirms Drude's behavior. ![Fig. 7- (a) Real part of dielectric constant, ε<sup>1</sup> (b) Imaginary part of dielectric constant, ε<sup>2</sup> (c) Reflectivity, R and (d) Loss function, (e) The coefficient of absorption, (α) (f) Refractive index, (n) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) for [100] electric field directions.](path) The reflectivity of a material measures the percentage of incident light energy reflected off it. Equation (S24) was employed to compute reflectivity using the dielectric function, as shown in Fig.7(c). The spectra's visible and infrared (IR) portions consistently exhibit values exceeding 55%. Consequently, the materials under examination are expected to appear metallic gray. In the infrared (IR) region, the maximum reflectivity for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> is 65% and 74%, respectively, occurring at 0.33 eV. Conversely, for Ta2GaB2, the maximum reflectivity is 59% at 14.02 eV, while for Ta2GeB2, it is 65% at 14.78 eV in the UV region. After 18 eV, the reflectance decreases significantly. According to reports, substances with an average reflectivity value above 44% in the visible light region can effectively reduce solar heating by reflecting a significant amount of the solar spectrum. Among all the compounds we examined, Ta2GaB<sup>2</sup> has a reflectivity in the visible range of more than 44% [78]. Therefore, Ta2GaB<sup>2</sup> can be utilized as a coating material and should be able to mitigate solar heating. plasma frequencies (ωp). The plasma frequency for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is observed at 23 eV, 25 eV, 21 eV, and 23 eV, respectively. At this specific frequency, the absorption coefficient rapidly decreases, ε<sup>1</sup> crosses zero from the negative side, and the reflectance R(ω) displays a falling tail. Above this distinctive frequency, the materials are transparent to the incident electromagnetic radiation. Fig. 7(e) illustrates the absorbance coefficient (α) of compounds M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) determined using equation (S26). As α begins to rise from zero photon energy, the metallic nature of the substances under study is again indicated. The visible light region experiences a sharp increase in absorption, peaking in the UV region at around 14 eV and then progressively declining. The IR region exhibits negligible absorption. However, the materials mentioned above appear to have a significant absorption band primarily located in the visible and ultraviolet spectrum. This suggests that the materials can be used in UV surface-disinfection devices, medical sterilization equipment, and other optoelectronic device designs. Its refractive index is crucial to a material's potential application in optical devices like waveguides and photonic crystals. Equations (S27) and (S28) were used to derive the refractive index (n) and extinction coefficient (k) for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), as depicted in Fig.7(c) and Fig.S-1(a). The variations of n and k in MAX phase carbides with incident photon energy closely resemble ε1(ω) and ε2(ω). The static refractive index n(0) for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is 9.4,13.07,7 and 9.2, respectively, and decreases gradually with the increase in photon energy. The extinction coefficient for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2 gradually increases in the IR region, reaching their maximum values of 2.56, 4.27, 2.42, and 3.28, respectively. Following this, they slowly decrease in the visible and UV regions. Figure S-1(b) illustrates the photoconductivity (σ) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) across various photon energies. The photoconductivity (σ) parameter quantifies the impact of photon irradiation on a material's electrical conductivity. Similar to the absorbance coefficient (α) spectrum, the σ spectrum aligns with the metallic nature of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). #### 4 Conclusions In summary, we employed DFT calculations to explore four 212 MAB phases, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), and investigated the structural, electronic, mechanical, lattice dynamical, and optical properties to predict their possible applications. The phonon dispersion curves, formation energy, and elastic constants collectively suggest that the M2AB<sup>2</sup> boride maintains dynamic, mechanical, chemical, and thermodynamic stability. The electronic band structure and density of states (DOS) offer evidence supporting the metallic nature of the compounds under investigation. Concurrently, the charge density mapping and atomic Mulliken population both confirm the presence of a strong B-B covalent bond. The stiffness constants, elastic moduli, findex, fracture toughness (KIC), Pugh's ratio (G/B), hardness parameters, and Cauchy Pressure (CP) of M2AB<sup>2</sup> were computed and compared with those of their 211 equivalents. We found that the values of the 212-phase borides are higher than those of the 211-phase carbides or borides. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are identified as brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit ductile characteristics, as indicated by Poisson's ratio (ν) and Pugh's ratio (B/G or G/B). The elastic characteristics display anisotropy due to the distinct atomic configurations along the aand c-directions. Vickers hardness calculations are considered reliable indicators of material hardness. The results suggest that Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> possess more pliable characteristics than Ta2GaB<sup>2</sup> and Ta2GeB2. The high hardness values of Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compared to the Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> due to higher BOP values of M-B bonding in Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> than in Mo2GaB<sup>2</sup> and Mo2GeB2. The value of minimum thermal conductivity (Kmin), thermal expansion coefficient, and melting temperature (Tm) collectively suggest the potential suitability of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) as a material for thermal barrier coating (TBC) applications in high-temperature devices. The optical conductivity and absorption coefficient corroborate the findings of the electronic band structure. Reflectivity is notably high in infrared (IR) regions and remains nearly constant in the visible and moderate ultraviolet (UV) regions, with an average value exceeding 44% for Ta2GaB2. This suggests that Ta2GaB<sup>2</sup> can be effectively utilized as a coating material to reduce solar heating. We expect that the comprehensive analysis of the diverse physical characteristics of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) presented in this study will establish a robust foundation for future theoretical and experimental explorations of these fascinating MAB phases. ## CRediT author statement A. K. M Naim Ishtiaq and Md Nasir Uddin: Data curation, Writing- Original draft preparation. Md. Rasel Rana, Shariful Islam and Noor Afsary: Reviewing and Editing. Md. 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Fig. 4 – TDOS and PDOS of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB2 compounds.	# Exploration of new 212 MAB phases: M2AB2 (M=Mo, Ta; A=Ga, Ge) via DFT calculations #### Abstract The recently developed MAB phases, an extension of the MAX phase, have sparked interest in research among scientists because of their better thermo-mechanical properties. In this paper, we have explored four new MAB phases M2AB2 (M=Mo, Ta and A=Ga, Ge) and studied the elastic, electronic, thermal, and optical properties to predict the possible applications. The stability of the new phases has been confirmed by calculating formation energy (Ef), formation enthalpy (∆H), phonon dispersion curve (PDC), and elastic constant (Cij). The study reveals that M2AB<sup>2</sup> (M=Mo, Ta and A=Ga, Ge) exhibit significantly higher elastic constants, elastic moduli, and Vickers hardness values than their counterpart 211 borides. Higher Vickers hardness values of Ta2AB2 (A=Ga, Ge) than Mo2AB2 (A=Ga, Ge) have been explained based on the values of the bond overlap population. The analysis of the density of states and electronic band structure revealed the metallic nature of the borides under examination. The thermodynamic characteristics of M2AB2 (M=Mo, Ta and A=Ga, Ge) under high temperatures (0–1000 K) are investigated using the quasi-harmonic Debye model. Critical thermal properties such as melting temperature (Tm), Grüneisen parameter (γ), minimum thermal conductivity (Kmin), Debye temperature (ΘD), and others are also computed. Compared with 211 MAX phases, the 212 phases exhibit higher values of (ΘD) and Tm, along with a lower value of Kmin. These findings suggest that the studied compounds exhibit superior thermal properties that are suitable for practical applications. The optical characteristics have been examined, and the reflectance spectrum indicates that the materials have the potential to mitigate solar heating across various energy regions. #First two authors contributed equally. ### 1. Introduction The MAX phase has garnered significant attention in the present era due to its outstanding mechanical and thermal characteristics at high temperatures, showcasing attributes shared by both metals and ceramics. The increased interest in MAX phase materials can be traced back to Barsoum's noteworthy contributions [1], [2]. The term MAX phase represents a family of multilayer solids where M is an earlier transition metal, A is an element from the IIIA or IVA group of the periodic table, and X is an atom of C/N/B [3]. MAX phase materials showcase metallic behavior due to alternate metallic A-layers and ceramic behavior attributed to the MX layers [4]. Like most metals and alloys, MAX phase materials have excellent thermal shock resistance, superior machinability, and enhanced thermal and electrical conductivities. On the other hand, they have high melting or decomposition temperatures and strong elastic stiffness, similar to many ceramics[5]. The unique combination of metallic and ceramic properties makes them versatile, with applications ranging from high-temperature coatings to nuclear accidenttolerant fuel (ATF), concentrated solar power (CSP), catalysis, and as precursors for MXenes [6], [7], [8], [9]. The diversity of MAX phases was confined to C and N as X elements for years (from 1960 to 2014). However, recent advancements have overcome this limitation by successfully synthesizing MAX phases that contain B [10]. The physical and chemical properties of B and Bcontaining compounds highlight the potential of MAX phase borides, which replace C/N with boron [11]. Due to the presence of B in the composition, they are also called the MAB phase. Furthermore, scientific communities are actively working to broaden the diversity of MAX phases by introducing structural changes, as seen in examples like Cr3AlB<sup>4</sup> (space group Immm), Cr4AlB<sup>6</sup> (space group Cmmm), Cr4AlB<sup>4</sup> [12], [13], [14]. Khazaei et al. reported on the first investigation of the theoretical MAX phase borides M2AlB (where M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta). They evaluated these compounds' electronic structure, mechanical characteristics, and dynamical stability in their investigation [15]. The diversity of MAX phases is a subject of research interest due to their structure variations and the number of atoms in the compounds, resulting in changes to their characteristics. In examples like the 211 MAX phase [16], 312 MAX phase [8][17], 212 MAX phase [3], and i-MAX phase [18], the recent additions to the MAX family include the 314 MAX phase and 212 MAX phase, where the element B has been incorporated as an X element [3] [19]. This introduces a novel aspect to the MAX phase family, contributing to its diversity. The unique structural features of MAB phases include both orthorhombic and hexagonal symmetry observed in crystals. These distinctive symmetries set MAB phases apart from the typical MAX phases. Yinqiao Liu's synthesis of the orthorhombic phase of M2AlB2, particularly the 212-MAB phases with M = Sc, Ti, Zr, Hf, V, Nb, Cr, Mo, W, Mn, Tc, Fe, Co, and Ni, has revealed unique structural stability and notable electrical and mechanical properties [20]. The MAB phase structure slightly differs from the standard MAX phases, which typically crystallize in the hexagonal system with a space group -6<sup>2</sup> (No. 187). Ali et al [3]. investigated the diverse physical properties of Zr2AB<sup>2</sup> (A = In, Tl). Martin Ade et al [13] synthesized ternary borides, namely Cr2AlB2, Cr3AlB4, and Cr4AlB6, and subsequently compared their mechanical properties. Qureshi et al. [19] investigated the 314 Zr3CdB<sup>4</sup> MAX phase boride, calculating its mechanical, thermodynamic, and optical properties. The 314 MAX phase Hf3PB<sup>4</sup> has been thoroughly studied using Density Functional Theory (DFT) and revealed that it was the hardest MAX phase compound discovered until that date [21]. The structure of 212 phases exhibits a slight deviation from 211 MAX phases. In the case of 212 phases, a 2D layer of B is situated between M layers, featuring an additional B atom at the X position, unlike in 211 phases [22] [23], [24]. The B-B bonding in 212 MAX phase borides has improved mechanical and thermal properties. To date, the physical properties of Zr2AB<sup>2</sup> (A = In, Tl) [25] and Hf2AB<sup>2</sup> (A = In, Sn) [26], M2AB (M= Ti, Zr, Hf; A=Al, Ga, In) [27] MAX phases, as well as Nb2AC (A = Ga, Ge, Tl, Zn, P, In, and Cd) MAX phases, have been investigated using density functional theory (DFT). In each instance, the mechanical properties of B-containing compounds show significant improvement compared to their traditional C/N containing 211 MAX phases. The Debye temperature and melting temperature are higher for boron-containing 212 phases than for 211 carbides/nitrides, while the minimum thermal conductivity is lower. The thermal expansion coefficient of borides remains well-suited for use as coating materials. Consequently, the superior thermomechanical properties of B-containing 212 MAX phases demonstrate their suitability for high-temperature technological applications, surpassing the commonly used 211 MAX phase carbides. It should be noted that the 212 MAB phase with a hexagonal structure has already been synthesized [24]. In addition, Ga and Ge-based MAX phases have also been synthesized previously [28]. Thus, the reports on the synthesis of 212 phase and Ga and Gebased MAX phases motivated us to select the 212 MAB phases: M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) for our present study, and we have performed an in-depth investigation of their physical properties through DFT method. Therefore, in this paper, the first-time prediction of the stability and mechanical, electronic, thermal, and optical properties of M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) phases has been presented. The results revealed that M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) compounds are stable and suitable for thermal barrier coating (TBC) and reflection coating applications. Additionally, to provide a comparison, the properties determined for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) are compared with those of other 212 and 211 compounds MAX phase borides. ### 2. Methods of calculations First principles density-functional theory (DFT) computations are conducted utilizing the Cambridge Serial Total Energy Package (CASTEP) module integrated within Materials Studio 2017 [29], [30]. The exchange-correlation function is estimated using the Generalized Gradient Approximation (GGA) method, originally suggested by Perdew, Burke, and Ernzerhof [31]. Pseudo-atomic simulations accounted for electronic orbitals corresponding to B (2s 2 2p 1 ), Ga (3d <sup>10</sup> 4s 2 4p 1 ), Ge (4s 2 4p 2 ), Mo (4s 2 4p 6 4d 5 5s 1 ), and Ta (5d 3 6s 2 ). The energy cutoff and k-point grids were established at 650 eV and 11 × 11 × 4, respectively. The structural relaxation was performed utilizing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique [32], while the electronic structure was computed employing density mixing. The parameters for relaxed structures incorporate the following tolerance thresholds: the self-consistent convergence of the total energy is set at 5 × 10-6 eV/atom, the maximum force exerted on the atom is limited to 0.01 eV/Å, the maximum ionic displacement is constrained to 5 × 10-4 Å, and a maximum stress threshold of 0.02 GPa is imposed. The finite strain method [33], grounded in density functional theory (DFT), is utilized to compute the elastic properties within this framework. All necessary equations for determining various properties are provided in the supplementary document. #### 3 Results and discussion #### 3.1 Structural properties The M2AB<sup>2</sup> compounds (where M=Mo or Ta; A=Ga or Ge) belong to the P6m<sup>2</sup> (No. 187)[23] space group and crystallize in the hexagonal system. Unlike conventional MAX phases, which typically belong to the P63/mmc (194) space group, the 212 MAX phases exhibit distinct characteristics. In Fig. 1, the unit cell structure of Mo2GaB<sup>2</sup> is depicted as a representative of M2AB<sup>2</sup> alongside Mo2GaB, facilitating a comparison to discern their differences easily. The atomic positions are as follows: M (Mo or Ta) at (0.3333, 0.6667, 0.6935), A at (0.6667, 0.3333, 0.0), and two B atoms positioned at (0.6667, 0.3333, 0.5) and (0.0, 0.0, 0.5). The B components are arranged at the corners of the unit cell in typical 211 MAX phases, but in 212 boride MAX phases, they form a 2D layer between the M layers. This structural arrangement results in B-B covalent bonds in the 2D layer, enhancing stability compared to conventional 211 MAX phases. ![Fig. 1 - The schematic unit cell of (a) Mo2GaB<sup>2</sup> and (b) Mo2GaB compound.](path) The calculated lattice parameters of M2AB<sup>2</sup> are presented in Table 1 alongside those of other 211 and 212 MAX phases for comparison, demonstrating consistency with prior results [16], [34], [35] and affirming the accuracy of the computational methodology used. The primary distinction between 212- and 211-unit cell structures arises from differences in the lattice parameter c, where the c value for 211 exceeds that of 212. Moreover, the volumes of Mo- and Ta-based 211 MAX phase borides surpass those of 212 MAX phase borides. #### 3.2 Stability Examining a compound's stability is significant for multiple purposes, as it yields valuable information regarding the compound's synthesis parameters and aids in assessing the material's resilience across diverse environments, including thermal, compressive, and mechanical pressures. In this section, we delve into a comprehensive theoretical analysis concerning the chemical, dynamic, and mechanical stability of M2AB<sup>2</sup> compounds. The compound's chemical stability is determined by computing its formation energy by the following equation [36]: 2<sup>2</sup> = 22−( + + ) ++ . In the context provided, 22 represents the total energy of the compound after optimizing the unit cell. , , and denote the energies of the individual elements M, A, and B, respectively. The variables x, y, and z correspond to the number of atoms in the unit cell for M, Ga, and B, respectively. For Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, the calculated formation energy (Ef) are -1.8079 eV/atom, -1.8859 eV/atom, -2.1252 eV/atom, and -2.1933 eV/atom, respectively. The negative values signify the chemical stability of all compounds. The order of chemical stability can be expressed as Ta2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Mo2GaB2, indicating that Ta2GeB<sup>2</sup> is the most stable. Additionally, it's observed that MAX phase borides containing Ta are more stable than those containing Mo. Negative formation energy alone may not fully explain the chemical stability of M2AB2 (M=Mo, Ta and A=Ga, Ge). We calculated its formation enthalpy by examining potential pathways to evaluate its thermodynamic stability. For this analysis, we used the experimentally identified stable phases of MoB[37] and TaB[38], Ga4Mo[39], and B2Mo[40]. The potential decomposition pathways for our compounds, as determined from the Open Quantum Materials Database (OQMD), are outlined below. ![We have calculated the reaction energy as follows [41]:](path) Where, M=Mo, Ta and A=Ga, Ge. The following formula can calculate the decomposition energy associated with the reaction energy. = . Where n is the number of participating atoms. The calculated decomposition energies for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> are -0.16, -0.17, -0.26, and -0.28 meV/atom, respectively. So, we can state that the M2AB2 (M=Mo, Ta and A=Ga, Ge) system exhibits thermodynamic stability. Phonon dispersion curves (PDCs) have been calculated at the ground state utilizing the density functional perturbation theory (DFPT) linear-response approach to evaluate the dynamic stability of the MAX phase borides under investigation [42]. The PDCs, depicting the phonon dispersion along the high symmetry directions of the crystal Brillouin zone (BZ), along with the total phonon density of states (PHDOS) of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, are illustrated in Fig. 2(a, b, c and d). Analysis of the PDCs reveals no negative phonon frequencies for any of the compounds, indicating their dynamic stability. The PHDOS of the M2AB<sup>2</sup> compounds are derived from the PDCs and are presented alongside the PDCs in Fig. 2(a, b, c, and d), facilitating band identification through comparison of corresponding peaks. From Fig. 2, it is observed that in Mo2GaB2, the flatness of the bands for the Transverse Optical (TO) modes results in a prominent peak in the PHDOS, whereas non-flat bands for the Longitudinal Optical (LO) modes lead to weaker peaks in the PHDOS. Similar trends are observed in Mo2GeB2, Ta2GaB2, and Ta2GeB2. Notably, a distinct discrepancy arises between the optical and acoustic branches, with the top of the LO and bottom of the TO modes situated at the G point, with separations of 7.49 THz, 6.39 THz, 9.39 THz, and 8.51 THz for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, respectively. ![Fig. 2 – Phonon DOS and Phonon dispersion curves of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) Materials are subjected to various forces and loads in practical applications, necessitating understanding their mechanical stability. The mechanical stability of a compound can be assessed using stiffness constants. For a hexagonal system, the conditions for mechanical stability are as follows [43]: C<sup>11</sup> > 0, C<sup>11</sup> > C12, C<sup>44</sup> > 0, and (C<sup>11</sup> + C12)C<sup>33</sup> - 2(C13)² > 0. As indicated in Table 2, the Cij values of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) satisfy these conditions, thus confirming the mechanical stability of herein predicted phases: M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge). #### 3.3 Electronic properties Analyzing the electronic band structure (EBS) is crucial for gaining insights into the electronic behavior of a compound. The EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) MAX phases are depicted in Fig. 3(a, b, c, and d), with the Fermi energy (EF) level set at 0 eV, represented by a horizontal line. Observing the EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge), it is evident that the conduction band overlaps with the valence band, indicating the absence of a band gap. This observation confirms that the M2AB<sup>2</sup> compounds exhibit metallic behavior, which aligns with conventional MAX phases. The red lines illustrate the overlapping band at the Fermi level. Fig. 3 (a, b) shows that for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> compounds, the maximum band overlap occurs along the A-H path. Conversely, in Fig. 3(c, d), the maximum band overlap is observed along the G-M paths. Utilizing the band structure, we can analyze the electrical anisotropy of M2AB<sup>2</sup> MAX phase compounds. The anisotropic nature can be understood by analyzing the energy dispersion in the basal plane and along the c-axis. The paths G-A, H-K, and M-L show energy dispersion along the c-direction, and A-H, K-G, G-M, and L-H show energy dispersion in the basal plane. In comparison to the paths A-H, K-G, G-M, and L-H (basal plane), there is less energy dispersion along the lines G-A, H–K, and M-L (c-direction), as shown by Fig. 2(a, b, c and d). Lower energy dispersion in the c-direction results from a higher effective mass [44], indicating the strong electronic anisotropy of the M2AB<sup>2</sup> MAX phase compound. Consequently, conductivity along the c-axis is expected to be lower than in the basal planes. These findings are consistent with prior studies [34], [45]. ![Fig. 3 – Band structure of (a) Mo2GaB2, (b) Mo2GeB2, (c) Ta2GaB2, and (d) Ta2GeB<sup>2</sup> compounds.](path) To investigate the bonding nature and electronic conductivity, the total density of states (TDOS) and partial density of states (PDOS) of M2AB<sup>2</sup> compounds were computed. Figure 4 (a, b, c, and d) illustrates the TDOS and PDOS of these compounds, with the Fermi energy (EF) set at zero energy level, indicated by a straight line. These profiles exhibit typical characteristics of MAX phase materials. The Mo or Ta-d electronic states predominantly contribute to the Fermi level, with a minor contribution from the B-p and Ga or Ge-p electronic states. To ascertain the hybridization characteristics of various electronic states within the valence band, the energy spectrum of the valence band has been partitioned into two distinct segments. The first segment encompasses the lower valence band region spanning from -7 eV to -3.5 eV, originating from the hybridization of Mo-p, Mo-d, and B-s orbitals in the case of the Mo2GaB<sup>2</sup> compound. Conversely, for the Mo2GeB2 compound, the lower valence band region arises from the hybridization of Mo-d, Ge-p, and B-s orbitals. Notably, for both the Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compounds, the dominance of the B-p state characterizes the lower valence band region. The second segment pertains to the upper valence band region from -3.5 eV to 0 eV. In the case of the Mo2GaB<sup>2</sup> compound, this region arises from the hybridization of Mo-p and Mo-d orbitals. However, for the Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, the upper valence band region results from the hybridization of Mo-d orbitals and (Ga/Ge)-p orbitals. Notably, the Fermi level of M2AB<sup>2</sup> resides near the pseudogap in the TDOS profile, indicating a high level of electronic stability. Similar trends are observed in other compounds like Zr2AlN, V2AlN, Sc2AlB, Sc2GaB, Ta2GaB, and Hf2GaB<sup>2</sup> [15], [35], [45]. ![Fig. 4 – TDOS and PDOS of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) The electron charge density mapping is helpful in understanding the distribution of electron densities linked to chemical bonds. It delineates areas of positive and negative charge densities, signifying the development and exhaustion of electrical charges, respectively. As depicted in the map, covalent bonds become apparent by accumulating charges between two atoms. Furthermore, the presence of ionic bonds can be inferred from a balance between negative and positive charges at specific atom positions[46]. The valence electronic CDM, denoted in units of eÅ-3 , for M2AB<sup>2</sup> (where M = Mo, Ta; A = Ga, Ge) is showcased in Fig. 5(a, b, c, and d) along the (110) crystallographic plane. The accompanying scale illustrates the intensity of electronic charge density, with red and blue colors indicating low and high electronic charge density, respectively. As depicted in Fig. 5(a, b, c, and d), it is evident that charges accumulate in the regions between the B sites. Consequently, it is anticipated that strong covalent B‒B bonding occurs through the formation of two center-two electron (2c‒2e) bonds in the M2AB<sup>2</sup> (where M=Mo, Ta; A=Ga, Ge) compound, similar to other 212 MAX phase borides like Ti2PB2, Zr2PbB2, Nb2SB2, Zr2GaB<sup>2</sup> and Hf2GaB<sup>2</sup> [34], [45]. Mulliken analysis has corroborated the charge transfer from Mo/Ta atoms to B atoms. The charge received from Mo/Ta atoms is distributed among the B atoms positioned at the transitions and those located at the edges, facilitating the formation of a two center-two electron (2c‒2e) bond between B atoms within the 2D layer of B, as illustrated in Fig. 1(a). The hardness of each bond value presented in Table 5 also aligns with the results obtained from charge density mapping (CDM) and our analysis using moduli and elastic stiffness constants. ![Fig. 5 – The charge density mapping of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) ## 3.4.1 Mechanical properties The mechanical properties of materials play a pivotal role in determining their potential applications, serving as crucial indicators of their behavior and suitability in materials engineering endeavors. These properties are equally applicable to MAX phase materials. Initially, to assess the mechanical properties of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phases, we employed the stress-strain method within the CASTEP code to compute the elastic constants (Cij) [22], [47], [48]. These calculated elastic constants (Cij) are presented in Table 2, alongside those of other 212 and 211 MAX phases. Due to the hexagonal crystal structure of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phase borides, five stiffness constants emerge: C11, C12, C13, C33, and C<sup>44</sup> [49]. The mechanical stability was evaluated using these stiffness constants in the preceding section. For instance, C<sup>11</sup> and C<sup>33</sup> determine stiffness when stress is exerted along the (100) and (001) directions, respectively, whereas C<sup>44</sup> evaluates resistance to shear deformation on the (100) and (001) planes. The stiffness constants C<sup>11</sup> and C<sup>12</sup> directly reflect the strength of atomic bonds along the a- and c-axes. When C<sup>11</sup> exceeds C<sup>33</sup> (or vice versa), it signifies stronger atomic bonding along the a-axis (or c-axis). In Table 2, for Mo2GaB2, Mo2GeB2, and Ta2GeB<sup>2</sup> compounds, C<sup>11</sup> surpasses C33, indicating superior atomic bonding along the a-axis compared to the c-axis. This robust bonding along the a-axis suggests heightened resistance to a-axial deformation. Conversely, in Ta2GeB2, where C<sup>11</sup> < C33, stronger atomic bonding along the c-axis translates to increased resistance against c-axial deformation. Analysis of Table-2 reveals that the values of C<sup>11</sup> and C<sup>22</sup> are notably higher for 212 MAX phase borides than 211 MAX phase borides. Consequently, it can be inferred that 212 MAX phases exhibit stronger resistance to axial deformation when juxtaposed with 211 MAX phases. C<sup>44</sup> is commonly utilized to gauge shear deformation tolerance among the elastic constants. Notably, Ta2GaB exhibits superior shear deformation resistance owing to its highest C<sup>44</sup> value among the studied compounds. Furthermore, the individual elastic constants C12, C13, and C<sup>44</sup> denote shear deformation response under external stress. The observation that C<sup>11</sup> and C<sup>33</sup> possess larger magnitudes than C<sup>44</sup> implies that shear deformation is more facile than axial strain. Another crucial parameter derived from the stiffness constants is the Cauchy pressure (CP), calculated as C<sup>12</sup> - C44, which provides vital insights relevant to the practical applications of solids [51]. Pettifor [51] emphasized the significance of Cauchy pressure (CP) in discerning the chemical bonding and ductile/brittle properties of solids. A negative CP value indicates covalently bonded brittle solids, whereas a positive value signifies isotropic ionic ductile solids. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> fall into covalently bonded brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit metallic ductile characteristics. Similarly, like Ga-containing 211 MAX phase borides, Ga-containing 212 MAX phases also demonstrate negative CP values and brittle behavior [52][53]. The elastic constants obtained are used to produce several bulk elastic parameters that are used to characterize polycrystalline materials, such as Young's modulus (Y), bulk modulus (B), and shear modulus (G). In Table 2, the bulk modulus (B) and shear modulus (G) calculated using Hill's approximation [54] are also presented. Hill's values represent the average of the upper limit (Voigt [55]) and lower limit (Reuss [56]) of B. The necessary equations for these calculations are provided in the supplementary document (S1). Young's modulus (Y) is a crucial indicator of material stiffness. A higher Y value indicates a stiffer material. It can be observed from the table that Mo2GaB<sup>2</sup> exhibits the highest Y value, signifying its greater stiffness compared to others. According to the sequence of Y values, stiffness can be ranked as follows: Mo2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Ta2GeB2. Therefore, Mo2GaB2, with its higher Young's modulus, is anticipated to demonstrate superior mechanical stability and deformation resistance compared to the other compounds under investigation. This quality is essential for aircraft parts or high-performance machinery applications where critical dimensional stability and structural integrity are required [57], [58]. Table 2 shows that 212 MAX phase materials exhibit larger Young's modulus (Y) values than 211 MAX phases. This suggests that 212 MAX phase borides are stiffer than their 211 MAX phase counterparts. Young's modulus (E) also correlates well with thermal shock resistance (R): R ∝ 1/E [59]. Lower Young's modulus values correspond to higher thermal shock resistance. Therefore, materials with higher thermal shock resistance (i.e., lower Young's modulus) are more suitable for use as Thermal Barrier Coating (TBC) materials. Given that Ta2GaB<sup>2</sup> possesses the lowest Y value among the materials studied, it should be considered a superior candidate for TBC material due to its higher thermal shock resistance. The material's ability to withstand shape distortion is elucidated by the shear modulus (G). On the other hand, the bulk modulus (B) indicates the strength of a material's chemical bonds and its ability to withstand uniform compression or volume change. We computed Young's modulus (Y), bulk modulus (B), and shear modulus for our analysis. Based on the bulk modulus values presented in Table 2, the sequence of a material that resists compression when pressure is applied can be outlined as follows: Mo2GeB<sup>2</sup> > Ta2GeB<sup>2</sup> > Mo2GaB<sup>2</sup> > Ta2GeB2. Mo2GeB2, boasting a higher bulk modulus, may exhibit reduced plastic deformation and superior resistance to stress-induced deformation compared to other compounds examined. The ductile and brittle characteristics of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds can also be assessed through Poisson's ratio (ν) and Pugh's ratio (G/B). A compound demonstrates ductile (brittle) behavior if the ν value surpasses (falls below) 0.26 [60]. Furthermore, if the G/B value exceeds (is less than) 0.571, then the compound exhibits brittle (ductile) behavior [19]. According to both criteria, Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are classified as brittle compounds, whereas Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> are categorized as ductile compounds. The ductile nature of Ge-based MAX phases has been reported previously [61]. Fracture toughness (KIC) is a vital property that gauges a material's ability to resist crack propagation. In the case of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, Equation (S2) was employed to determine KIC. The KIC values are documented in Table 2, and notably, these values surpass those of 211 MAX phases [16], [35]. Another parameter, the "f-index," characterizes the isotropic nature and strength of atom-atom bonds within a single hexagonal crystalline lattice along the a- and c-directions. If the f-index is less than 1, chemical bonds exhibit greater rigidity along the c-axis; conversely, if f exceeds 1, bonds are more rigid in the ab-plane. When the value of f is set to 1, atomic bonds exhibit similar strength and uniformity in all directions [15]. The fvalue is computed using Equation (S3) and displayed in Table 2. Table 2 shows that the f-values of Mo2GeB2, Ta2GaB2, and Ta2GeB2, which are close to one, indicate a slight anisotropic bonding strength. Nevertheless, substances with robust bonds in the horizontal plane (ab plane) (f > 1), such as Mo2GaB2, are deemed optimal candidates for the exfoliation process. In engineering applications, the hardness of a solid material serves as a valuable criterion for designing various devices. The elastic properties of polycrystalline materials can be utilized to calculate hardness values, as the ability to resist indentation is closely linked to a material's hardness. Both micro-hardness (Hmicro) and macro-hardness (Hmacro) were computed using Equation (S4) and are presented in Table 2. Based on the values of Hmicro and Hmacro, Mo2GaB<sup>2</sup> emerges as the toughest among the studied phases. The order of hardness is as follows: Mo2GaB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GeB2. Interestingly, Ga-containing MAX phase borides exhibit superior hardness compared to Ge-containing MAX phase borides. In Table 2, we compare our findings with previously reported MAX phases and observe that our data align perfectly with the earlier results. #### 3.4.2 Elastic anisotropy Additionally, anisotropy is linked to other crucial events like anisotropic plastic deformation and the development and spread of micro-cracks within mechanical stress. By providing directiondependent elastic constants, the understanding of anisotropy also offers a framework for improving the mechanical stability of materials in extreme circumstances. The following formulae are used for hexagonal structures to calculate the various anisotropic variables from elastic constants Cij [62]. $A\_{1} = \frac{\frac{1}{6}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{44}};$ $A\_{2} = \frac{2\mathcal{C}\_{44}}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ and $A\_{3} = A\_{1}.$ $A\_{2} = \frac{\frac{1}{3}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ Table 3 lists every anisotropy parameter. Since the value of A<sup>i</sup> should be 1 to be isotropic, the computed value of A<sup>i</sup> (i = 1-3) indicates that all of the compounds under research exhibit anisotropic behavior [63]. An additional method for estimating elastic anisotropy is to use the percentage anisotropy to compressibility and shear (A<sup>B</sup> & AG). This gives polycrystalline materials a helpful way to measure elastic anisotropy. They have been described as [45]; Zero values for AB, AG, and the universal anisotropy factor (AU) show elastic isotropy; the maximum amount of anisotropy is represented by a value of 1. Table 3 shows that, compared to other compounds, the values of AB, AG, and A U for Mo2GeB<sup>2</sup> and Ta2GaB<sup>2</sup> are incredibly close to zero, suggesting that these compounds have nearly isotropic characteristics. #### 3.4.3 Mulliken Populations The Mulliken charge assigned to an atomic species quantifies the effective valence by calculating the absolute difference between the formal ionic charge and the Mulliken charge. Equations (S5) and (S6) are employed to ascertain the Mulliken charge for each atom (α). Table-4 provides the Mulliken atomic population and effective valence charge. Transition metals Mo and Ta in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have pure valence states of 4d<sup>5</sup> and 5d<sup>3</sup> , respectively. The d- orbital electrons of transition metals have been found to influence their effective valence charge significantly. A non-zero positive value indicates a combination of covalent and ionic attributes within chemical bonds. As this value decreases towards zero, it signifies a rise in ionicity. A zero value suggests an ideal ionic character in the bond. Conversely, a progression from zero with a positive value indicates an elevation in the covalency level of the bonds. Based on their effective valence, M atoms move from the left to the right in the periodic table, increasing the covalency of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). Table-4 reveals that the Mulliken atomic charge ascribed to the B atoms is solely negative. Conversely, positive Mulliken atomic charges are associated with transition metals (M) and A. This suggests a charge transfer from M and A to B for each compound within M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge), thereby fostering ionic chemical bonds among these atoms. Bond population serves as another indicator of bond covalency within a crystal, as a high value of bond population essentially signifies a heightened degree of covalency within the chemical bond. The bonding and anti-bonding states influence the populations with positive and negative bond overlap. As demonstrated in Table-5, the B-B bond exhibits greater covalency compared to any other bond in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The presence of an antibonding state between two relevant atoms, which subsequently decreases their chemical bonding, is indicated by a hostile bond overlap population. In Mo2GeB<sup>2</sup> and Ta2GeB2, a hostile bond overlap population is observed in the Ge-Mo and Ge-Ta bonds, indicating the presence of an antibonding state. Therefore, the existence of ionic bonding is guaranteed by electronic charge transfer. On the other hand, the high positive value of the bond overlap population (BOP) denotes the presence of covalent bonding, a feature shared by materials in the MAX phase. ## 3.4.4 Theoretical Vickers Hardness Vickers hardness, derived from the atomic bonds found in solids, indicates how resistant a material is to deformation in extreme circumstances. Several variables influence this feature, such as the crystal flaws, solid structure, atomic arrangement, and bond strength. The Vickers hardness of the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) MAX phases is determined using the Mulliken bond population method, as described by Gou et al. [66] using the formula (S7-S10). This method is particularly suitable for partial metallic systems such as MAX phases. Table-5 lists the computed Vickers hardness values for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The calculated values are 3.35 GPa, 4.76 GPa, 8.21 GPa, and 8.74 GPa for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, respectively. We observed that Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> have much higher hardness values than Mo2GaB<sup>2</sup> and Mo2GeB2. These values are also higher than that of other 212 phases, like, Ti2InB<sup>2</sup> (4.05 GPa) [50], Hf2InB<sup>2</sup> (3.94 GPa), and Hf2SnB<sup>2</sup> (4.41 GPa) [26], Zr2InB<sup>2</sup> (2.92 GPa) and Zr2TlB<sup>2</sup> (2.19 GPa) [25], Zr2GaB<sup>2</sup> (2.53GPa), Zr2GeB<sup>2</sup> (3.31GPa), Hf2GaB<sup>2</sup> (4.73GPa) and Hf2GeB2(4.83GPa) [67]The Hv calculated by the geometrical average of the individual bonding, where the bonding strength mainly determined by the BOP values. In the case of Ta2GaB<sup>2</sup> and Ta2GeB2, the BOP of M-B bonding is much higher compared to Mo2GaB<sup>2</sup> and Mo2GeB2. Even though the BOP of M-B for Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> is higher than that of the other 212 phases mentioned earlier. Therefore, higher values of Hv are expected for Ta2GaB<sup>2</sup> and Ta2GeB2. Additionally, we looked at the Vickers hardness value between 211 and 212 compounds and discovered that the 212 MAX phase compounds had a higher H<sup>V</sup> value. This is because a 2D layer of B atoms is positioned between the M atoms. The B atoms share two center-two electrons to form an extremely strong B-B bond [3]. Table 5 Calculated data for Mulliken bond number (n μ ), bond length (d μ ), bond overlap populations BOP, (P μ ), metallic populations (P μˊ), Vickers hardness (HV) M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. # 3.5 Thermal Properties MAX phases are ideal for high-temperature applications due to their exceptional mechanical qualities at elevated temperatures. As a result, researching the fundamental parameters necessary for predicting their application is of great interest, and these can be obtained from the vibrations of atoms or phonons. The Debye temperature (ΘD) of a solid is directly connected to its bonding strength, melting temperature, thermal expansion, and conductivity. Using the sound velocity and Anderson's technique [68], the Θ<sup>D</sup> of the phases under study has been computed using the formula (S11). Equation (S12) can be used to get the average sound velocity (Vm) from the longitudinal and transverse sound velocities. Equations (S13–14) were used to determine v<sup>l</sup> and vt. The calculated values of Debye's temperature are shown in Table 6, where Mo2GaB<sup>2</sup> has the highest Θ<sup>D</sup> and Ta2GeB<sup>2</sup> has the lowest. If we rank them, it is as follows: Mo2GaB<sup>2</sup> < Mo2GeB<sup>2</sup> < Ta2GaB2 < Ta2GeB2. Hadi et al. recently reported a MAX phase (V2SnC) as a TBC material with a Θ<sup>D</sup> value of 472 K [69]. Thus, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) exhibit encouraging potential as TBC materials, as shown in Table 6. Table 6-Data for density (ρ), longitudinal, transverse, and average sound velocities (vl, vt, and vm), Debye temperature (ΘD), minimum thermal conductivity (Kmin), Grüneisen parameter (γ), thermal expansion coefficient (TEC) at 300K and melting temperature (Tm) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The constant thermal conductivity value at high temperatures is the minimum thermal conductivity (Kmin). As the name implies, this conductivity is minimal because, at high temperatures, phonon coupling breaks. The formula (S15) for the minimum thermal conductivity of solids was derived using the Clarke model [70]. Table 6 lists the calculated value of kmin, with Ta2GaB<sup>2</sup> having the lowest value and Mo2GaB<sup>2</sup> having the highest. When selecting suitable materials for TBC applications, a minimum thermal conductivity of 1.25 W/mK is used as a screening criterion [71]. Our compounds exhibit lower minimum thermal conductivity values, holding promising potential as TBC materials. Gd2Zr2O<sup>7</sup> and Y2SiO5, two recently developed thermal barrier coating (TBC) materials, have minimal thermal conductivities (Kmin) of 1.22 W/m.K. and 1.3 W/m.K. [72], respectively, as confirmed by experiment. These numbers roughly match the values we computed for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The Grüneisen parameter (γ) is a crucial thermal parameter that helps explain the anharmonic effects of lattice dynamics; solids utilized at high temperatures are expected to have lower anharmonic effects. The Grüneisen parameter (γ) can be determined with the help of Poisson's ratio using equation (S16) [73]. The computed γ values, as shown in Table 6, suggest that the compounds under investigation exhibit a weak anharmonic effect. Additionally, for solids with a Poisson's ratio between 0.05 and 0.46, the values similarly fall within the range of 0.85 and 3.53 [74]. The melting temperature (Tm) of the compounds under investigation has been calculated using the following formula (S17). The strength of atomic bonding is the primary factor determining the melting temperature of solids; the higher the Tm, the stronger the atomic bonding. The order of T<sup>m</sup> for the titled phases is found to mirror the Y-based (Young's modulus) order, indicating a close link between T<sup>m</sup> and Y [75]. As observed in Table 6, our compounds roughly follow the Ybased ranking with Mo2GeB<sup>2</sup> < Mo2GaB<sup>2</sup> < Ta2GeB<sup>2</sup> < Ta2GaB2. The T<sup>m</sup> value for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) is also comparable with the TBC material Y4Al2O<sup>9</sup> (2000 K)[75]. ![Fig. 6: Temperature dependence of the calculated thermodynamic parameters. (a) Specific heat at constant pressure, C<sup>p</sup> (b) Specific heat at constant volume, C<sup>v</sup> (c) Free energy, F (d) Entropy, S (e) Internal energy, E and (f) Thermal expansion coefficient, TEC of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge).](path) The temperature dependence of specific heats, Cv, Cp, for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds was obtained using the formulas (S19-S20), as shown in Fig.6(a,b). Assuming that the quasi-harmonic model is accurate and that phase transitions are not anticipated for the compounds under study, these properties are approximated across a temperature range of 0 to 1000 K. Because phonon thermal softening occurs at higher temperatures, the heat capacity rises with temperature. Heat capacities increase quickly and follow the Debye-T 3 power law at lower temperatures. At higher temperature regimes, where C<sup>v</sup> and C<sup>p</sup> do not greatly depend on temperature, they approach the Dulong-Petit (3) limit.[76]. Using the quasi-harmonic approximation, various temperature-dependent thermodynamic potential functions for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have been estimated at zero pressure and displayed in Fig.6 [77]. These functions include the Helmholtz free energy (F), internal energy (E), and entropy (S) within 0-1000 K temperature using equation (S21-S23). The free energy progressively decreases as the temperature rises, as shown in Fig.6(c). Free energy typically declines, and this trend becomes increasingly negative as a natural process proceeds. As demonstrated in Fig. 6(d), the internal energy (E) shows a rising trend with temperature, in contrast to the free energy. Since thermal agitation creates disorder, a system's entropy rises as temperature rises. This is illustrated in Fig.6(d). A material's thermal expansion coefficient (TEC) is derived from the anharmonicity in the lattice dynamics and can be found using equation (S18). The measure of a material's capacity to expand or contract with heat or cold is called the Thermal Expansion Coefficient, or TEC. As observed in Fig. 5(f), the Thermal Expansion Coefficient (TEC) increases rapidly up to 365 K. Then it approaches a constant value, which indicates lower saturations in the materials with temperature changes. The materials under study have a very low TEC value, a crucial characteristic of materials intended for application in high-temperature technology. To be effective as thermal barrier coating (TBC) materials, compounds must exhibit a low thermal conductivity (Kmin) to impede heat transfer, a high melting temperature to withstand extreme heat, and a low thermal expansion coefficient (TEC) to maintain dimensional stability under thermal stress. The compounds M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge) possess these properties, making them suitable candidates for use as TBC materials #### 3.6 Optical Properties Different materials exhibit unique behaviors when exposed to electromagnetic radiation. The optical constants determine the overall response of the sample to the incident radiation. The complex dielectric function, defined as ε(ω)=ε1(ω)+iε2(ω), is one of the main optical characteristics of solids. The following formula determines the imaginary part of the dielectric function ε2(ω) from the momentum matrix element between the occupied and unoccupied electronic states. In this formula, e stands for an electronic charge, ω for light angular frequency, u for the polarization vector of the incident electric field, and and for the conduction and valence band wave functions, respectively, at k. The Kramers-Kronig equation can estimate the real part of the dielectric function, ε1(ω). In contrast, ε2(ω) and ε1(ω) are utilized to evaluate all other optical parameters, such as the absorption coefficient, photoconductivity, reflectivity, and loss function [78]. In this part, several energy-dependent optical properties of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) 212 MAX phases are calculated and analyzed in detail for the photon energy range of 0 to 30 eV, for [100] plane enabling the first assessment of the compounds' practical applicability. Given the metallic conductivity of the MAX compounds' electronic structure, additional parameters were chosen to analyze the optical properties. These include a plasma frequency of 3 eV, damping of 0.05 eV, and Gaussian smearing of 0.5 eV [79]. Figure 7(a) shows the real component of the dielectric function, ε1, which exhibits metallic behavior. In metallic systems, ε<sup>1</sup> has a considerably high negative value in the low-energy range, with the real component reaching negative, which aligns with the band structure finding. Fig. 7(b) depicts the imaginary part of the dielectric function, ε2(ω), representing dielectric losses about frequency. Mo2GeB<sup>2</sup> demonstrates the highest peak in the low-energy region, with all compounds approaching zero from above at around 17 eV. This observation confirms Drude's behavior. ![Fig. 7- (a) Real part of dielectric constant, ε<sup>1</sup> (b) Imaginary part of dielectric constant, ε<sup>2</sup> (c) Reflectivity, R and (d) Loss function, (e) The coefficient of absorption, (α) (f) Refractive index, (n) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) for [100] electric field directions.](path) The reflectivity of a material measures the percentage of incident light energy reflected off it. Equation (S24) was employed to compute reflectivity using the dielectric function, as shown in Fig.7(c). The spectra's visible and infrared (IR) portions consistently exhibit values exceeding 55%. Consequently, the materials under examination are expected to appear metallic gray. In the infrared (IR) region, the maximum reflectivity for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> is 65% and 74%, respectively, occurring at 0.33 eV. Conversely, for Ta2GaB2, the maximum reflectivity is 59% at 14.02 eV, while for Ta2GeB2, it is 65% at 14.78 eV in the UV region. After 18 eV, the reflectance decreases significantly. According to reports, substances with an average reflectivity value above 44% in the visible light region can effectively reduce solar heating by reflecting a significant amount of the solar spectrum. Among all the compounds we examined, Ta2GaB<sup>2</sup> has a reflectivity in the visible range of more than 44% [78]. Therefore, Ta2GaB<sup>2</sup> can be utilized as a coating material and should be able to mitigate solar heating. plasma frequencies (ωp). The plasma frequency for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is observed at 23 eV, 25 eV, 21 eV, and 23 eV, respectively. At this specific frequency, the absorption coefficient rapidly decreases, ε<sup>1</sup> crosses zero from the negative side, and the reflectance R(ω) displays a falling tail. Above this distinctive frequency, the materials are transparent to the incident electromagnetic radiation. Fig. 7(e) illustrates the absorbance coefficient (α) of compounds M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) determined using equation (S26). As α begins to rise from zero photon energy, the metallic nature of the substances under study is again indicated. The visible light region experiences a sharp increase in absorption, peaking in the UV region at around 14 eV and then progressively declining. The IR region exhibits negligible absorption. However, the materials mentioned above appear to have a significant absorption band primarily located in the visible and ultraviolet spectrum. This suggests that the materials can be used in UV surface-disinfection devices, medical sterilization equipment, and other optoelectronic device designs. Its refractive index is crucial to a material's potential application in optical devices like waveguides and photonic crystals. Equations (S27) and (S28) were used to derive the refractive index (n) and extinction coefficient (k) for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), as depicted in Fig.7(c) and Fig.S-1(a). The variations of n and k in MAX phase carbides with incident photon energy closely resemble ε1(ω) and ε2(ω). The static refractive index n(0) for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is 9.4,13.07,7 and 9.2, respectively, and decreases gradually with the increase in photon energy. The extinction coefficient for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2 gradually increases in the IR region, reaching their maximum values of 2.56, 4.27, 2.42, and 3.28, respectively. Following this, they slowly decrease in the visible and UV regions. Figure S-1(b) illustrates the photoconductivity (σ) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) across various photon energies. The photoconductivity (σ) parameter quantifies the impact of photon irradiation on a material's electrical conductivity. Similar to the absorbance coefficient (α) spectrum, the σ spectrum aligns with the metallic nature of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). #### 4 Conclusions In summary, we employed DFT calculations to explore four 212 MAB phases, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), and investigated the structural, electronic, mechanical, lattice dynamical, and optical properties to predict their possible applications. The phonon dispersion curves, formation energy, and elastic constants collectively suggest that the M2AB<sup>2</sup> boride maintains dynamic, mechanical, chemical, and thermodynamic stability. The electronic band structure and density of states (DOS) offer evidence supporting the metallic nature of the compounds under investigation. Concurrently, the charge density mapping and atomic Mulliken population both confirm the presence of a strong B-B covalent bond. The stiffness constants, elastic moduli, findex, fracture toughness (KIC), Pugh's ratio (G/B), hardness parameters, and Cauchy Pressure (CP) of M2AB<sup>2</sup> were computed and compared with those of their 211 equivalents. We found that the values of the 212-phase borides are higher than those of the 211-phase carbides or borides. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are identified as brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit ductile characteristics, as indicated by Poisson's ratio (ν) and Pugh's ratio (B/G or G/B). The elastic characteristics display anisotropy due to the distinct atomic configurations along the aand c-directions. Vickers hardness calculations are considered reliable indicators of material hardness. The results suggest that Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> possess more pliable characteristics than Ta2GaB<sup>2</sup> and Ta2GeB2. The high hardness values of Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compared to the Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> due to higher BOP values of M-B bonding in Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> than in Mo2GaB<sup>2</sup> and Mo2GeB2. The value of minimum thermal conductivity (Kmin), thermal expansion coefficient, and melting temperature (Tm) collectively suggest the potential suitability of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) as a material for thermal barrier coating (TBC) applications in high-temperature devices. The optical conductivity and absorption coefficient corroborate the findings of the electronic band structure. Reflectivity is notably high in infrared (IR) regions and remains nearly constant in the visible and moderate ultraviolet (UV) regions, with an average value exceeding 44% for Ta2GaB2. This suggests that Ta2GaB<sup>2</sup> can be effectively utilized as a coating material to reduce solar heating. We expect that the comprehensive analysis of the diverse physical characteristics of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) presented in this study will establish a robust foundation for future theoretical and experimental explorations of these fascinating MAB phases. ## CRediT author statement A. K. M Naim Ishtiaq and Md Nasir Uddin: Data curation, Writing- Original draft preparation. Md. Rasel Rana, Shariful Islam and Noor Afsary: Reviewing and Editing. Md. 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Fig. 2 – Phonon DOS and Phonon dispersion curves of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB² compounds.	# Exploration of new 212 MAB phases: M2AB2 (M=Mo, Ta; A=Ga, Ge) via DFT calculations #### Abstract The recently developed MAB phases, an extension of the MAX phase, have sparked interest in research among scientists because of their better thermo-mechanical properties. In this paper, we have explored four new MAB phases M2AB2 (M=Mo, Ta and A=Ga, Ge) and studied the elastic, electronic, thermal, and optical properties to predict the possible applications. The stability of the new phases has been confirmed by calculating formation energy (Ef), formation enthalpy (∆H), phonon dispersion curve (PDC), and elastic constant (Cij). The study reveals that M2AB<sup>2</sup> (M=Mo, Ta and A=Ga, Ge) exhibit significantly higher elastic constants, elastic moduli, and Vickers hardness values than their counterpart 211 borides. Higher Vickers hardness values of Ta2AB2 (A=Ga, Ge) than Mo2AB2 (A=Ga, Ge) have been explained based on the values of the bond overlap population. The analysis of the density of states and electronic band structure revealed the metallic nature of the borides under examination. The thermodynamic characteristics of M2AB2 (M=Mo, Ta and A=Ga, Ge) under high temperatures (0–1000 K) are investigated using the quasi-harmonic Debye model. Critical thermal properties such as melting temperature (Tm), Grüneisen parameter (γ), minimum thermal conductivity (Kmin), Debye temperature (ΘD), and others are also computed. Compared with 211 MAX phases, the 212 phases exhibit higher values of (ΘD) and Tm, along with a lower value of Kmin. These findings suggest that the studied compounds exhibit superior thermal properties that are suitable for practical applications. The optical characteristics have been examined, and the reflectance spectrum indicates that the materials have the potential to mitigate solar heating across various energy regions. #First two authors contributed equally. ### 1. Introduction The MAX phase has garnered significant attention in the present era due to its outstanding mechanical and thermal characteristics at high temperatures, showcasing attributes shared by both metals and ceramics. The increased interest in MAX phase materials can be traced back to Barsoum's noteworthy contributions [1], [2]. The term MAX phase represents a family of multilayer solids where M is an earlier transition metal, A is an element from the IIIA or IVA group of the periodic table, and X is an atom of C/N/B [3]. MAX phase materials showcase metallic behavior due to alternate metallic A-layers and ceramic behavior attributed to the MX layers [4]. Like most metals and alloys, MAX phase materials have excellent thermal shock resistance, superior machinability, and enhanced thermal and electrical conductivities. On the other hand, they have high melting or decomposition temperatures and strong elastic stiffness, similar to many ceramics[5]. The unique combination of metallic and ceramic properties makes them versatile, with applications ranging from high-temperature coatings to nuclear accidenttolerant fuel (ATF), concentrated solar power (CSP), catalysis, and as precursors for MXenes [6], [7], [8], [9]. The diversity of MAX phases was confined to C and N as X elements for years (from 1960 to 2014). However, recent advancements have overcome this limitation by successfully synthesizing MAX phases that contain B [10]. The physical and chemical properties of B and Bcontaining compounds highlight the potential of MAX phase borides, which replace C/N with boron [11]. Due to the presence of B in the composition, they are also called the MAB phase. Furthermore, scientific communities are actively working to broaden the diversity of MAX phases by introducing structural changes, as seen in examples like Cr3AlB<sup>4</sup> (space group Immm), Cr4AlB<sup>6</sup> (space group Cmmm), Cr4AlB<sup>4</sup> [12], [13], [14]. Khazaei et al. reported on the first investigation of the theoretical MAX phase borides M2AlB (where M = Sc, Ti, Cr, Zr, Nb, Mo, Hf, or Ta). They evaluated these compounds' electronic structure, mechanical characteristics, and dynamical stability in their investigation [15]. The diversity of MAX phases is a subject of research interest due to their structure variations and the number of atoms in the compounds, resulting in changes to their characteristics. In examples like the 211 MAX phase [16], 312 MAX phase [8][17], 212 MAX phase [3], and i-MAX phase [18], the recent additions to the MAX family include the 314 MAX phase and 212 MAX phase, where the element B has been incorporated as an X element [3] [19]. This introduces a novel aspect to the MAX phase family, contributing to its diversity. The unique structural features of MAB phases include both orthorhombic and hexagonal symmetry observed in crystals. These distinctive symmetries set MAB phases apart from the typical MAX phases. Yinqiao Liu's synthesis of the orthorhombic phase of M2AlB2, particularly the 212-MAB phases with M = Sc, Ti, Zr, Hf, V, Nb, Cr, Mo, W, Mn, Tc, Fe, Co, and Ni, has revealed unique structural stability and notable electrical and mechanical properties [20]. The MAB phase structure slightly differs from the standard MAX phases, which typically crystallize in the hexagonal system with a space group -6<sup>2</sup> (No. 187). Ali et al [3]. investigated the diverse physical properties of Zr2AB<sup>2</sup> (A = In, Tl). Martin Ade et al [13] synthesized ternary borides, namely Cr2AlB2, Cr3AlB4, and Cr4AlB6, and subsequently compared their mechanical properties. Qureshi et al. [19] investigated the 314 Zr3CdB<sup>4</sup> MAX phase boride, calculating its mechanical, thermodynamic, and optical properties. The 314 MAX phase Hf3PB<sup>4</sup> has been thoroughly studied using Density Functional Theory (DFT) and revealed that it was the hardest MAX phase compound discovered until that date [21]. The structure of 212 phases exhibits a slight deviation from 211 MAX phases. In the case of 212 phases, a 2D layer of B is situated between M layers, featuring an additional B atom at the X position, unlike in 211 phases [22] [23], [24]. The B-B bonding in 212 MAX phase borides has improved mechanical and thermal properties. To date, the physical properties of Zr2AB<sup>2</sup> (A = In, Tl) [25] and Hf2AB<sup>2</sup> (A = In, Sn) [26], M2AB (M= Ti, Zr, Hf; A=Al, Ga, In) [27] MAX phases, as well as Nb2AC (A = Ga, Ge, Tl, Zn, P, In, and Cd) MAX phases, have been investigated using density functional theory (DFT). In each instance, the mechanical properties of B-containing compounds show significant improvement compared to their traditional C/N containing 211 MAX phases. The Debye temperature and melting temperature are higher for boron-containing 212 phases than for 211 carbides/nitrides, while the minimum thermal conductivity is lower. The thermal expansion coefficient of borides remains well-suited for use as coating materials. Consequently, the superior thermomechanical properties of B-containing 212 MAX phases demonstrate their suitability for high-temperature technological applications, surpassing the commonly used 211 MAX phase carbides. It should be noted that the 212 MAB phase with a hexagonal structure has already been synthesized [24]. In addition, Ga and Ge-based MAX phases have also been synthesized previously [28]. Thus, the reports on the synthesis of 212 phase and Ga and Gebased MAX phases motivated us to select the 212 MAB phases: M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) for our present study, and we have performed an in-depth investigation of their physical properties through DFT method. Therefore, in this paper, the first-time prediction of the stability and mechanical, electronic, thermal, and optical properties of M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) phases has been presented. The results revealed that M2AB<sup>2</sup> (M = Mo, Ta; A = Ga, Ge) compounds are stable and suitable for thermal barrier coating (TBC) and reflection coating applications. Additionally, to provide a comparison, the properties determined for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) are compared with those of other 212 and 211 compounds MAX phase borides. ### 2. Methods of calculations First principles density-functional theory (DFT) computations are conducted utilizing the Cambridge Serial Total Energy Package (CASTEP) module integrated within Materials Studio 2017 [29], [30]. The exchange-correlation function is estimated using the Generalized Gradient Approximation (GGA) method, originally suggested by Perdew, Burke, and Ernzerhof [31]. Pseudo-atomic simulations accounted for electronic orbitals corresponding to B (2s 2 2p 1 ), Ga (3d <sup>10</sup> 4s 2 4p 1 ), Ge (4s 2 4p 2 ), Mo (4s 2 4p 6 4d 5 5s 1 ), and Ta (5d 3 6s 2 ). The energy cutoff and k-point grids were established at 650 eV and 11 × 11 × 4, respectively. The structural relaxation was performed utilizing the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique [32], while the electronic structure was computed employing density mixing. The parameters for relaxed structures incorporate the following tolerance thresholds: the self-consistent convergence of the total energy is set at 5 × 10-6 eV/atom, the maximum force exerted on the atom is limited to 0.01 eV/Å, the maximum ionic displacement is constrained to 5 × 10-4 Å, and a maximum stress threshold of 0.02 GPa is imposed. The finite strain method [33], grounded in density functional theory (DFT), is utilized to compute the elastic properties within this framework. All necessary equations for determining various properties are provided in the supplementary document. #### 3 Results and discussion #### 3.1 Structural properties The M2AB<sup>2</sup> compounds (where M=Mo or Ta; A=Ga or Ge) belong to the P6m<sup>2</sup> (No. 187)[23] space group and crystallize in the hexagonal system. Unlike conventional MAX phases, which typically belong to the P63/mmc (194) space group, the 212 MAX phases exhibit distinct characteristics. In Fig. 1, the unit cell structure of Mo2GaB<sup>2</sup> is depicted as a representative of M2AB<sup>2</sup> alongside Mo2GaB, facilitating a comparison to discern their differences easily. The atomic positions are as follows: M (Mo or Ta) at (0.3333, 0.6667, 0.6935), A at (0.6667, 0.3333, 0.0), and two B atoms positioned at (0.6667, 0.3333, 0.5) and (0.0, 0.0, 0.5). The B components are arranged at the corners of the unit cell in typical 211 MAX phases, but in 212 boride MAX phases, they form a 2D layer between the M layers. This structural arrangement results in B-B covalent bonds in the 2D layer, enhancing stability compared to conventional 211 MAX phases. ![Fig. 1 - The schematic unit cell of (a) Mo2GaB<sup>2</sup> and (b) Mo2GaB compound.](path) The calculated lattice parameters of M2AB<sup>2</sup> are presented in Table 1 alongside those of other 211 and 212 MAX phases for comparison, demonstrating consistency with prior results [16], [34], [35] and affirming the accuracy of the computational methodology used. The primary distinction between 212- and 211-unit cell structures arises from differences in the lattice parameter c, where the c value for 211 exceeds that of 212. Moreover, the volumes of Mo- and Ta-based 211 MAX phase borides surpass those of 212 MAX phase borides. #### 3.2 Stability Examining a compound's stability is significant for multiple purposes, as it yields valuable information regarding the compound's synthesis parameters and aids in assessing the material's resilience across diverse environments, including thermal, compressive, and mechanical pressures. In this section, we delve into a comprehensive theoretical analysis concerning the chemical, dynamic, and mechanical stability of M2AB<sup>2</sup> compounds. The compound's chemical stability is determined by computing its formation energy by the following equation [36]: 2<sup>2</sup> = 22−( + + ) ++ . In the context provided, 22 represents the total energy of the compound after optimizing the unit cell. , , and denote the energies of the individual elements M, A, and B, respectively. The variables x, y, and z correspond to the number of atoms in the unit cell for M, Ga, and B, respectively. For Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, the calculated formation energy (Ef) are -1.8079 eV/atom, -1.8859 eV/atom, -2.1252 eV/atom, and -2.1933 eV/atom, respectively. The negative values signify the chemical stability of all compounds. The order of chemical stability can be expressed as Ta2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Mo2GaB2, indicating that Ta2GeB<sup>2</sup> is the most stable. Additionally, it's observed that MAX phase borides containing Ta are more stable than those containing Mo. Negative formation energy alone may not fully explain the chemical stability of M2AB2 (M=Mo, Ta and A=Ga, Ge). We calculated its formation enthalpy by examining potential pathways to evaluate its thermodynamic stability. For this analysis, we used the experimentally identified stable phases of MoB[37] and TaB[38], Ga4Mo[39], and B2Mo[40]. The potential decomposition pathways for our compounds, as determined from the Open Quantum Materials Database (OQMD), are outlined below. ![We have calculated the reaction energy as follows [41]:](path) Where, M=Mo, Ta and A=Ga, Ge. The following formula can calculate the decomposition energy associated with the reaction energy. = . Where n is the number of participating atoms. The calculated decomposition energies for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> are -0.16, -0.17, -0.26, and -0.28 meV/atom, respectively. So, we can state that the M2AB2 (M=Mo, Ta and A=Ga, Ge) system exhibits thermodynamic stability. Phonon dispersion curves (PDCs) have been calculated at the ground state utilizing the density functional perturbation theory (DFPT) linear-response approach to evaluate the dynamic stability of the MAX phase borides under investigation [42]. The PDCs, depicting the phonon dispersion along the high symmetry directions of the crystal Brillouin zone (BZ), along with the total phonon density of states (PHDOS) of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, are illustrated in Fig. 2(a, b, c and d). Analysis of the PDCs reveals no negative phonon frequencies for any of the compounds, indicating their dynamic stability. The PHDOS of the M2AB<sup>2</sup> compounds are derived from the PDCs and are presented alongside the PDCs in Fig. 2(a, b, c, and d), facilitating band identification through comparison of corresponding peaks. From Fig. 2, it is observed that in Mo2GaB2, the flatness of the bands for the Transverse Optical (TO) modes results in a prominent peak in the PHDOS, whereas non-flat bands for the Longitudinal Optical (LO) modes lead to weaker peaks in the PHDOS. Similar trends are observed in Mo2GeB2, Ta2GaB2, and Ta2GeB2. Notably, a distinct discrepancy arises between the optical and acoustic branches, with the top of the LO and bottom of the TO modes situated at the G point, with separations of 7.49 THz, 6.39 THz, 9.39 THz, and 8.51 THz for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, respectively. ![Fig. 2 – Phonon DOS and Phonon dispersion curves of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) Materials are subjected to various forces and loads in practical applications, necessitating understanding their mechanical stability. The mechanical stability of a compound can be assessed using stiffness constants. For a hexagonal system, the conditions for mechanical stability are as follows [43]: C<sup>11</sup> > 0, C<sup>11</sup> > C12, C<sup>44</sup> > 0, and (C<sup>11</sup> + C12)C<sup>33</sup> - 2(C13)² > 0. As indicated in Table 2, the Cij values of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) satisfy these conditions, thus confirming the mechanical stability of herein predicted phases: M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge). #### 3.3 Electronic properties Analyzing the electronic band structure (EBS) is crucial for gaining insights into the electronic behavior of a compound. The EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) MAX phases are depicted in Fig. 3(a, b, c, and d), with the Fermi energy (EF) level set at 0 eV, represented by a horizontal line. Observing the EBS of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge), it is evident that the conduction band overlaps with the valence band, indicating the absence of a band gap. This observation confirms that the M2AB<sup>2</sup> compounds exhibit metallic behavior, which aligns with conventional MAX phases. The red lines illustrate the overlapping band at the Fermi level. Fig. 3 (a, b) shows that for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> compounds, the maximum band overlap occurs along the A-H path. Conversely, in Fig. 3(c, d), the maximum band overlap is observed along the G-M paths. Utilizing the band structure, we can analyze the electrical anisotropy of M2AB<sup>2</sup> MAX phase compounds. The anisotropic nature can be understood by analyzing the energy dispersion in the basal plane and along the c-axis. The paths G-A, H-K, and M-L show energy dispersion along the c-direction, and A-H, K-G, G-M, and L-H show energy dispersion in the basal plane. In comparison to the paths A-H, K-G, G-M, and L-H (basal plane), there is less energy dispersion along the lines G-A, H–K, and M-L (c-direction), as shown by Fig. 2(a, b, c and d). Lower energy dispersion in the c-direction results from a higher effective mass [44], indicating the strong electronic anisotropy of the M2AB<sup>2</sup> MAX phase compound. Consequently, conductivity along the c-axis is expected to be lower than in the basal planes. These findings are consistent with prior studies [34], [45]. ![Fig. 3 – Band structure of (a) Mo2GaB2, (b) Mo2GeB2, (c) Ta2GaB2, and (d) Ta2GeB<sup>2</sup> compounds.](path) To investigate the bonding nature and electronic conductivity, the total density of states (TDOS) and partial density of states (PDOS) of M2AB<sup>2</sup> compounds were computed. Figure 4 (a, b, c, and d) illustrates the TDOS and PDOS of these compounds, with the Fermi energy (EF) set at zero energy level, indicated by a straight line. These profiles exhibit typical characteristics of MAX phase materials. The Mo or Ta-d electronic states predominantly contribute to the Fermi level, with a minor contribution from the B-p and Ga or Ge-p electronic states. To ascertain the hybridization characteristics of various electronic states within the valence band, the energy spectrum of the valence band has been partitioned into two distinct segments. The first segment encompasses the lower valence band region spanning from -7 eV to -3.5 eV, originating from the hybridization of Mo-p, Mo-d, and B-s orbitals in the case of the Mo2GaB<sup>2</sup> compound. Conversely, for the Mo2GeB2 compound, the lower valence band region arises from the hybridization of Mo-d, Ge-p, and B-s orbitals. Notably, for both the Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compounds, the dominance of the B-p state characterizes the lower valence band region. The second segment pertains to the upper valence band region from -3.5 eV to 0 eV. In the case of the Mo2GaB<sup>2</sup> compound, this region arises from the hybridization of Mo-p and Mo-d orbitals. However, for the Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> compounds, the upper valence band region results from the hybridization of Mo-d orbitals and (Ga/Ge)-p orbitals. Notably, the Fermi level of M2AB<sup>2</sup> resides near the pseudogap in the TDOS profile, indicating a high level of electronic stability. Similar trends are observed in other compounds like Zr2AlN, V2AlN, Sc2AlB, Sc2GaB, Ta2GaB, and Hf2GaB<sup>2</sup> [15], [35], [45]. ![Fig. 4 – TDOS and PDOS of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) The electron charge density mapping is helpful in understanding the distribution of electron densities linked to chemical bonds. It delineates areas of positive and negative charge densities, signifying the development and exhaustion of electrical charges, respectively. As depicted in the map, covalent bonds become apparent by accumulating charges between two atoms. Furthermore, the presence of ionic bonds can be inferred from a balance between negative and positive charges at specific atom positions[46]. The valence electronic CDM, denoted in units of eÅ-3 , for M2AB<sup>2</sup> (where M = Mo, Ta; A = Ga, Ge) is showcased in Fig. 5(a, b, c, and d) along the (110) crystallographic plane. The accompanying scale illustrates the intensity of electronic charge density, with red and blue colors indicating low and high electronic charge density, respectively. As depicted in Fig. 5(a, b, c, and d), it is evident that charges accumulate in the regions between the B sites. Consequently, it is anticipated that strong covalent B‒B bonding occurs through the formation of two center-two electron (2c‒2e) bonds in the M2AB<sup>2</sup> (where M=Mo, Ta; A=Ga, Ge) compound, similar to other 212 MAX phase borides like Ti2PB2, Zr2PbB2, Nb2SB2, Zr2GaB<sup>2</sup> and Hf2GaB<sup>2</sup> [34], [45]. Mulliken analysis has corroborated the charge transfer from Mo/Ta atoms to B atoms. The charge received from Mo/Ta atoms is distributed among the B atoms positioned at the transitions and those located at the edges, facilitating the formation of a two center-two electron (2c‒2e) bond between B atoms within the 2D layer of B, as illustrated in Fig. 1(a). The hardness of each bond value presented in Table 5 also aligns with the results obtained from charge density mapping (CDM) and our analysis using moduli and elastic stiffness constants. ![Fig. 5 – The charge density mapping of (a) Mo2GaB2, (b) Mo2GeB2, (c)Ta2GaB2, and (d)Ta2GeB<sup>2</sup> compounds.](path) ## 3.4.1 Mechanical properties The mechanical properties of materials play a pivotal role in determining their potential applications, serving as crucial indicators of their behavior and suitability in materials engineering endeavors. These properties are equally applicable to MAX phase materials. Initially, to assess the mechanical properties of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phases, we employed the stress-strain method within the CASTEP code to compute the elastic constants (Cij) [22], [47], [48]. These calculated elastic constants (Cij) are presented in Table 2, alongside those of other 212 and 211 MAX phases. Due to the hexagonal crystal structure of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) phase borides, five stiffness constants emerge: C11, C12, C13, C33, and C<sup>44</sup> [49]. The mechanical stability was evaluated using these stiffness constants in the preceding section. For instance, C<sup>11</sup> and C<sup>33</sup> determine stiffness when stress is exerted along the (100) and (001) directions, respectively, whereas C<sup>44</sup> evaluates resistance to shear deformation on the (100) and (001) planes. The stiffness constants C<sup>11</sup> and C<sup>12</sup> directly reflect the strength of atomic bonds along the a- and c-axes. When C<sup>11</sup> exceeds C<sup>33</sup> (or vice versa), it signifies stronger atomic bonding along the a-axis (or c-axis). In Table 2, for Mo2GaB2, Mo2GeB2, and Ta2GeB<sup>2</sup> compounds, C<sup>11</sup> surpasses C33, indicating superior atomic bonding along the a-axis compared to the c-axis. This robust bonding along the a-axis suggests heightened resistance to a-axial deformation. Conversely, in Ta2GeB2, where C<sup>11</sup> < C33, stronger atomic bonding along the c-axis translates to increased resistance against c-axial deformation. Analysis of Table-2 reveals that the values of C<sup>11</sup> and C<sup>22</sup> are notably higher for 212 MAX phase borides than 211 MAX phase borides. Consequently, it can be inferred that 212 MAX phases exhibit stronger resistance to axial deformation when juxtaposed with 211 MAX phases. C<sup>44</sup> is commonly utilized to gauge shear deformation tolerance among the elastic constants. Notably, Ta2GaB exhibits superior shear deformation resistance owing to its highest C<sup>44</sup> value among the studied compounds. Furthermore, the individual elastic constants C12, C13, and C<sup>44</sup> denote shear deformation response under external stress. The observation that C<sup>11</sup> and C<sup>33</sup> possess larger magnitudes than C<sup>44</sup> implies that shear deformation is more facile than axial strain. Another crucial parameter derived from the stiffness constants is the Cauchy pressure (CP), calculated as C<sup>12</sup> - C44, which provides vital insights relevant to the practical applications of solids [51]. Pettifor [51] emphasized the significance of Cauchy pressure (CP) in discerning the chemical bonding and ductile/brittle properties of solids. A negative CP value indicates covalently bonded brittle solids, whereas a positive value signifies isotropic ionic ductile solids. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> fall into covalently bonded brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit metallic ductile characteristics. Similarly, like Ga-containing 211 MAX phase borides, Ga-containing 212 MAX phases also demonstrate negative CP values and brittle behavior [52][53]. The elastic constants obtained are used to produce several bulk elastic parameters that are used to characterize polycrystalline materials, such as Young's modulus (Y), bulk modulus (B), and shear modulus (G). In Table 2, the bulk modulus (B) and shear modulus (G) calculated using Hill's approximation [54] are also presented. Hill's values represent the average of the upper limit (Voigt [55]) and lower limit (Reuss [56]) of B. The necessary equations for these calculations are provided in the supplementary document (S1). Young's modulus (Y) is a crucial indicator of material stiffness. A higher Y value indicates a stiffer material. It can be observed from the table that Mo2GaB<sup>2</sup> exhibits the highest Y value, signifying its greater stiffness compared to others. According to the sequence of Y values, stiffness can be ranked as follows: Mo2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GaB<sup>2</sup> > Ta2GeB2. Therefore, Mo2GaB2, with its higher Young's modulus, is anticipated to demonstrate superior mechanical stability and deformation resistance compared to the other compounds under investigation. This quality is essential for aircraft parts or high-performance machinery applications where critical dimensional stability and structural integrity are required [57], [58]. Table 2 shows that 212 MAX phase materials exhibit larger Young's modulus (Y) values than 211 MAX phases. This suggests that 212 MAX phase borides are stiffer than their 211 MAX phase counterparts. Young's modulus (E) also correlates well with thermal shock resistance (R): R ∝ 1/E [59]. Lower Young's modulus values correspond to higher thermal shock resistance. Therefore, materials with higher thermal shock resistance (i.e., lower Young's modulus) are more suitable for use as Thermal Barrier Coating (TBC) materials. Given that Ta2GaB<sup>2</sup> possesses the lowest Y value among the materials studied, it should be considered a superior candidate for TBC material due to its higher thermal shock resistance. The material's ability to withstand shape distortion is elucidated by the shear modulus (G). On the other hand, the bulk modulus (B) indicates the strength of a material's chemical bonds and its ability to withstand uniform compression or volume change. We computed Young's modulus (Y), bulk modulus (B), and shear modulus for our analysis. Based on the bulk modulus values presented in Table 2, the sequence of a material that resists compression when pressure is applied can be outlined as follows: Mo2GeB<sup>2</sup> > Ta2GeB<sup>2</sup> > Mo2GaB<sup>2</sup> > Ta2GeB2. Mo2GeB2, boasting a higher bulk modulus, may exhibit reduced plastic deformation and superior resistance to stress-induced deformation compared to other compounds examined. The ductile and brittle characteristics of the M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds can also be assessed through Poisson's ratio (ν) and Pugh's ratio (G/B). A compound demonstrates ductile (brittle) behavior if the ν value surpasses (falls below) 0.26 [60]. Furthermore, if the G/B value exceeds (is less than) 0.571, then the compound exhibits brittle (ductile) behavior [19]. According to both criteria, Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are classified as brittle compounds, whereas Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> are categorized as ductile compounds. The ductile nature of Ge-based MAX phases has been reported previously [61]. Fracture toughness (KIC) is a vital property that gauges a material's ability to resist crack propagation. In the case of M2AB<sup>2</sup> (M=Mo, Ta; A=Ga, Ge) compounds, Equation (S2) was employed to determine KIC. The KIC values are documented in Table 2, and notably, these values surpass those of 211 MAX phases [16], [35]. Another parameter, the "f-index," characterizes the isotropic nature and strength of atom-atom bonds within a single hexagonal crystalline lattice along the a- and c-directions. If the f-index is less than 1, chemical bonds exhibit greater rigidity along the c-axis; conversely, if f exceeds 1, bonds are more rigid in the ab-plane. When the value of f is set to 1, atomic bonds exhibit similar strength and uniformity in all directions [15]. The fvalue is computed using Equation (S3) and displayed in Table 2. Table 2 shows that the f-values of Mo2GeB2, Ta2GaB2, and Ta2GeB2, which are close to one, indicate a slight anisotropic bonding strength. Nevertheless, substances with robust bonds in the horizontal plane (ab plane) (f > 1), such as Mo2GaB2, are deemed optimal candidates for the exfoliation process. In engineering applications, the hardness of a solid material serves as a valuable criterion for designing various devices. The elastic properties of polycrystalline materials can be utilized to calculate hardness values, as the ability to resist indentation is closely linked to a material's hardness. Both micro-hardness (Hmicro) and macro-hardness (Hmacro) were computed using Equation (S4) and are presented in Table 2. Based on the values of Hmicro and Hmacro, Mo2GaB<sup>2</sup> emerges as the toughest among the studied phases. The order of hardness is as follows: Mo2GaB<sup>2</sup> > Ta2GaB<sup>2</sup> > Mo2GeB<sup>2</sup> > Ta2GeB2. Interestingly, Ga-containing MAX phase borides exhibit superior hardness compared to Ge-containing MAX phase borides. In Table 2, we compare our findings with previously reported MAX phases and observe that our data align perfectly with the earlier results. #### 3.4.2 Elastic anisotropy Additionally, anisotropy is linked to other crucial events like anisotropic plastic deformation and the development and spread of micro-cracks within mechanical stress. By providing directiondependent elastic constants, the understanding of anisotropy also offers a framework for improving the mechanical stability of materials in extreme circumstances. The following formulae are used for hexagonal structures to calculate the various anisotropic variables from elastic constants Cij [62]. $A\_{1} = \frac{\frac{1}{6}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{44}};$ $A\_{2} = \frac{2\mathcal{C}\_{44}}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ and $A\_{3} = A\_{1}.$ $A\_{2} = \frac{\frac{1}{3}(\mathcal{C}\_{11} + \mathcal{C}\_{12} + 2\mathcal{C}\_{33} - 4\mathcal{C}\_{13})}{\mathcal{C}\_{11} - \mathcal{C}\_{12}}$ Table 3 lists every anisotropy parameter. Since the value of A<sup>i</sup> should be 1 to be isotropic, the computed value of A<sup>i</sup> (i = 1-3) indicates that all of the compounds under research exhibit anisotropic behavior [63]. An additional method for estimating elastic anisotropy is to use the percentage anisotropy to compressibility and shear (A<sup>B</sup> & AG). This gives polycrystalline materials a helpful way to measure elastic anisotropy. They have been described as [45]; Zero values for AB, AG, and the universal anisotropy factor (AU) show elastic isotropy; the maximum amount of anisotropy is represented by a value of 1. Table 3 shows that, compared to other compounds, the values of AB, AG, and A U for Mo2GeB<sup>2</sup> and Ta2GaB<sup>2</sup> are incredibly close to zero, suggesting that these compounds have nearly isotropic characteristics. #### 3.4.3 Mulliken Populations The Mulliken charge assigned to an atomic species quantifies the effective valence by calculating the absolute difference between the formal ionic charge and the Mulliken charge. Equations (S5) and (S6) are employed to ascertain the Mulliken charge for each atom (α). Table-4 provides the Mulliken atomic population and effective valence charge. Transition metals Mo and Ta in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have pure valence states of 4d<sup>5</sup> and 5d<sup>3</sup> , respectively. The d- orbital electrons of transition metals have been found to influence their effective valence charge significantly. A non-zero positive value indicates a combination of covalent and ionic attributes within chemical bonds. As this value decreases towards zero, it signifies a rise in ionicity. A zero value suggests an ideal ionic character in the bond. Conversely, a progression from zero with a positive value indicates an elevation in the covalency level of the bonds. Based on their effective valence, M atoms move from the left to the right in the periodic table, increasing the covalency of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). Table-4 reveals that the Mulliken atomic charge ascribed to the B atoms is solely negative. Conversely, positive Mulliken atomic charges are associated with transition metals (M) and A. This suggests a charge transfer from M and A to B for each compound within M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge), thereby fostering ionic chemical bonds among these atoms. Bond population serves as another indicator of bond covalency within a crystal, as a high value of bond population essentially signifies a heightened degree of covalency within the chemical bond. The bonding and anti-bonding states influence the populations with positive and negative bond overlap. As demonstrated in Table-5, the B-B bond exhibits greater covalency compared to any other bond in M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The presence of an antibonding state between two relevant atoms, which subsequently decreases their chemical bonding, is indicated by a hostile bond overlap population. In Mo2GeB<sup>2</sup> and Ta2GeB2, a hostile bond overlap population is observed in the Ge-Mo and Ge-Ta bonds, indicating the presence of an antibonding state. Therefore, the existence of ionic bonding is guaranteed by electronic charge transfer. On the other hand, the high positive value of the bond overlap population (BOP) denotes the presence of covalent bonding, a feature shared by materials in the MAX phase. ## 3.4.4 Theoretical Vickers Hardness Vickers hardness, derived from the atomic bonds found in solids, indicates how resistant a material is to deformation in extreme circumstances. Several variables influence this feature, such as the crystal flaws, solid structure, atomic arrangement, and bond strength. The Vickers hardness of the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) MAX phases is determined using the Mulliken bond population method, as described by Gou et al. [66] using the formula (S7-S10). This method is particularly suitable for partial metallic systems such as MAX phases. Table-5 lists the computed Vickers hardness values for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The calculated values are 3.35 GPa, 4.76 GPa, 8.21 GPa, and 8.74 GPa for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2, respectively. We observed that Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> have much higher hardness values than Mo2GaB<sup>2</sup> and Mo2GeB2. These values are also higher than that of other 212 phases, like, Ti2InB<sup>2</sup> (4.05 GPa) [50], Hf2InB<sup>2</sup> (3.94 GPa), and Hf2SnB<sup>2</sup> (4.41 GPa) [26], Zr2InB<sup>2</sup> (2.92 GPa) and Zr2TlB<sup>2</sup> (2.19 GPa) [25], Zr2GaB<sup>2</sup> (2.53GPa), Zr2GeB<sup>2</sup> (3.31GPa), Hf2GaB<sup>2</sup> (4.73GPa) and Hf2GeB2(4.83GPa) [67]The Hv calculated by the geometrical average of the individual bonding, where the bonding strength mainly determined by the BOP values. In the case of Ta2GaB<sup>2</sup> and Ta2GeB2, the BOP of M-B bonding is much higher compared to Mo2GaB<sup>2</sup> and Mo2GeB2. Even though the BOP of M-B for Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> is higher than that of the other 212 phases mentioned earlier. Therefore, higher values of Hv are expected for Ta2GaB<sup>2</sup> and Ta2GeB2. Additionally, we looked at the Vickers hardness value between 211 and 212 compounds and discovered that the 212 MAX phase compounds had a higher H<sup>V</sup> value. This is because a 2D layer of B atoms is positioned between the M atoms. The B atoms share two center-two electrons to form an extremely strong B-B bond [3]. Table 5 Calculated data for Mulliken bond number (n μ ), bond length (d μ ), bond overlap populations BOP, (P μ ), metallic populations (P μˊ), Vickers hardness (HV) M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. # 3.5 Thermal Properties MAX phases are ideal for high-temperature applications due to their exceptional mechanical qualities at elevated temperatures. As a result, researching the fundamental parameters necessary for predicting their application is of great interest, and these can be obtained from the vibrations of atoms or phonons. The Debye temperature (ΘD) of a solid is directly connected to its bonding strength, melting temperature, thermal expansion, and conductivity. Using the sound velocity and Anderson's technique [68], the Θ<sup>D</sup> of the phases under study has been computed using the formula (S11). Equation (S12) can be used to get the average sound velocity (Vm) from the longitudinal and transverse sound velocities. Equations (S13–14) were used to determine v<sup>l</sup> and vt. The calculated values of Debye's temperature are shown in Table 6, where Mo2GaB<sup>2</sup> has the highest Θ<sup>D</sup> and Ta2GeB<sup>2</sup> has the lowest. If we rank them, it is as follows: Mo2GaB<sup>2</sup> < Mo2GeB<sup>2</sup> < Ta2GaB2 < Ta2GeB2. Hadi et al. recently reported a MAX phase (V2SnC) as a TBC material with a Θ<sup>D</sup> value of 472 K [69]. Thus, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) exhibit encouraging potential as TBC materials, as shown in Table 6. Table 6-Data for density (ρ), longitudinal, transverse, and average sound velocities (vl, vt, and vm), Debye temperature (ΘD), minimum thermal conductivity (Kmin), Grüneisen parameter (γ), thermal expansion coefficient (TEC) at 300K and melting temperature (Tm) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds. The constant thermal conductivity value at high temperatures is the minimum thermal conductivity (Kmin). As the name implies, this conductivity is minimal because, at high temperatures, phonon coupling breaks. The formula (S15) for the minimum thermal conductivity of solids was derived using the Clarke model [70]. Table 6 lists the calculated value of kmin, with Ta2GaB<sup>2</sup> having the lowest value and Mo2GaB<sup>2</sup> having the highest. When selecting suitable materials for TBC applications, a minimum thermal conductivity of 1.25 W/mK is used as a screening criterion [71]. Our compounds exhibit lower minimum thermal conductivity values, holding promising potential as TBC materials. Gd2Zr2O<sup>7</sup> and Y2SiO5, two recently developed thermal barrier coating (TBC) materials, have minimal thermal conductivities (Kmin) of 1.22 W/m.K. and 1.3 W/m.K. [72], respectively, as confirmed by experiment. These numbers roughly match the values we computed for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). The Grüneisen parameter (γ) is a crucial thermal parameter that helps explain the anharmonic effects of lattice dynamics; solids utilized at high temperatures are expected to have lower anharmonic effects. The Grüneisen parameter (γ) can be determined with the help of Poisson's ratio using equation (S16) [73]. The computed γ values, as shown in Table 6, suggest that the compounds under investigation exhibit a weak anharmonic effect. Additionally, for solids with a Poisson's ratio between 0.05 and 0.46, the values similarly fall within the range of 0.85 and 3.53 [74]. The melting temperature (Tm) of the compounds under investigation has been calculated using the following formula (S17). The strength of atomic bonding is the primary factor determining the melting temperature of solids; the higher the Tm, the stronger the atomic bonding. The order of T<sup>m</sup> for the titled phases is found to mirror the Y-based (Young's modulus) order, indicating a close link between T<sup>m</sup> and Y [75]. As observed in Table 6, our compounds roughly follow the Ybased ranking with Mo2GeB<sup>2</sup> < Mo2GaB<sup>2</sup> < Ta2GeB<sup>2</sup> < Ta2GaB2. The T<sup>m</sup> value for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) is also comparable with the TBC material Y4Al2O<sup>9</sup> (2000 K)[75]. ![Fig. 6: Temperature dependence of the calculated thermodynamic parameters. (a) Specific heat at constant pressure, C<sup>p</sup> (b) Specific heat at constant volume, C<sup>v</sup> (c) Free energy, F (d) Entropy, S (e) Internal energy, E and (f) Thermal expansion coefficient, TEC of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge).](path) The temperature dependence of specific heats, Cv, Cp, for the M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) compounds was obtained using the formulas (S19-S20), as shown in Fig.6(a,b). Assuming that the quasi-harmonic model is accurate and that phase transitions are not anticipated for the compounds under study, these properties are approximated across a temperature range of 0 to 1000 K. Because phonon thermal softening occurs at higher temperatures, the heat capacity rises with temperature. Heat capacities increase quickly and follow the Debye-T 3 power law at lower temperatures. At higher temperature regimes, where C<sup>v</sup> and C<sup>p</sup> do not greatly depend on temperature, they approach the Dulong-Petit (3) limit.[76]. Using the quasi-harmonic approximation, various temperature-dependent thermodynamic potential functions for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) have been estimated at zero pressure and displayed in Fig.6 [77]. These functions include the Helmholtz free energy (F), internal energy (E), and entropy (S) within 0-1000 K temperature using equation (S21-S23). The free energy progressively decreases as the temperature rises, as shown in Fig.6(c). Free energy typically declines, and this trend becomes increasingly negative as a natural process proceeds. As demonstrated in Fig. 6(d), the internal energy (E) shows a rising trend with temperature, in contrast to the free energy. Since thermal agitation creates disorder, a system's entropy rises as temperature rises. This is illustrated in Fig.6(d). A material's thermal expansion coefficient (TEC) is derived from the anharmonicity in the lattice dynamics and can be found using equation (S18). The measure of a material's capacity to expand or contract with heat or cold is called the Thermal Expansion Coefficient, or TEC. As observed in Fig. 5(f), the Thermal Expansion Coefficient (TEC) increases rapidly up to 365 K. Then it approaches a constant value, which indicates lower saturations in the materials with temperature changes. The materials under study have a very low TEC value, a crucial characteristic of materials intended for application in high-temperature technology. To be effective as thermal barrier coating (TBC) materials, compounds must exhibit a low thermal conductivity (Kmin) to impede heat transfer, a high melting temperature to withstand extreme heat, and a low thermal expansion coefficient (TEC) to maintain dimensional stability under thermal stress. The compounds M2AB<sup>2</sup> (where M = Mo, Ta, and A = Ga, Ge) possess these properties, making them suitable candidates for use as TBC materials #### 3.6 Optical Properties Different materials exhibit unique behaviors when exposed to electromagnetic radiation. The optical constants determine the overall response of the sample to the incident radiation. The complex dielectric function, defined as ε(ω)=ε1(ω)+iε2(ω), is one of the main optical characteristics of solids. The following formula determines the imaginary part of the dielectric function ε2(ω) from the momentum matrix element between the occupied and unoccupied electronic states. In this formula, e stands for an electronic charge, ω for light angular frequency, u for the polarization vector of the incident electric field, and and for the conduction and valence band wave functions, respectively, at k. The Kramers-Kronig equation can estimate the real part of the dielectric function, ε1(ω). In contrast, ε2(ω) and ε1(ω) are utilized to evaluate all other optical parameters, such as the absorption coefficient, photoconductivity, reflectivity, and loss function [78]. In this part, several energy-dependent optical properties of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) 212 MAX phases are calculated and analyzed in detail for the photon energy range of 0 to 30 eV, for [100] plane enabling the first assessment of the compounds' practical applicability. Given the metallic conductivity of the MAX compounds' electronic structure, additional parameters were chosen to analyze the optical properties. These include a plasma frequency of 3 eV, damping of 0.05 eV, and Gaussian smearing of 0.5 eV [79]. Figure 7(a) shows the real component of the dielectric function, ε1, which exhibits metallic behavior. In metallic systems, ε<sup>1</sup> has a considerably high negative value in the low-energy range, with the real component reaching negative, which aligns with the band structure finding. Fig. 7(b) depicts the imaginary part of the dielectric function, ε2(ω), representing dielectric losses about frequency. Mo2GeB<sup>2</sup> demonstrates the highest peak in the low-energy region, with all compounds approaching zero from above at around 17 eV. This observation confirms Drude's behavior. ![Fig. 7- (a) Real part of dielectric constant, ε<sup>1</sup> (b) Imaginary part of dielectric constant, ε<sup>2</sup> (c) Reflectivity, R and (d) Loss function, (e) The coefficient of absorption, (α) (f) Refractive index, (n) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) for [100] electric field directions.](path) The reflectivity of a material measures the percentage of incident light energy reflected off it. Equation (S24) was employed to compute reflectivity using the dielectric function, as shown in Fig.7(c). The spectra's visible and infrared (IR) portions consistently exhibit values exceeding 55%. Consequently, the materials under examination are expected to appear metallic gray. In the infrared (IR) region, the maximum reflectivity for Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> is 65% and 74%, respectively, occurring at 0.33 eV. Conversely, for Ta2GaB2, the maximum reflectivity is 59% at 14.02 eV, while for Ta2GeB2, it is 65% at 14.78 eV in the UV region. After 18 eV, the reflectance decreases significantly. According to reports, substances with an average reflectivity value above 44% in the visible light region can effectively reduce solar heating by reflecting a significant amount of the solar spectrum. Among all the compounds we examined, Ta2GaB<sup>2</sup> has a reflectivity in the visible range of more than 44% [78]. Therefore, Ta2GaB<sup>2</sup> can be utilized as a coating material and should be able to mitigate solar heating. plasma frequencies (ωp). The plasma frequency for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is observed at 23 eV, 25 eV, 21 eV, and 23 eV, respectively. At this specific frequency, the absorption coefficient rapidly decreases, ε<sup>1</sup> crosses zero from the negative side, and the reflectance R(ω) displays a falling tail. Above this distinctive frequency, the materials are transparent to the incident electromagnetic radiation. Fig. 7(e) illustrates the absorbance coefficient (α) of compounds M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) determined using equation (S26). As α begins to rise from zero photon energy, the metallic nature of the substances under study is again indicated. The visible light region experiences a sharp increase in absorption, peaking in the UV region at around 14 eV and then progressively declining. The IR region exhibits negligible absorption. However, the materials mentioned above appear to have a significant absorption band primarily located in the visible and ultraviolet spectrum. This suggests that the materials can be used in UV surface-disinfection devices, medical sterilization equipment, and other optoelectronic device designs. Its refractive index is crucial to a material's potential application in optical devices like waveguides and photonic crystals. Equations (S27) and (S28) were used to derive the refractive index (n) and extinction coefficient (k) for M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), as depicted in Fig.7(c) and Fig.S-1(a). The variations of n and k in MAX phase carbides with incident photon energy closely resemble ε1(ω) and ε2(ω). The static refractive index n(0) for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB<sup>2</sup> is 9.4,13.07,7 and 9.2, respectively, and decreases gradually with the increase in photon energy. The extinction coefficient for Mo2GaB2, Mo2GeB2, Ta2GaB2, and Ta2GeB2 gradually increases in the IR region, reaching their maximum values of 2.56, 4.27, 2.42, and 3.28, respectively. Following this, they slowly decrease in the visible and UV regions. Figure S-1(b) illustrates the photoconductivity (σ) of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) across various photon energies. The photoconductivity (σ) parameter quantifies the impact of photon irradiation on a material's electrical conductivity. Similar to the absorbance coefficient (α) spectrum, the σ spectrum aligns with the metallic nature of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge). #### 4 Conclusions In summary, we employed DFT calculations to explore four 212 MAB phases, M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge), and investigated the structural, electronic, mechanical, lattice dynamical, and optical properties to predict their possible applications. The phonon dispersion curves, formation energy, and elastic constants collectively suggest that the M2AB<sup>2</sup> boride maintains dynamic, mechanical, chemical, and thermodynamic stability. The electronic band structure and density of states (DOS) offer evidence supporting the metallic nature of the compounds under investigation. Concurrently, the charge density mapping and atomic Mulliken population both confirm the presence of a strong B-B covalent bond. The stiffness constants, elastic moduli, findex, fracture toughness (KIC), Pugh's ratio (G/B), hardness parameters, and Cauchy Pressure (CP) of M2AB<sup>2</sup> were computed and compared with those of their 211 equivalents. We found that the values of the 212-phase borides are higher than those of the 211-phase carbides or borides. Mo2GaB<sup>2</sup> and Ta2GaB<sup>2</sup> are identified as brittle solids, while Mo2GeB<sup>2</sup> and Ta2GeB<sup>2</sup> exhibit ductile characteristics, as indicated by Poisson's ratio (ν) and Pugh's ratio (B/G or G/B). The elastic characteristics display anisotropy due to the distinct atomic configurations along the aand c-directions. Vickers hardness calculations are considered reliable indicators of material hardness. The results suggest that Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> possess more pliable characteristics than Ta2GaB<sup>2</sup> and Ta2GeB2. The high hardness values of Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> compared to the Mo2GaB<sup>2</sup> and Mo2GeB<sup>2</sup> due to higher BOP values of M-B bonding in Ta2GaB<sup>2</sup> and Ta2GeB<sup>2</sup> than in Mo2GaB<sup>2</sup> and Mo2GeB2. The value of minimum thermal conductivity (Kmin), thermal expansion coefficient, and melting temperature (Tm) collectively suggest the potential suitability of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) as a material for thermal barrier coating (TBC) applications in high-temperature devices. The optical conductivity and absorption coefficient corroborate the findings of the electronic band structure. Reflectivity is notably high in infrared (IR) regions and remains nearly constant in the visible and moderate ultraviolet (UV) regions, with an average value exceeding 44% for Ta2GaB2. This suggests that Ta2GaB<sup>2</sup> can be effectively utilized as a coating material to reduce solar heating. We expect that the comprehensive analysis of the diverse physical characteristics of M2AB<sup>2</sup> (M = Mo, Ta, and A = Ga, Ge) presented in this study will establish a robust foundation for future theoretical and experimental explorations of these fascinating MAB phases. ## CRediT author statement A. K. M Naim Ishtiaq and Md Nasir Uddin: Data curation, Writing- Original draft preparation. Md. Rasel Rana, Shariful Islam and Noor Afsary: Reviewing and Editing. Md. Ashraf Ali: Methodology, Reviewing and Editing, conceptualization, supervision; and Karimul Hoque: conceptualization, supervision, editing, and reviewing. # Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. # Acknowledgments The authors acknowledge Physics Discipline, Khulna University, Khulna for the logistic support and Advanced Computational Materials Research Laboratory (ACMRL), Department of Physics at Chittagong University of Engineering & Technology (CUET), Chattogram-4349, Bangladesh for laboratory facilities. - [1] M. W. Barsoum, G. Yaroschuk, and S. Tyagi, "Fabrication and characterization of M2SnC (M = Ti, Zr, Hf and Nb)," Scr. Mater., vol. 37, no. 10, pp. 1583–1591, 1997, doi: 10.1016/S1359-6462(97)00288-1. - [2] M. W. Barsoum and T. 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Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening.	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. Schnyder, Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery, Phys. Rev. Mat. 5, 124202 (2021). - 2. X. Zhou, Ch.-H. Hsu, H. Aramberri, M. Iraola, Ch.-Yi Huang, J. L. Mañes, M. G. Vergniory, H. Lin, and N. Kioussis, Novel family of topological semimetals with butterflylike nodal lines, Phys. Rev. B 104, 125135 (2021). - 3. J. Wang, H. Yuan, Z.-M. Yu, Z. Zhang, and X. Wang, Coexistence of symmetryenforced phononic Dirac nodal-line net and three-nodal surfaces phonons in solid-state materials: Theory and materials realization, Phys. Rev. Mat. 5, 124203 (2021). - 4. J. Bannies, E. Razzoli, M. Michiardi, H.-H. Kung, I. S. Elfimov, M. Yao, A. Fedorov, J. Fink , C. Jozwiak, A. Bostwick, E. Rotenberg, A. Damascelli , and C. Felser Extremely large magnetoresistance from electron-hole compensation in the nodal-loop semimetal ZrP2, Phys. Rev. B 103, 155144 (2021). - 5. A. S. Wadge, K. Zberecki, B. J. Kowalski, D. Jastrzebski, P. K. Tanwar, P. Iwanowski, R. Diduszko, A. Moosarikandy, M. Rosmus, N. Olszowska, and A. Wisniewski, Emergent impervious band crossing in the bulk in topological nodal line semimetal ZrAs<sup>2</sup> [https://arxiv.org/abs/2403.04434.](https://arxiv.org/abs/2403.04434) - 6. S. Nandi, B. B. Maity, V. Sharma, R. Verma, V. Saini, B. Singh, D. Aoki, and A. Thamizhavel, Magnetotransport and Fermi surface studies of a purported nodal line semimetal ZrAs2, Phys. Rev. B 109, 075155 (2024). - 8. G. B. Osterhoudt, Y. Wang, C. A. C. Garcia, V. M. Plisson, J. Gooth, C. Felser, P. Narang, and K. S. Burch, Evidence for Dominant Phonon-Electron Scattering in Weyl Semimetal WP2, Phys. Rev. X 11, 011017 (2021). - 9. K. Zhang, T. Wang, X. Pang, F. Han, S. Shang, N.T. Hung , Zi-Kui Liu, M. Li, R. Saito, and S. Huang, Anisotropic Fano resonance in the Weyl semimetal candidate LaAlSi Phys. Rev. B 102, 235162 (2020). - 10. K. Tanaka, T. Nishihara, A. Takakura,\|Y. Segawa, K. Matsuda, and Y. Miyauchi, Polarized Raman spectroscopy on topological semimetal Co3Sn2S<sup>2</sup> J. Raman Spectroscopy 54(1), 93 (2023). - 21. K. Zhang, X. Pang, T. Wang, F. Han, Shun-Li Shang, N. T. Hung, A.R.T. Nugraha, Zi-Kui Liu, M. Li, R, Saito, and S. Huang, Anomalous phonon-mode dependence in polarized Raman spectroscopy of the topological Weyl semimetal TaP, Phys. Rev. B 101, 014308 (2020). - 22. H. P. C. Miranda, S. Reichardt, G. Froehlicher, A. Molina-Sanchez, S. Berciaud, and L. Wirtz, Quantum Interference Effects in Resonant Raman Spectroscopy of Single- and Triple-Layer MoTe<sup>2</sup> from First-Principles, Nano Lett. 17, 2381 (2017). - 23. M. Osiekowicz, D. Staszczuk, K. Olkowska‑Pucko , Ł. Kipczak , M. Grzeszczyk, M. Zinkiewicz, K. Nogajewski, Z. R. Kudrynskyi, Z. D. Kovalyuk, A. Patané, A. Babiński, and M. R. Molas, Resonance and antiresonance in Raman scattering in GaSe and InSe crystals, Sci. Rep. 11, 924 (2021). - 24. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. 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Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![![](_page_14_Figure_1.jpeg)](path) ![Fig.2. Experiment geometry and crystal orientation. The crystal frame of reference (XYZ) and the lab frame of reference (X'Y'Z'). Light polarization (incident and scattered) is in the (X'Y) plane. The red arrow shows the light propagation vector, and the blue denotes the light polarization vector.](path) ![Fig.3. Room temperature Raman spectra for three different excitations (as measured). The continuous background is present with the dashed lines as reference levels for every plot. The green line is the fitted model of quasi-elastic scattering background using expression (3) to the 785 nm plot. The inset shows the temperature dependence of the integrated background intensity for the 785 nm excitation.](path) ![Fig.4. Comparison of the parallel (YY) and the (X'Y) crossed polarization spectra recorded with the 785 nm excitation at room temperature.](path) ![Fig. 5. Room temperature Raman spectra in the (YY) (a) and (X'X') (b) configurations for 473 nm, 633 nm and 785 nm excitation wavelengths.](path) ![Fig. 6. Band structure of ZrAs<sup>2</sup> with a few allowed electronic transitions to/from intermediate states in the Raman process, which are shown with the vertical arrows: black - 785 nm excitation, red 633 nm, and blue 473 nm (a); Density of states (b); Details of the electronic bands close to the Fermi level (c); shaded areas indicate regions of the Brillouin zone where phonon decay via e-h pairs excitation is energetically possible; Brillouin zone of ZrAs<sup>2</sup> (d).](path) ![Fig. 7. Fano profile fitted to the 171 cm-1 A<sup>g</sup> phonon peak measured with the 785 nm excitation at room temperature for two parallel light polarizations (YY) and (X'X').](path) ![Fig. 8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).](path) ![![](_page_20_Figure_1.jpeg)](path) Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening. # SUPPORTING INFORMATION # Temperature and excitation energy dependence of Raman scattering in nodal line Dirac semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland #### S1. Angular dependence of Raman scattering intensity According to the factor group analysis, 18 Raman active modes in ZrAs<sup>2</sup> are distributed as follows: 6A 3B 6B 3B g 1g 2g 3g + + + . In the identification process of modes symmetry, we measured angular dependence of the modes intensity as a function of the angle (Fig.2 in the main text) in parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) polarization configuration. We expect the following expressions for the intensity of the modes: Fitting the angular dependence of intensity with real elements of Raman tensor fails for A<sup>g</sup> modes. To properly describe this dependence, we have to assume complex Raman elements [S1,S2]: a i a a e = , b i b b e = c i c c e = . It leads to a complicated expression that cannot be deconvoluted to find modules or phase factors. Operationally, it comes down to expressions: For B modes phase factors do not play any role and the real values of Raman tensors are good enough to fit the angular dependence for parallel and perpendicular configurations. Figure S1 shows the angular dependence for six representative modes of different symmetry. The fitted value of the angle offset was 4. ![Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.](path) ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
Fig. 8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. 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Miyauchi, Polarized Raman spectroscopy on topological semimetal Co3Sn2S<sup>2</sup> J. Raman Spectroscopy 54(1), 93 (2023). - 21. K. Zhang, X. Pang, T. Wang, F. Han, Shun-Li Shang, N. T. Hung, A.R.T. Nugraha, Zi-Kui Liu, M. Li, R, Saito, and S. Huang, Anomalous phonon-mode dependence in polarized Raman spectroscopy of the topological Weyl semimetal TaP, Phys. Rev. B 101, 014308 (2020). - 22. H. P. C. Miranda, S. Reichardt, G. Froehlicher, A. Molina-Sanchez, S. Berciaud, and L. Wirtz, Quantum Interference Effects in Resonant Raman Spectroscopy of Single- and Triple-Layer MoTe<sup>2</sup> from First-Principles, Nano Lett. 17, 2381 (2017). - 23. M. Osiekowicz, D. Staszczuk, K. Olkowska‑Pucko , Ł. Kipczak , M. Grzeszczyk, M. Zinkiewicz, K. Nogajewski, Z. R. Kudrynskyi, Z. D. Kovalyuk, A. Patané, A. Babiński, and M. R. Molas, Resonance and antiresonance in Raman scattering in GaSe and InSe crystals, Sci. Rep. 11, 924 (2021). - 24. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. 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Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![![](_page_20_Figure_1.jpeg)](path) Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening. # SUPPORTING INFORMATION # Temperature and excitation energy dependence of Raman scattering in nodal line Dirac semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland #### S1. Angular dependence of Raman scattering intensity According to the factor group analysis, 18 Raman active modes in ZrAs<sup>2</sup> are distributed as follows: 6A 3B 6B 3B g 1g 2g 3g + + + . In the identification process of modes symmetry, we measured angular dependence of the modes intensity as a function of the angle (Fig.2 in the main text) in parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) polarization configuration. We expect the following expressions for the intensity of the modes: Fitting the angular dependence of intensity with real elements of Raman tensor fails for A<sup>g</sup> modes. To properly describe this dependence, we have to assume complex Raman elements [S1,S2]: a i a a e = , b i b b e = c i c c e = . It leads to a complicated expression that cannot be deconvoluted to find modules or phase factors. Operationally, it comes down to expressions: For B modes phase factors do not play any role and the real values of Raman tensors are good enough to fit the angular dependence for parallel and perpendicular configurations. Figure S1 shows the angular dependence for six representative modes of different symmetry. The fitted value of the angle offset was 4. ![Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.](path) ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
Fig.3. Room temperature Raman spectra for three different excitations (as measured). The continuous background is present with the dashed lines as reference levels for every plot. The green line is the fitted model of quasi-elastic scattering background using expression (3) to the 785 nm plot. The inset shows the temperature dependence of the integrated background intensity for the 785 nm excitation.	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. 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Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![Fig.4. Comparison of the parallel (YY) and the (X'Y) crossed polarization spectra recorded with the 785 nm excitation at room temperature.](path) ![Fig. 5. Room temperature Raman spectra in the (YY) (a) and (X'X') (b) configurations for 473 nm, 633 nm and 785 nm excitation wavelengths.](path) ![Fig. 6. Band structure of ZrAs<sup>2</sup> with a few allowed electronic transitions to/from intermediate states in the Raman process, which are shown with the vertical arrows: black - 785 nm excitation, red 633 nm, and blue 473 nm (a); Density of states (b); Details of the electronic bands close to the Fermi level (c); shaded areas indicate regions of the Brillouin zone where phonon decay via e-h pairs excitation is energetically possible; Brillouin zone of ZrAs<sup>2</sup> (d).](path) ![Fig. 7. Fano profile fitted to the 171 cm-1 A<sup>g</sup> phonon peak measured with the 785 nm excitation at room temperature for two parallel light polarizations (YY) and (X'X').](path) ![Fig. 8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).](path) ![![](_page_20_Figure_1.jpeg)](path) Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening. # SUPPORTING INFORMATION # Temperature and excitation energy dependence of Raman scattering in nodal line Dirac semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland #### S1. Angular dependence of Raman scattering intensity According to the factor group analysis, 18 Raman active modes in ZrAs<sup>2</sup> are distributed as follows: 6A 3B 6B 3B g 1g 2g 3g + + + . In the identification process of modes symmetry, we measured angular dependence of the modes intensity as a function of the angle (Fig.2 in the main text) in parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) polarization configuration. We expect the following expressions for the intensity of the modes: Fitting the angular dependence of intensity with real elements of Raman tensor fails for A<sup>g</sup> modes. To properly describe this dependence, we have to assume complex Raman elements [S1,S2]: a i a a e = , b i b b e = c i c c e = . It leads to a complicated expression that cannot be deconvoluted to find modules or phase factors. Operationally, it comes down to expressions: For B modes phase factors do not play any role and the real values of Raman tensors are good enough to fit the angular dependence for parallel and perpendicular configurations. Figure S1 shows the angular dependence for six representative modes of different symmetry. The fitted value of the angle offset was 4. ![Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.](path) ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
Fig. 1. Orthorhombic unit cell of ZrAs2.	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. Schnyder, Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery, Phys. Rev. Mat. 5, 124202 (2021). - 2. X. Zhou, Ch.-H. Hsu, H. Aramberri, M. Iraola, Ch.-Yi Huang, J. L. Mañes, M. G. Vergniory, H. Lin, and N. Kioussis, Novel family of topological semimetals with butterflylike nodal lines, Phys. Rev. B 104, 125135 (2021). - 3. J. Wang, H. Yuan, Z.-M. Yu, Z. Zhang, and X. Wang, Coexistence of symmetryenforced phononic Dirac nodal-line net and three-nodal surfaces phonons in solid-state materials: Theory and materials realization, Phys. Rev. Mat. 5, 124203 (2021). - 4. J. Bannies, E. Razzoli, M. Michiardi, H.-H. Kung, I. S. Elfimov, M. Yao, A. Fedorov, J. Fink , C. Jozwiak, A. Bostwick, E. Rotenberg, A. Damascelli , and C. Felser Extremely large magnetoresistance from electron-hole compensation in the nodal-loop semimetal ZrP2, Phys. Rev. B 103, 155144 (2021). - 5. A. S. Wadge, K. Zberecki, B. J. Kowalski, D. Jastrzebski, P. K. Tanwar, P. Iwanowski, R. Diduszko, A. Moosarikandy, M. Rosmus, N. Olszowska, and A. Wisniewski, Emergent impervious band crossing in the bulk in topological nodal line semimetal ZrAs<sup>2</sup> [https://arxiv.org/abs/2403.04434.](https://arxiv.org/abs/2403.04434) - 6. S. Nandi, B. B. Maity, V. Sharma, R. Verma, V. Saini, B. Singh, D. Aoki, and A. Thamizhavel, Magnetotransport and Fermi surface studies of a purported nodal line semimetal ZrAs2, Phys. Rev. B 109, 075155 (2024). - 8. G. B. Osterhoudt, Y. Wang, C. A. C. Garcia, V. M. Plisson, J. Gooth, C. Felser, P. Narang, and K. S. Burch, Evidence for Dominant Phonon-Electron Scattering in Weyl Semimetal WP2, Phys. Rev. X 11, 011017 (2021). - 9. K. Zhang, T. Wang, X. Pang, F. Han, S. Shang, N.T. Hung , Zi-Kui Liu, M. Li, R. Saito, and S. Huang, Anisotropic Fano resonance in the Weyl semimetal candidate LaAlSi Phys. Rev. B 102, 235162 (2020). - 10. K. Tanaka, T. Nishihara, A. Takakura,\|Y. Segawa, K. Matsuda, and Y. Miyauchi, Polarized Raman spectroscopy on topological semimetal Co3Sn2S<sup>2</sup> J. Raman Spectroscopy 54(1), 93 (2023). - 21. K. Zhang, X. Pang, T. Wang, F. Han, Shun-Li Shang, N. T. Hung, A.R.T. Nugraha, Zi-Kui Liu, M. Li, R, Saito, and S. Huang, Anomalous phonon-mode dependence in polarized Raman spectroscopy of the topological Weyl semimetal TaP, Phys. Rev. B 101, 014308 (2020). - 22. H. P. C. Miranda, S. Reichardt, G. Froehlicher, A. Molina-Sanchez, S. Berciaud, and L. Wirtz, Quantum Interference Effects in Resonant Raman Spectroscopy of Single- and Triple-Layer MoTe<sup>2</sup> from First-Principles, Nano Lett. 17, 2381 (2017). - 23. M. Osiekowicz, D. Staszczuk, K. Olkowska‑Pucko , Ł. Kipczak , M. Grzeszczyk, M. Zinkiewicz, K. Nogajewski, Z. R. Kudrynskyi, Z. D. Kovalyuk, A. Patané, A. Babiński, and M. R. Molas, Resonance and antiresonance in Raman scattering in GaSe and InSe crystals, Sci. Rep. 11, 924 (2021). - 24. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron– phonon interactions in the Weyl semimetal NbIrTe4, Sci. Rep. 11, 8155 (2021). - 25. S. Nath Gupta, D. V. S. Muthu, C. Shekhar, R. Sankar, C. Felser, and A. K. Sood Pressure-induced electronic and structural phase transitions in Dirac semimetal Cd3As2: Raman study, EPL, 120, 57003 (2017). - 26. D. Wulferding, P. Lemmens, F. Büscher, D. Schmeltzer, C. Felser, and Ch. Shekhar, Effect of topology on quasiparticle interactions in the Weyl semimetal WP<sup>2</sup> Phys. Rev. B 102, 075116 (2020). - 32. J. Coulter, G. B. Osterhoudt, C A. C. Garcia, Y. Wang, V. M. Plisson, B. Shen, N. Ni, K. S. Burch, and P. Narang, Uncovering electron-phonon scattering and phonon dynamics in type-I Weyl semimetals, Phys. Rev. B 100, 220301(R) (2019). - 33. Xu, Y.M. Da, L.X. Zha, K. Wang, R. Yang, W. Zhang, J.Y. Liu, H. Xiao, G.F. Chen, S.A. Trugman, J.-X. Zhu, A.J. Taylor, D.A. Yarotski, R.P. Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![![](_page_14_Figure_1.jpeg)](path) ![Fig.2. Experiment geometry and crystal orientation. The crystal frame of reference (XYZ) and the lab frame of reference (X'Y'Z'). Light polarization (incident and scattered) is in the (X'Y) plane. The red arrow shows the light propagation vector, and the blue denotes the light polarization vector.](path) ![Fig.3. Room temperature Raman spectra for three different excitations (as measured). The continuous background is present with the dashed lines as reference levels for every plot. The green line is the fitted model of quasi-elastic scattering background using expression (3) to the 785 nm plot. The inset shows the temperature dependence of the integrated background intensity for the 785 nm excitation.](path) ![Fig.4. Comparison of the parallel (YY) and the (X'Y) crossed polarization spectra recorded with the 785 nm excitation at room temperature.](path) ![Fig. 5. Room temperature Raman spectra in the (YY) (a) and (X'X') (b) configurations for 473 nm, 633 nm and 785 nm excitation wavelengths.](path) ![Fig. 6. Band structure of ZrAs<sup>2</sup> with a few allowed electronic transitions to/from intermediate states in the Raman process, which are shown with the vertical arrows: black - 785 nm excitation, red 633 nm, and blue 473 nm (a); Density of states (b); Details of the electronic bands close to the Fermi level (c); shaded areas indicate regions of the Brillouin zone where phonon decay via e-h pairs excitation is energetically possible; Brillouin zone of ZrAs<sup>2</sup> (d).](path) ![Fig. 7. Fano profile fitted to the 171 cm-1 A<sup>g</sup> phonon peak measured with the 785 nm excitation at room temperature for two parallel light polarizations (YY) and (X'X').](path) ![Fig. 8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).](path) ![![](_page_20_Figure_1.jpeg)](path) Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening. # SUPPORTING INFORMATION # Temperature and excitation energy dependence of Raman scattering in nodal line Dirac semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland #### S1. Angular dependence of Raman scattering intensity According to the factor group analysis, 18 Raman active modes in ZrAs<sup>2</sup> are distributed as follows: 6A 3B 6B 3B g 1g 2g 3g + + + . In the identification process of modes symmetry, we measured angular dependence of the modes intensity as a function of the angle (Fig.2 in the main text) in parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) polarization configuration. We expect the following expressions for the intensity of the modes: Fitting the angular dependence of intensity with real elements of Raman tensor fails for A<sup>g</sup> modes. To properly describe this dependence, we have to assume complex Raman elements [S1,S2]: a i a a e = , b i b b e = c i c c e = . It leads to a complicated expression that cannot be deconvoluted to find modules or phase factors. Operationally, it comes down to expressions: For B modes phase factors do not play any role and the real values of Raman tensors are good enough to fit the angular dependence for parallel and perpendicular configurations. Figure S1 shows the angular dependence for six representative modes of different symmetry. The fitted value of the angle offset was 4. ![Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.](path) ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
Fig. 6. Band structure of ZrAs<sup>2</sup> with a few allowed electronic transitions to/from intermediate states in the Raman process, which are shown with the vertical arrows: black - 785 nm excitation, red 633 nm, and blue 473 nm (a); Density of states (b); Details of the electronic bands close to the Fermi level (c); shaded areas indicate regions of the Brillouin zone where phonon decay via e-h pairs excitation is energetically possible; Brillouin zone of ZrAs<sup>2</sup> (d).	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. 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Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron– phonon interactions in the Weyl semimetal NbIrTe4, Sci. Rep. 11, 8155 (2021). - 25. S. Nath Gupta, D. V. S. Muthu, C. Shekhar, R. Sankar, C. Felser, and A. K. Sood Pressure-induced electronic and structural phase transitions in Dirac semimetal Cd3As2: Raman study, EPL, 120, 57003 (2017). - 26. D. Wulferding, P. Lemmens, F. Büscher, D. Schmeltzer, C. Felser, and Ch. Shekhar, Effect of topology on quasiparticle interactions in the Weyl semimetal WP<sup>2</sup> Phys. Rev. B 102, 075116 (2020). - 32. J. Coulter, G. B. Osterhoudt, C A. C. Garcia, Y. Wang, V. M. Plisson, B. Shen, N. Ni, K. S. Burch, and P. Narang, Uncovering electron-phonon scattering and phonon dynamics in type-I Weyl semimetals, Phys. Rev. B 100, 220301(R) (2019). - 33. Xu, Y.M. Da, L.X. Zha, K. Wang, R. Yang, W. Zhang, J.Y. Liu, H. Xiao, G.F. Chen, S.A. Trugman, J.-X. Zhu, A.J. Taylor, D.A. Yarotski, R.P. Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![![](_page_14_Figure_1.jpeg)](path) ![Fig.2. Experiment geometry and crystal orientation. The crystal frame of reference (XYZ) and the lab frame of reference (X'Y'Z'). Light polarization (incident and scattered) is in the (X'Y) plane. The red arrow shows the light propagation vector, and the blue denotes the light polarization vector.](path) ![Fig.3. Room temperature Raman spectra for three different excitations (as measured). The continuous background is present with the dashed lines as reference levels for every plot. The green line is the fitted model of quasi-elastic scattering background using expression (3) to the 785 nm plot. The inset shows the temperature dependence of the integrated background intensity for the 785 nm excitation.](path) ![Fig.4. Comparison of the parallel (YY) and the (X'Y) crossed polarization spectra recorded with the 785 nm excitation at room temperature.](path) ![Fig. 5. Room temperature Raman spectra in the (YY) (a) and (X'X') (b) configurations for 473 nm, 633 nm and 785 nm excitation wavelengths.](path) ![Fig. 6. Band structure of ZrAs<sup>2</sup> with a few allowed electronic transitions to/from intermediate states in the Raman process, which are shown with the vertical arrows: black - 785 nm excitation, red 633 nm, and blue 473 nm (a); Density of states (b); Details of the electronic bands close to the Fermi level (c); shaded areas indicate regions of the Brillouin zone where phonon decay via e-h pairs excitation is energetically possible; Brillouin zone of ZrAs<sup>2</sup> (d).](path) ![Fig. 7. Fano profile fitted to the 171 cm-1 A<sup>g</sup> phonon peak measured with the 785 nm excitation at room temperature for two parallel light polarizations (YY) and (X'X').](path) ![Fig. 8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).](path) ![![](_page_20_Figure_1.jpeg)](path) Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening. # SUPPORTING INFORMATION # Temperature and excitation energy dependence of Raman scattering in nodal line Dirac semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland #### S1. Angular dependence of Raman scattering intensity According to the factor group analysis, 18 Raman active modes in ZrAs<sup>2</sup> are distributed as follows: 6A 3B 6B 3B g 1g 2g 3g + + + . In the identification process of modes symmetry, we measured angular dependence of the modes intensity as a function of the angle (Fig.2 in the main text) in parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) polarization configuration. We expect the following expressions for the intensity of the modes: Fitting the angular dependence of intensity with real elements of Raman tensor fails for A<sup>g</sup> modes. To properly describe this dependence, we have to assume complex Raman elements [S1,S2]: a i a a e = , b i b b e = c i c c e = . It leads to a complicated expression that cannot be deconvoluted to find modules or phase factors. Operationally, it comes down to expressions: For B modes phase factors do not play any role and the real values of Raman tensors are good enough to fit the angular dependence for parallel and perpendicular configurations. Figure S1 shows the angular dependence for six representative modes of different symmetry. The fitted value of the angle offset was 4. ![Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.](path) ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. 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Miyauchi, Polarized Raman spectroscopy on topological semimetal Co3Sn2S<sup>2</sup> J. Raman Spectroscopy 54(1), 93 (2023). - 21. K. Zhang, X. Pang, T. Wang, F. Han, Shun-Li Shang, N. T. Hung, A.R.T. Nugraha, Zi-Kui Liu, M. Li, R, Saito, and S. Huang, Anomalous phonon-mode dependence in polarized Raman spectroscopy of the topological Weyl semimetal TaP, Phys. Rev. B 101, 014308 (2020). - 22. H. P. C. Miranda, S. Reichardt, G. Froehlicher, A. Molina-Sanchez, S. Berciaud, and L. Wirtz, Quantum Interference Effects in Resonant Raman Spectroscopy of Single- and Triple-Layer MoTe<sup>2</sup> from First-Principles, Nano Lett. 17, 2381 (2017). - 23. M. Osiekowicz, D. Staszczuk, K. Olkowska‑Pucko , Ł. Kipczak , M. Grzeszczyk, M. Zinkiewicz, K. Nogajewski, Z. R. Kudrynskyi, Z. D. Kovalyuk, A. Patané, A. Babiński, and M. R. Molas, Resonance and antiresonance in Raman scattering in GaSe and InSe crystals, Sci. Rep. 11, 924 (2021). - 24. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron– phonon interactions in the Weyl semimetal NbIrTe4, Sci. Rep. 11, 8155 (2021). - 25. S. Nath Gupta, D. V. S. Muthu, C. Shekhar, R. Sankar, C. Felser, and A. K. Sood Pressure-induced electronic and structural phase transitions in Dirac semimetal Cd3As2: Raman study, EPL, 120, 57003 (2017). - 26. D. Wulferding, P. Lemmens, F. Büscher, D. Schmeltzer, C. Felser, and Ch. Shekhar, Effect of topology on quasiparticle interactions in the Weyl semimetal WP<sup>2</sup> Phys. Rev. B 102, 075116 (2020). - 32. J. Coulter, G. B. Osterhoudt, C A. C. Garcia, Y. Wang, V. M. Plisson, B. Shen, N. Ni, K. S. Burch, and P. Narang, Uncovering electron-phonon scattering and phonon dynamics in type-I Weyl semimetals, Phys. Rev. B 100, 220301(R) (2019). - 33. Xu, Y.M. Da, L.X. Zha, K. Wang, R. Yang, W. Zhang, J.Y. Liu, H. Xiao, G.F. Chen, S.A. Trugman, J.-X. Zhu, A.J. Taylor, D.A. Yarotski, R.P. Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
Fig.8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).	# Temperature and excitation energy dependence of Raman scattering in nodal line semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland 3 # ABSTRACT We present a Raman study of ZrAs<sup>2</sup> single crystals, a nodal line semimetal with symmetryenforced Dirac-like band crossings. We identified the symmetry of phonon modes by polarized light measurements and comparison with calculated phonon frequencies. Significant dependence of peak intensities on the excitation wavelength was observed, indicating quantum interference effects. Phonon peaks in the spectra are superimposed on the electronic background, with quasi-elastic scattering observed for the 785 nm excitation. We identified the Fano shape of the 171 cm-1 A<sup>g</sup> mode due to interference of the phonon state with the electronic continuum. The temperature dependence of phonon peaks linewidth indicates that the electronphonon coupling plays an essential role in phonon decay. # INTRODUCTION Topological semimetals have attracted considerable scientific interest in recent years owing to their unique electronic structure and the specific role of symmetry. In most Dirac and Weyl semimetals, the energy bands intersect at a point in k-space. Nodal line semimetals exhibit band crossings that extend on a one-dimensional line or loop in k-space [1]. The nonsymmorphic symmetries can play an important role in protecting the crossings [2,3]. Transition metal dipnictides ZrP<sup>2</sup> [4] and ZrAs<sup>2</sup> [5] have been found to represent nodal line semimetals. Bannies et al. [4] using angle-resolved photoemission spectroscopy (ARPES) and magnetotransport studies found that ZrP<sup>2</sup> exhibits an extremely large and unsaturated magnetoresistance (MR) of up to 40 000 % at 2 K, which originates from an almost perfect electron-hole compensation. Their band structure calculations and ARPES studies showed that ZrP<sup>2</sup> hosts a topological nodal loop in proximity to the Fermi level. Very recently, magnetotransport studies of ZrAs2 have been reported by Nandi et al. [6]. They observed large MR with quadratic field dependence, unsaturated up to magnetic field of 14 T. Their electronic structure analysis demonstrates the coexistence of electron and hole pockets at the Fermi surface. The carrier concentration was estimated from the field-dependent Hall resistivity, and it was found that the charge carriers are nearly compensated, which results in a large MR. Wadge et al. [5], reported results for the ZrAs<sup>2</sup> single crystals, obtained using ARPES technique and DFT calculations. In ARPES scans, a distinctive nodal loop structure was observed at lower photon energies of 30 and 50 eV. Furthermore, DFT calculations unveiled symmetry-enforced band crossings anchored at specific points in the Brillouin zone. Raman scattering studies of topological semimetals offer insight into lattice dynamics, electronic structure, and electron-phonon interaction [7-10]. This paper presents the Raman study of zirconium di-arsenide ZrAs<sup>2</sup> single crystals. Since, to our knowledge, no Raman studies on ZrAs<sup>2</sup> have been published yet, we performed angle-resolved polarization measurements to assign symmetry to the observed Raman modes. With the support of the phonon frequencies ab initio calculations, it was possible to identify all Raman modes in ZrAs2. The Raman spectra depend substantially on the excitation energy, with some modes visible only for specific excitation. We ascribe this to resonance-like/interference effects related to the complicated band structure of ZrAs2. We also analyzed the electronic Raman scattering, which produces a pronounced continuum background in the Raman spectra. An essential aspect of the Raman spectroscopy of semimetals is the role of electron-phonon coupling. Our study shows the electron-phonon coupling effects in ZrAs2, manifesting in Fano resonance, and the temperature dependence of Raman peaks linewidth. ### EXPERIMENTAL DETAILS ZrAs<sup>2</sup> crystallizes in the PbCl<sup>2</sup> – type structure with the centrosymmetric, nonsymorphic space group Pnma (D2h16). Orthorhombic cell parameters are [11]: a = 6.8006 Å, b = 3.6883 Å, c = 9.0328(4) Å. The unit cell of ZrAs<sup>2</sup> contains 4 formula units (Fig. 1), all atoms occupy 4c Wycoff positions. Needle-like crystals of ZrAs2 have been grown by the iodine transport method, with the crystallographic b axis along the needle length. The chains of covalently bonded As atoms along the b-axis provide preferred direction of crystal growth [11]. Raman measurements were performed on two spectrometers: Horiba Jobin Ivon Aramis spectrometer with 2400 l/mm diffraction grating (appr. resolution 1 cm-1 ), used for measurements with 473 nm and 633 nm excitation lasers; Renishaw Quanta spectrometer with diffraction grating 1800 l/mm and approximate resolution of 2 cm-1 (785 nm laser). The beam power was kept low in all cases to prevent sample heating. The spectra were measured in the temperature range 80 – 443 K. Raman spectra were measured in backscattering geometry with parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) configurations, where i e ˆ , e ˆ s are polarization vectors of incident and scattered light, respectively. For most of the measurements the natural (101) face of the crystal was used, with light impinging along the Z' axis, perpendicular to the (101) face (Fig. 2). We use Porto's notation throughout the paper, e.g. Z'(YY)Z' configuration, which we write in short as (YY). ## COMPUTATIONAL DETAILS The calculations of the band structure were performed within density functional theory (DFT) as implemented in the VASP Package [12-15] with Projector augmented wave pseudopotentials (PAW) [16, 17]. In all cases, the PBE (GGA) functional [18] has been used. For the sampling of the Brillouin zone, a dense 8 × 8 × 8 grid was used, while the plane wave energy cutoff was set to 500 eV. All the structures were optimized until the force exerted on each atom was smaller than 10−5 eV/Å. Phonon dispersion calculations were made using, the frozen phonon method as implemented in the phonopy code [19]. ## RESULTS AND DISCUSSION ## Room temperature studies $$\mathbf{A}\_{\mathbf{g}} = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \quad \mathbf{B}\_{\mathbf{l}\_{\mathbf{g}}} = \begin{pmatrix} 0 & d & 0 \\ d & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad \mathbf{B}\_{\mathbf{2g}} = \begin{pmatrix} 0 & 0 & e \\ 0 & 0 & 0 \\ e & 0 & 0 \end{pmatrix} \qquad \mathbf{B}\_{\mathbf{3g}} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & f \\ 0 & f & 0 \end{pmatrix} \tag{2}$$ Figure 3 presents room temperature Raman spectra recorded at the (YY) geometry for three excitation energies: 2.62 eV (473 nm), 1.96 eV (633 nm), and 1.58 eV (785 nm) with continuous background dependent on excitation energy. The background in spectra will be discussed further below. Phonon modes positions, linewidths and intensities were analyzed after background subtraction. Generally, the Raman spectra measured in this study are consistent with selection rules stemming from the Raman tensor with some "leaking" effect due to resonant conditions. Apparent difference is observed for the lowest frequency A<sup>g</sup> mode at 94.5 cm-1 , which has a significant intensity in the forbidden cross polarizer configuration (Fig. 4). Table 1 gives the assignment of the observed phonon modes according to their symmetry compared with calculated frequencies (VASP\_FP). The polarized angular-resolved Raman spectra confirm this assignment (Fig.S1. in the Supplement presents the polar graphs for selected modes, as a function of the angle between the light polarization vector and the baxis. Figure 5 shows spectra for (YY) and (X'X') polarization for different excitation wavelengths. Intensities of several Raman peaks have distinctive excitation wavelength dependence. The A<sup>g</sup> symmetry modes at 129 cm-1 and 171 cm-1 have much higher intensity for the 633 and 785 nm excitations than for the 473 nm excitation. The A<sup>g</sup> mode at 276 cm-1 does not appear in the 473 nm spectrum. Mode at 244 cm-1 (B2g) is the most intense for the 473 nm excitation in the (X'X') configuration and appears as a small kink in the 785 nm plot (it is not present in the 633 nm plot). Usually, in resonant Raman spectra, all modes are seen to increase their intensity when the excitation energy is close to a characteristic electronic transition (Van Hove singularity or band nesting). In the Raman spectra of ZrAs2, we observe both resonance and antiresonance effects depending on mode. This means that different electronic intermediate states are involved in the Raman scattering. Since in the band structure of ZrAs<sup>2</sup> (Fig. 6a), many bands are available for such transitions, different bands in the whole Brillouin zone take part in the light scattering process. It leads to a quantum interference effect, when electronic transitions in different parts of the Brillouin zone enhance or quench each other [21-24]. Due to the complicated band structure of ZrAs2, we cannot identify the regions that contribute constructively to the Raman amplitude and those that interfere destructively. It is worth noting that differences in electronphonon matrix elements for phonon modes can also account for observed excitation dependence [24]. As shown in Fig. 3, the continuous background is present in the ZrAs<sup>2</sup> Raman scattering spectra. The background shows a quasi-elastic scattering (QES) wing and a flat finite energy continuum extending up to 1500 cm-1 . It is polarization-dependent and has the highest intensity in the (YY) configuration, i.e. for light polarization along the b-axis, where arsenic atoms form covalently bonded chain. It also depends on excitation energy and has the highest intensity with an intense QES part for the lowest used excitation energy of 1.58 eV (785 nm). For the higher excitation energies, 1.96 eV and 2.62 eV, the intensity of the quasi-elastic component is weaker, and the broad energy range background is observed. The background intensity measured with the 785 nm excitation decreases with increasing temperature (inset in Fig. 3). A broad band of continuous energy Raman scattering was observed for some topological semimetals: e.g. in Cd3As<sup>2</sup> [7,25], WP<sup>2</sup> [26], LaAlSi [21]. It is attributed to electronic Raman scattering (ERS) due to electron density fluctuations. Such a background was first observed in heavily doped semiconductors [27, 28]. It was interpreted as scattering by intervalley density fluctuations in the collision-limited regime [29]. Zawadowski and Cardona [30] have shown, that in Raman spectra of metals a background can be due to a scattering between different parts of Fermi surface. Intense QES is also observed in the metallic phase of La0.7Sr0.3MnO<sup>3</sup> [31]. The quasi-elastic part of the continuum background is often described by the Lorentzian profile for zero energy: Equation (3) gives a good fit to the QES wing in the 785 nm excitation spectra (Fig. 3). The intensity of this background decreases with temperature in the range of 80 – 443 K (inset in Fig. 3). el is almost independent of temperature and has an average value of (230 ± 20) cm-1 . Due to the presence of an electronic background in Raman scattering spectra, one can expect effects of interference between discrete phonon modes and electron continuum states resulting in asymmetric Fano-type profiles of Raman peaks intensity: where is a spectral width, ω<sup>0</sup> is a peak position. The 1/q is the asymmetry factor that accounts for the strength of electron-phonon interaction. Identifying the asymmetric Fano shape for closely spaced, partially overlapped peaks was impossible. For all but one of the separated Raman peaks in ZrAs2, symmetric (Lorentz or Gaussian) line shape provided better fits to experimental data than the Fano profile (value of 1/q was practically 0). Only for the 171 cm-1 A<sup>g</sup> mode in (YY) configuration, the peak shape described by the Fano profile was found in the spectra excited by 633 nm and 785 nm lasers. Figure 7 presents comparison of the (YY) and the (X'X') spectra with the fitted Fano profile. In the (YY) configuration the fitted value of asymmetry factor is 1/q = -0.19 ± 0.05, and it is nearly independent of temperature. This indicates no significant change in the electronic structure at the Fermi surface as a function of temperature. The (X'X') plot is almost symmetric with 1/q value close to zero. It is in keeping with a small relative value of background intensity for the (X'X') spectra compared to the (YY) spectra. The scarcity of Fano-like peaks in Raman spectra of ZrAs<sup>2</sup> resembles the situation of another topological semimetal TaAs, where the electron-phonon coupling was identified in temperature evolution of the Raman modes linewidth, but no asymmetric Fano-like Raman peaks were reported [32]. However, asymmetric Fano profiles have been found in the infrared spectra of TaAs [33]. ### Effect of temperature on Raman scattering in ZrAs<sup>2</sup> Figure 8 shows the evolution of the Raman spectra measured in the parallel configuration for = 45 in the temperature range 80 K – 443 K for the 2.62 eV (473 nm) excitation. Spectra for the 1.96 eV (633 nm) excitation are shown in Fig.S3 in the Supplement. Most peaks are visible for the whole temperature range. However, some peaks for the 2.62 eV excitation lose their intensity with increasing temperature, e.g. the A<sup>g</sup> mode at 280 cm-1 (at 80 K) is hardly visible for temperatures above 300 K. To find the basic parameters of the phonon lines (position, linewidth, area) the fitting procedure has been performed. The best fit was achieved with the pseudo-Voigt function, which is a weighted sum of the Lorentz and Gaussian profiles. Attempts to fit spectra with the Voigt profile give unreliable results because of too low signal-to-noise ratio [34]. The temperature dependence of the optical phonon frequency and linewidth is usually ascribed to two effects: the anharmonic effect due to the phonon-phonon coupling and quasi-harmonic effect due to thermal expansion of the crystal lattice. Anharmonic interaction is analyzed within the extended Klemens model [35, 36], which assumes that phonon decays into two or three acoustic phonons. The change of phonon frequency is given as: Where is phonon energy, / 2 B x k T = , / 3 B y k T = and A and B denote anharmonic constants related to three phonon processes (decay of optical phonon into two phonons) and four phonon processes, respectively. Similar expression describes the temperature evolution of the phonon linewidth due to the anharmonic interaction: Figure 9 presents the temperature dependence of the position and the linewidth of several phonon peaks in ZrAs2. Results for the 473 and 633 nm excitation agree well. In the temperature range used in our experiment, the redshift of phonon frequency is for most modes close to the linear formula (7), with slight deviation at temperatures near 80 K. Fitting with the anharmonic expression (5) also produces almost straight line, so these effects are nearly indistinguishable. However, for the modes at 166 cm-1 and 245 cm-1 , the temperature redshift of phonon frequency is distinctly nonlinear and follows the expression (5). The temperature dependence of the linewidths of Raman peaks is often successfully described by an anharmonic expression (6). It predicts an increase in linewidth with increasing temperature. However, several phonon modes in Raman spectra show a reduction in linewidth for increasing temperature (Fig. 9). It is due to important contribution of the electron-phonon coupling, which is particularly important in semimetals. In this interaction phonons decay into electron-hole pairs via intra- or inter-band transitions close to the Fermi level (EF) [8,37,38]. Temperature dependence of the linewidth is determined by the difference in occupation of electronic states below and above EF. For increasing temperature, the occupation of the electron states below E<sup>F</sup> decreases, while the occupation of the states above E<sup>F</sup> increases. It leads to decreasing number of available electronic states for the phonon induced transitions and results in a decrease of the linewidth with increasing temperature. This behavior is quantitively expressed by the formula [37, 38]: It is worth noting that taking into account of a finite chemical potential in (8) can result in a nonmonotonic temperature dependence of ep<sup>h</sup> [8, 32]. The decay of optical phonon with zero wavevectors via the creation of the electronhole pairs depends on the energy and symmetry of phonon mode. Since the highest energy phonons at the center of the Brillouin zone of ZrAs<sup>2</sup> have an energy of 34 meV, electron-phonon coupling is possible for pairs of electron bands below and above the Fermi energy, which are closer to each other than 34 meV. It can happen only with an electron k-vector along the - Z line and around the T point in the Brillouin zone (Fig. 6c). However, symmetry-based selection rules (with and without spin-orbit coupling) do not allow Raman active phonons to induce interband transitions at the T point. Due to spin-orbit coupling, such transitions are allowed near the T point and along the - Z line. We cannot exclude that conditions for effective electron-phonon coupling may also exist at general k-points in the Brillouin zone. In an analysis of the temperature dependence of the linewidth, we used an expression with two contributions: anharmonic term anh - equation (6) (we put D = 0, since four phonon processes give negligible contribution), and the electron-phonon coupling term eph according to the expression (8). We investigated the temperature dependence of the linewidth of phonon peaks represented by the full width at half maximum (FWHM) for two excitation energies, 1.96 eV (633 nm) and 2.61 eV (473 nm), whenever the peak was present and measurable in the spectra for a given excitation energy. For low intensity modes and overlapping peaks it was impossible to get reliable values of the linewidth. Temperature dependence of the linewidth measured for the two excitations exhibits significant differences (Fig. 9). In the spectra recorded with 633 nm laser we observe for most modes almost linear linewidth increase in accord with anharmonic model, apart from the 94.5 cm-1 mode, when the 633 nm plot follows the 473 nm line. It is unclear to us what the source of these discrepancies is; one of the possible reasons can be a difference in light penetration depth for the used excitation energies. For the 473 nm excitation, the contribution of the electron-phonon interaction to phonon decay is seen for several modes as a monotonous decrease or a minimum in the temperature dependence of the linewidth. This type of temperature dependence is observed for the modes of A<sup>g</sup> symmetry at 94.5 cm-1 , 129 cm-1 , and 223 cm-1 , for two modes of B2g symmetry at 150 and 245 cm-1 , and the 166 cm-1 B3g mode. The dominance of the electron-phonon coupling over anharmonic decay is observed for the modes at 94.5 and 166 cm-1 , where no contribution from the anharmonic term is needed (C = 0) to fit the experimental data. This observation indicates no clear-cut correlation between the strength of the electron-phonon coupling in phonon decay, and phonon symmetry or frequency. It is an individual property of the phonon mode. The values of the eph parameter, characterizing the strength of electron-phonon coupling for phonon modes (Table 2), are smaller than the values reported for some phonon modes in NiTe<sup>2</sup> - 5.41 cm-‑1 [39] and PdTe<sup>2</sup> - 28.8 cm-1 [40] as well as in graphene and graphite - appr. 10 cm- <sup>1</sup> [38]. Still, the electron-phonon coupling plays a significant role in phonon decay processes in ZrAs2. # CONCLUSIONS We have investigated Raman scattering in a Dirac nodal line semimetal ZrAs2. Raman spectra have been recorded for several excitation laser energies at different light polarizations. The calculated zero wavevector phonon frequencies and polarization dependence of the Raman peaks enabled symmetry identification of all observed phonon modes. Due to an interference between excitation paths with different intermediate states, significant peak intensity differences exist for different excitation wavelengths. The polarization-dependent electronic background is present in the Raman spectra, with an intense quasi-elastic scattering in the spectra recorded with the 785 nm excitation. The asymmetric Fano peak is observed for the 171 cm-1 phonon mode due to interference of a phonon with intense electronic background observed for light polarization along the b axis of the ZrAs2. Effects of electron-phonon interaction manifest themselves in decreasing peak linewidth with increasing temperature for modes of different symmetry, indicating differences in the electron-phonon coupling. We identified points in the Brillouin zone at the Fermi level where optical phonons can decay via e-h pair creation: only for the electronic states on the - Z high symmetry line and in the vicinity of the T point, phonons are allowed to create e-h pairs. Since in ZrAs<sup>2</sup> there are no Dirac points close to the Fermi energy, it is hard to expect a significant influence of these points on the electron-phonon coupling. # ACKNOWLEDGEMENTS This research was partially supported by the Foundation for Polish Science, facilitated by the IRA Programme and co-financed by the European Union within the framework of the Smart Growth Operational Programme (Grant No. MAB/2017/1). # REFERENCES - 1. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. Schnyder, Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery, Phys. Rev. Mat. 5, 124202 (2021). - 2. X. Zhou, Ch.-H. Hsu, H. Aramberri, M. Iraola, Ch.-Yi Huang, J. L. Mañes, M. G. Vergniory, H. Lin, and N. Kioussis, Novel family of topological semimetals with butterflylike nodal lines, Phys. Rev. B 104, 125135 (2021). - 3. J. Wang, H. Yuan, Z.-M. Yu, Z. Zhang, and X. Wang, Coexistence of symmetryenforced phononic Dirac nodal-line net and three-nodal surfaces phonons in solid-state materials: Theory and materials realization, Phys. Rev. Mat. 5, 124203 (2021). - 4. J. Bannies, E. Razzoli, M. Michiardi, H.-H. Kung, I. S. Elfimov, M. Yao, A. Fedorov, J. Fink , C. Jozwiak, A. Bostwick, E. Rotenberg, A. Damascelli , and C. Felser Extremely large magnetoresistance from electron-hole compensation in the nodal-loop semimetal ZrP2, Phys. Rev. B 103, 155144 (2021). - 5. A. S. Wadge, K. Zberecki, B. J. Kowalski, D. Jastrzebski, P. K. 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Miyauchi, Polarized Raman spectroscopy on topological semimetal Co3Sn2S<sup>2</sup> J. Raman Spectroscopy 54(1), 93 (2023). - 21. K. Zhang, X. Pang, T. Wang, F. Han, Shun-Li Shang, N. T. Hung, A.R.T. Nugraha, Zi-Kui Liu, M. Li, R, Saito, and S. Huang, Anomalous phonon-mode dependence in polarized Raman spectroscopy of the topological Weyl semimetal TaP, Phys. Rev. B 101, 014308 (2020). - 22. H. P. C. Miranda, S. Reichardt, G. Froehlicher, A. Molina-Sanchez, S. Berciaud, and L. Wirtz, Quantum Interference Effects in Resonant Raman Spectroscopy of Single- and Triple-Layer MoTe<sup>2</sup> from First-Principles, Nano Lett. 17, 2381 (2017). - 23. M. Osiekowicz, D. Staszczuk, K. Olkowska‑Pucko , Ł. Kipczak , M. Grzeszczyk, M. Zinkiewicz, K. Nogajewski, Z. R. Kudrynskyi, Z. D. Kovalyuk, A. Patané, A. Babiński, and M. R. Molas, Resonance and antiresonance in Raman scattering in GaSe and InSe crystals, Sci. Rep. 11, 924 (2021). - 24. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron– phonon interactions in the Weyl semimetal NbIrTe4, Sci. Rep. 11, 8155 (2021). - 25. S. Nath Gupta, D. V. S. Muthu, C. Shekhar, R. Sankar, C. Felser, and A. K. Sood Pressure-induced electronic and structural phase transitions in Dirac semimetal Cd3As2: Raman study, EPL, 120, 57003 (2017). - 26. D. Wulferding, P. Lemmens, F. Büscher, D. Schmeltzer, C. Felser, and Ch. Shekhar, Effect of topology on quasiparticle interactions in the Weyl semimetal WP<sup>2</sup> Phys. Rev. B 102, 075116 (2020). - 32. J. Coulter, G. B. Osterhoudt, C A. C. Garcia, Y. Wang, V. M. Plisson, B. Shen, N. Ni, K. S. Burch, and P. Narang, Uncovering electron-phonon scattering and phonon dynamics in type-I Weyl semimetals, Phys. Rev. B 100, 220301(R) (2019). - 33. Xu, Y.M. Da, L.X. Zha, K. Wang, R. Yang, W. Zhang, J.Y. Liu, H. Xiao, G.F. Chen, S.A. Trugman, J.-X. Zhu, A.J. Taylor, D.A. Yarotski, R.P. Prasankumar, and X.G. Qiu, Temperature-tunable Fano resonance induced by strong coupling between Weyl fermions and phonons in TaAs, Nat. Comm. 8, 14933 (2017). - 39. M. Theres James, S. Mandal, N. K Sebastian, P. Mishra, R Ganesan, and P. S. Anil Kumar, Probing electron-phonon and phonon-phonon coupling in type-II Dirac semimetal NiTe<sup>2</sup> via temperature-dependent Raman spectroscopy, J. Phys.: Condensed Matter 35, 125701 (2023). - 40. M. M. AL-Makeen, X. Guo, Y. Wang, D. Yang, S. You, M. Yassine, J. Jiang, P.-Bo Song, Y.-G. Shi, H. Xie, J.-Q. Meng, and H. Huang, Investigation of the Electron– Phonon Coupling in Dirac Semimetal PdTe<sup>2</sup> via Temperature-Dependent Raman Spectroscopy, Phys. Stat. Solidi RRL 16, 2200257 (2022). # Tables Table 1. Symmetry assignment of the observed Raman modes in ZrAs2. The mode frequencies were determined from the room temperature spectra. Accuracy of mode frequency values is appr. 1 cm-1 . Theoretical values were calculated using the DFT method. Table 2. Fitting parameters of the temperature dependence of peak position: coefficient equation (7), and linewidth: anharmonic parameter C (6), electron-phonon coupling parameter eph (8) for the 473 nm excitation. Additionally, for the 94.5 cm-1 mode parameters from the fit for the 633 nm excitation spectra are given. ![![](_page_14_Figure_1.jpeg)](path) ![Fig.2. Experiment geometry and crystal orientation. The crystal frame of reference (XYZ) and the lab frame of reference (X'Y'Z'). Light polarization (incident and scattered) is in the (X'Y) plane. The red arrow shows the light propagation vector, and the blue denotes the light polarization vector.](path) ![Fig.3. Room temperature Raman spectra for three different excitations (as measured). The continuous background is present with the dashed lines as reference levels for every plot. The green line is the fitted model of quasi-elastic scattering background using expression (3) to the 785 nm plot. The inset shows the temperature dependence of the integrated background intensity for the 785 nm excitation.](path) ![Fig.4. Comparison of the parallel (YY) and the (X'Y) crossed polarization spectra recorded with the 785 nm excitation at room temperature.](path) ![Fig. 5. Room temperature Raman spectra in the (YY) (a) and (X'X') (b) configurations for 473 nm, 633 nm and 785 nm excitation wavelengths.](path) ![Fig. 6. Band structure of ZrAs<sup>2</sup> with a few allowed electronic transitions to/from intermediate states in the Raman process, which are shown with the vertical arrows: black - 785 nm excitation, red 633 nm, and blue 473 nm (a); Density of states (b); Details of the electronic bands close to the Fermi level (c); shaded areas indicate regions of the Brillouin zone where phonon decay via e-h pairs excitation is energetically possible; Brillouin zone of ZrAs<sup>2</sup> (d).](path) ![Fig. 7. Fano profile fitted to the 171 cm-1 A<sup>g</sup> phonon peak measured with the 785 nm excitation at room temperature for two parallel light polarizations (YY) and (X'X').](path) ![Fig. 8. Temperature dependence of the Raman spectra measured with the 473 nm excitation for parallel configuration and = 45 (for the 633 nm excitation see the Supplement).](path) ![![](_page_20_Figure_1.jpeg)](path) Fig.9. Temperature dependence of phonon modes frequency in ZrAs<sup>2</sup> (left panels) and the linewidth (right panels) for two excitation wavelengths: 473 nm and 633 nm. The red line is a fit of the expression (5) to the mode frequency change. At the right panels, blue line is a fit of the model where both anharmonic effects and electron-phonon interaction are included. The black line corresponds to the linewidth model of anharmonic interaction only (6). The linewidth results were not corrected for the instrumental broadening. # SUPPORTING INFORMATION # Temperature and excitation energy dependence of Raman scattering in nodal line Dirac semimetal ZrAs<sup>2</sup> 1 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL- 00-662, Warsaw, Poland 2 International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland #### S1. Angular dependence of Raman scattering intensity According to the factor group analysis, 18 Raman active modes in ZrAs<sup>2</sup> are distributed as follows: 6A 3B 6B 3B g 1g 2g 3g + + + . In the identification process of modes symmetry, we measured angular dependence of the modes intensity as a function of the angle (Fig.2 in the main text) in parallel ( i e ˆ \|\| e ˆ s ) and crossed ( i e ˆ ⊥ <sup>e</sup> ˆ s ) polarization configuration. We expect the following expressions for the intensity of the modes: Fitting the angular dependence of intensity with real elements of Raman tensor fails for A<sup>g</sup> modes. To properly describe this dependence, we have to assume complex Raman elements [S1,S2]: a i a a e = , b i b b e = c i c c e = . It leads to a complicated expression that cannot be deconvoluted to find modules or phase factors. Operationally, it comes down to expressions: For B modes phase factors do not play any role and the real values of Raman tensors are good enough to fit the angular dependence for parallel and perpendicular configurations. Figure S1 shows the angular dependence for six representative modes of different symmetry. The fitted value of the angle offset was 4. ![Fig.S1. Polar plots of Raman intensity as a function of the angle between the b-axis of ZrAs2 crystal (it is also growth direction of needle-like crystals) and the incident light polarization vector i e ˆ for two configurations parallel i e ˆ \|\| e ˆ s and perpendicular i e ˆ ⊥ e ˆ s . Square symbol and red line correspond to parallel configuration, circle symbol and blue line denote the perpendicular configuration.](path) ![Fig.S2. Phonon dispersion calculated using the frozen phonon method, as implemented in the phonopy code [19].](path) ### S2. Temperature dependence of the Raman spectra for the 633 nm excitation ![Fig.S3. Temperature dependence of the Raman spectra measured with the 633 nm excitation for parallel polarization configuration with = 45. Periodically distributed peaks below 100 cm-1 are due to the air scattering.](path) ### REFERENCES - S1. I. Abbasian Shojaei, S. Pournia, C. Le, B. R. Ortiz, G. Jnawali, Fu‑Ch. Zhang, S. D. Wilson, H. E. Jackson, and L. M. Smith, A Raman probe of phonons and electron–phonon interactions in the Weyl semimetal NbIrTe<sup>4</sup> Sci. Rep. 11, 8155 (2021) - S2 H.B. Ribeiro, M.A. Pimento, C.J.S. de Matos, R.L. Moreira, A.S. Rodin, J.D. Zapata, E.A.T. de Souza, and A.H.Castro Neto, Unusual Angular Dependence of the Raman Response in Black Phosphorus ACS Nano 9, 4276 (2015)
FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 5. Positron diffraction analysis with the Al-embedded silicene model. (a,b) Total-reflection high-energy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with an 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions. (c) Schematic of the model structure. A 3×3 unit cell is depicted.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. Figure 5(a,b) is a set of TRHEPD rocking curves of the Al(111)3×3-Si surface, with comparisons to simulation curves (z1−<sup>8</sup> = 2.40 ˚A) of the flat Al-embedded silicene model that replaces one of the Si atoms with an Al atom [Fig. 5 (c)]. The results show good agreements that confirm the Al-Si mixed model. A R-factor value of the Al-embedded model was R = 2.12%, which was slightly smaller than R = 2.38% for the flat silicene model (Fig. 2). The R-factor values derived from Figs.4 and 5 were different because the former TRHEPD results were obtained with only a specular beam, while the latter were obtained with many diffracted beams. Previous reports proposed a "kagome-like silicene" model as the surface structure[\[21\]](#page-20-10). However, the TRHEPD analysis here was in disagreement, as shown in Fig. 12 in the Appendix. The first-principles calculation with OpenMX has also resulted in the flat structure for the Al-embedded silicene layer on Al(111) after the optimization. A standard derivation of the height (SDH) of the Al-embedded silicene layer on the Al(111) was only 0.06 ˚A. The result is consistent to the flat surface structure, as determined by the TRHEPD analyses. ## B. Core-level spectra Positron diffraction experiments revealed that the Al(111)3×3-Si surface structure could be described by a flat honeycomb lattice model. However, substitution of Al atoms in the Si overlayer was uncertain because of indistinguishable positron scattering. The Si atoms could be bonded with neighboring Si and/or Al atoms. This creates different chemical environments for these Si sites that should result in corresponding chemical shifts in Si core-level photoemission spectra[\[1\]](#page-19-0). Figure 6(a) shows Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface. There were at least five peaks, as reported previously[\[21\]](#page-20-10). Because a single chemical site for Si generates a doublet state, Si 2p3/<sup>2</sup> and 2p1/2, from spin-orbit splitting in the binding energy range, the spectrum indicated the existence of multiple chemical environments for Si atoms at the surface. To reveal the components, spectral curve-fittings were conducted using Doniach-Sunjic functions for various emission angles, θ's. The spin-orbit splitting and the branching ratio were fixed at 0.4 eV and 0.5, respectively. The Doniach-Sunjic width was fixed at 0.0425 eV to be consistent to the 0.085 eV reference Lorentzian width reported previously[\[44\]](#page-22-4). The curve-fit spectra are shown in Fig. 6(b-d), with parameters given in Table I. The results indicated four Si components, or four chemical sites, on the Al(111)3×3-Si surface. To analyze chemical environments at the Al(111)3×3-Si surface, the first-principles spectral simulations were calculated using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0). Figure 7(a) shows four prominent peaks in a simulated spectrum for the silicene layer on Al(111). Because of spin-orbit splitting, each Si 2p core-level appeared as a doublet state, and the simulations indicated two ![FIG. 6. Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface, taken at emission angles of (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with 150 eV photons. In (b-d), experimental spectra are dotted lines, while curve-fit spectra of the Si components are given by green and red curves, respectively. The curve-fit parameters are summarized in Table I.](path) TABLE I. Fitting parameters of the Si 2p core-level spectra for the Al(111)3×3-Si surface, shown in Fig. 6. The spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.6 eV, 0.0425 eV, and 0.5, respectively. Si components for the model structure. As depicted in the inset, one was attributed to Si atoms at atop Al sites and the other to bridge Al sites. While the calculations successfully linked the surface structure and the core-level spectra, the simulated results for the silicene layer were completely different from the experimental data (Fig. 6). Thus, it was necessity to consider a surface structure beyond a simple silicene layer for the Al(111)3×3-Si surface. ![FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦ , (c) θ = 30◦ , and (d) θ = 60◦ . In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b-d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.](path) Si sites in the Al-embedded silicene layer model (Fig. 8). As noted earlier, Si atoms in the surface layer can bond with different numbers of intralayer Si and/or Al atoms. For example, in Fig. 8(a), one Si atom used for the calculations was bonded with two Si atoms and one Al nearest neighbor (NN). Table II lists binding energies of the Si 2p3/<sup>2</sup> level for various structure models of silicene and Al-embedded silicene layers on Al(111). The core-level binding energies varied with respect to the number of NN Al atoms in the Si-Al bonding. Smaller NN values had higher binding energies, which was consistent with the silicene model results given in Table II. Moreover, there was a quantitative relationship between the binding energy and the NN value: 98.4±0.1eV (NN=3), 98.6±0.1eV (NN=2), 98.7±0.1eV (NN=1), and 98.8±0.1eV (NN=0). Because the model calculations in Fig. 8 considered only a single (atop) site on the Al substrate, differences in the substitution sites should be considered in TABLE III. Simulation parameters for the Si 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 7. The spectral components were given by Voigt functions with 0.085 eV Lorentzian widths and Gaussian widths given in the table. Spin-orbit splitting and a branching ratio were set at 0.6 eV and 0.5, respectively. The simulation was performed using OpenMX[\[34,](#page-21-8) [38\]](#page-22-0). On the basis of the calculated binding energies, the experimental spectra of the Al(111)3×3- Si surface were fairly well reproduced by five components, S1, S2, S2', S3 and S4, with the parameters in Table III. From the binding energies, S3 and S4 could be assigned to Si atoms that bonded only with NN Si atoms, while S1, S2, and S2' were also bonded to NN Al ![FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.](path) atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 6. Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface, taken at emission angles of (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with 150 eV photons. In (b-d), experimental spectra are dotted lines, while curve-fit spectra of the Si components are given by green and red curves, respectively. The curve-fit parameters are summarized in Table I.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. TABLE I. Fitting parameters of the Si 2p core-level spectra for the Al(111)3×3-Si surface, shown in Fig. 6. The spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.6 eV, 0.0425 eV, and 0.5, respectively. Si components for the model structure. As depicted in the inset, one was attributed to Si atoms at atop Al sites and the other to bridge Al sites. While the calculations successfully linked the surface structure and the core-level spectra, the simulated results for the silicene layer were completely different from the experimental data (Fig. 6). Thus, it was necessity to consider a surface structure beyond a simple silicene layer for the Al(111)3×3-Si surface. ![FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦ , (c) θ = 30◦ , and (d) θ = 60◦ . In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b-d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.](path) Si sites in the Al-embedded silicene layer model (Fig. 8). As noted earlier, Si atoms in the surface layer can bond with different numbers of intralayer Si and/or Al atoms. For example, in Fig. 8(a), one Si atom used for the calculations was bonded with two Si atoms and one Al nearest neighbor (NN). Table II lists binding energies of the Si 2p3/<sup>2</sup> level for various structure models of silicene and Al-embedded silicene layers on Al(111). The core-level binding energies varied with respect to the number of NN Al atoms in the Si-Al bonding. Smaller NN values had higher binding energies, which was consistent with the silicene model results given in Table II. Moreover, there was a quantitative relationship between the binding energy and the NN value: 98.4±0.1eV (NN=3), 98.6±0.1eV (NN=2), 98.7±0.1eV (NN=1), and 98.8±0.1eV (NN=0). Because the model calculations in Fig. 8 considered only a single (atop) site on the Al substrate, differences in the substitution sites should be considered in TABLE III. Simulation parameters for the Si 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 7. The spectral components were given by Voigt functions with 0.085 eV Lorentzian widths and Gaussian widths given in the table. Spin-orbit splitting and a branching ratio were set at 0.6 eV and 0.5, respectively. The simulation was performed using OpenMX[\[34,](#page-21-8) [38\]](#page-22-0). On the basis of the calculated binding energies, the experimental spectra of the Al(111)3×3- Si surface were fairly well reproduced by five components, S1, S2, S2', S3 and S4, with the parameters in Table III. From the binding energies, S3 and S4 could be assigned to Si atoms that bonded only with NN Si atoms, while S1, S2, and S2' were also bonded to NN Al ![FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.](path) atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 4. (a) A contour plot of the R-factor with respect to z positions of one (z<sup>7</sup> in Fig. 2) and the other Si atoms (z1,z2,z3,z4,z5,z6,z<sup>8</sup> in Fig. 2) in the Si honeycomb lattice layer on Al(111). (b) Total-reflection high-energy positron diffraction rocking curves for the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam under the one-beam condition.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. Figure 4(a) shows the 2D R-factor results for TRHEPD rocking curves under the onebeam condition [Fig. 4(b)] for the silicene model on Al(111), calculated via massively parallel data-analysis using 2DMAT[\[32\]](#page-21-5). In the model, there were eight atoms in the 3×3 unit cell and the height was z<sup>i</sup> for the i th atom (i = 1 ∼ 8). To examine deformation of the flat structure, the z<sup>i</sup> parameters were classified into two groups, one atom (z7) and the other containing atoms with the same height (z1=z2=z3=z4=z5=z6=z8). R-factor values were then plotted with respect to these heights [Fig. 4(a)]. In the 2D diagram, the minimum R-factor of 1.26 % was found at z<sup>7</sup> = 2.40 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.35 ˚A. The heights, z<sup>i</sup> 's, were essentially equal at each Si site. Further searches for optimum values around the minimum of R-factors were performed using Nelder-Mead methods with 2DMAT[\[32\]](#page-21-5). The optimum silicene atomic positions had nearly the same heights (z<sup>7</sup> = 2.38 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.37 ˚A). As shown in Fig. 4(b), the TRHEPD rocking curve calculated from the optimum heights was in good agreement with the experimental curve. Consequently, the analysis confirmed that the silicene layer was flat with no buckling structure. ![FIG. 5. Positron diffraction analysis with the Al-embedded silicene model. (a,b) Total-reflection high-energy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with an 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions. (c) Schematic of the model structure. A 3×3 unit cell is depicted.](path) Figure 5(a,b) is a set of TRHEPD rocking curves of the Al(111)3×3-Si surface, with comparisons to simulation curves (z1−<sup>8</sup> = 2.40 ˚A) of the flat Al-embedded silicene model that replaces one of the Si atoms with an Al atom [Fig. 5 (c)]. The results show good agreements that confirm the Al-Si mixed model. A R-factor value of the Al-embedded model was R = 2.12%, which was slightly smaller than R = 2.38% for the flat silicene model (Fig. 2). The R-factor values derived from Figs.4 and 5 were different because the former TRHEPD results were obtained with only a specular beam, while the latter were obtained with many diffracted beams. Previous reports proposed a "kagome-like silicene" model as the surface structure[\[21\]](#page-20-10). However, the TRHEPD analysis here was in disagreement, as shown in Fig. 12 in the Appendix. The first-principles calculation with OpenMX has also resulted in the flat structure for the Al-embedded silicene layer on Al(111) after the optimization. A standard derivation of the height (SDH) of the Al-embedded silicene layer on the Al(111) was only 0.06 ˚A. The result is consistent to the flat surface structure, as determined by the TRHEPD analyses. ## B. Core-level spectra Positron diffraction experiments revealed that the Al(111)3×3-Si surface structure could be described by a flat honeycomb lattice model. However, substitution of Al atoms in the Si overlayer was uncertain because of indistinguishable positron scattering. The Si atoms could be bonded with neighboring Si and/or Al atoms. This creates different chemical environments for these Si sites that should result in corresponding chemical shifts in Si core-level photoemission spectra[\[1\]](#page-19-0). Figure 6(a) shows Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface. There were at least five peaks, as reported previously[\[21\]](#page-20-10). Because a single chemical site for Si generates a doublet state, Si 2p3/<sup>2</sup> and 2p1/2, from spin-orbit splitting in the binding energy range, the spectrum indicated the existence of multiple chemical environments for Si atoms at the surface. To reveal the components, spectral curve-fittings were conducted using Doniach-Sunjic functions for various emission angles, θ's. The spin-orbit splitting and the branching ratio were fixed at 0.4 eV and 0.5, respectively. The Doniach-Sunjic width was fixed at 0.0425 eV to be consistent to the 0.085 eV reference Lorentzian width reported previously[\[44\]](#page-22-4). The curve-fit spectra are shown in Fig. 6(b-d), with parameters given in Table I. The results indicated four Si components, or four chemical sites, on the Al(111)3×3-Si surface. To analyze chemical environments at the Al(111)3×3-Si surface, the first-principles spectral simulations were calculated using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0). Figure 7(a) shows four prominent peaks in a simulated spectrum for the silicene layer on Al(111). Because of spin-orbit splitting, each Si 2p core-level appeared as a doublet state, and the simulations indicated two ![FIG. 6. Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface, taken at emission angles of (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with 150 eV photons. In (b-d), experimental spectra are dotted lines, while curve-fit spectra of the Si components are given by green and red curves, respectively. The curve-fit parameters are summarized in Table I.](path) TABLE I. Fitting parameters of the Si 2p core-level spectra for the Al(111)3×3-Si surface, shown in Fig. 6. The spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.6 eV, 0.0425 eV, and 0.5, respectively. Si components for the model structure. As depicted in the inset, one was attributed to Si atoms at atop Al sites and the other to bridge Al sites. While the calculations successfully linked the surface structure and the core-level spectra, the simulated results for the silicene layer were completely different from the experimental data (Fig. 6). Thus, it was necessity to consider a surface structure beyond a simple silicene layer for the Al(111)3×3-Si surface. ![FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦ , (c) θ = 30◦ , and (d) θ = 60◦ . In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b-d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.](path) Si sites in the Al-embedded silicene layer model (Fig. 8). As noted earlier, Si atoms in the surface layer can bond with different numbers of intralayer Si and/or Al atoms. For example, in Fig. 8(a), one Si atom used for the calculations was bonded with two Si atoms and one Al nearest neighbor (NN). Table II lists binding energies of the Si 2p3/<sup>2</sup> level for various structure models of silicene and Al-embedded silicene layers on Al(111). The core-level binding energies varied with respect to the number of NN Al atoms in the Si-Al bonding. Smaller NN values had higher binding energies, which was consistent with the silicene model results given in Table II. Moreover, there was a quantitative relationship between the binding energy and the NN value: 98.4±0.1eV (NN=3), 98.6±0.1eV (NN=2), 98.7±0.1eV (NN=1), and 98.8±0.1eV (NN=0). Because the model calculations in Fig. 8 considered only a single (atop) site on the Al substrate, differences in the substitution sites should be considered in TABLE III. Simulation parameters for the Si 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 7. The spectral components were given by Voigt functions with 0.085 eV Lorentzian widths and Gaussian widths given in the table. Spin-orbit splitting and a branching ratio were set at 0.6 eV and 0.5, respectively. The simulation was performed using OpenMX[\[34,](#page-21-8) [38\]](#page-22-0). On the basis of the calculated binding energies, the experimental spectra of the Al(111)3×3- Si surface were fairly well reproduced by five components, S1, S2, S2', S3 and S4, with the parameters in Table III. From the binding energies, S3 and S4 could be assigned to Si atoms that bonded only with NN Si atoms, while S1, S2, and S2' were also bonded to NN Al ![FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.](path) atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 1. Low-energy electron diffraction patterns acquired with 88.2 eV electrons. (a) A pristine Al(111) surface, and (b) the Al(111)3×3-Si surface[22], .	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. ![FIG. 1. Low-energy electron diffraction patterns acquired with 88.2 eV electrons. (a) A pristine Al(111) surface, and (b) the Al(111)3×3-Si surface[\[22\]](#page-20-9), .](path) # II. EXPERIMENTS AND SIMULATIONS ## A. Sample preparation The 3×3 phase was prepared via Si deposition on the Al(111) substrate in an ultrahigh vacuum chamber that had a base pressure of 2 × 10<sup>−</sup><sup>10</sup> mbar. A clean Al(111) surface was prepared via cycles of Ar<sup>+</sup> sputtering and annealing at 670 K. A clean and ordered surface was confirmed by elemental analyses via X-ray photoelectron spectroscopy and electron (positron) diffraction patterns, respectively. Here, various diffraction methods were used to characterize the sample surface: low-energy electron diffraction (LEED), reflection highenergy electron diffraction, and total-reflection high-energy positron diffraction (TRHEPD). Figure 1(a) shows a LEED pattern of a pristine Al(111) surface. Si was then deposited on this Al(111) substrate at 350 K. The Si source was based on sublimation of a Si wafer via Joule heating. Formation of the Al(111)3×3-Si phase was checked with a electron diffraction pattern, as shown in Fig. 1(b). The sample surface was then transferred in situ to perform positron diffraction and core-level spectroscopy. #### B. Positron diffraction The Al(111)3×3-Si structure was examined with TRHEPD [\[1,](#page-19-0) [11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1) performed at the Slow Positron Facility at KEK[\[28\]](#page-21-2). Details of the experimental setup were reported previously[\[29\]](#page-21-3). The energy of the incident positron beam was 10 keV, and rocking curves and spot intensities as a function of glancing angle of the beam were measured by rotating the sample holder up to 6◦ with step sizes of 0.1 ◦ . The surface structure was determined by searching for a model that had good agreement between experimental and simulated rocking curves. It was also quantitatively judged by a R-factor with the minimum value for the most appropriate structure model[\[11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1). The R-factor (R) was defined by: The data-driven analyses used TRHEPD results taken under the one-beam condition, and simulation results obtained with the 2DMAT software package[\[32\]](#page-21-5). Bayesian inference was performed via the massively parallel Monte Carlo method on the "Fugaku" supercomputer. #### C. Core-level photoemission spectroscopy Core-level photoemission spectroscopy measurements were performed at a vacuum ultraviolet photoemission beamline (Elettra, Trieste). Details were described previously[\[1,](#page-19-0) [12,](#page-20-4) [13\]](#page-20-11). The beamline covered a photon energy range of 20 eV to 750 eV, and the base pressure was 1 × 10<sup>−</sup><sup>11</sup> mbar. Simulations of the spectroscopy were performed with OpenMX[\[33–](#page-21-6)[38\]](#page-22-0). In the calculations, 3,3,2 optimized radial functions were allocated for Al s, p, and d orbitals with a cutoff radius of 7 Bohr. For Si atoms, 2, 2, 1 optimized radial functions were allocated for s, p, and d orbitals with a cutoff radius of 7 Bohr. A Perdew-Burke-Ernzerhof generalized gradient approximation was used as the exchange correlation functional[\[39\]](#page-22-1). In the spectral simulations, both atomic orbitals and pseudo-potentials included contributions from the 2p orbital. The energy cutoff and the energy convergence criterion were 220 Rydberg and 1.0× 10<sup>−</sup><sup>8</sup> Hartree, respectively. For the wavevector space, a 6×6×1 grid was used. In the calculations, an Al(111) slab of seven layers was used, where one side was covered with the Si(Al) surface layer. The slabs were repeatedly separated by a vacuum at a distance of 17 ˚A and facing the bare and overlayer surfaces. The binding energy of each core-level was obtained by calculating the total energy difference before and after creation of the core-hole state[\[38\]](#page-22-0). #### D. Electronic structure calculations Calculations of structural optimizations and band dispersion curves were performed using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0) with the same conditions used for the spectral simulations described above for optimized radial functions, a cutoff radius, and an exchange correlation functional. The energy cutoff was 300 Rydberg, while the energy convergence criterion was 1.0× 10<sup>−</sup><sup>8</sup> Hartree. A grid of 6×6×1 was used for the wavevector space. In the calculations, an Al(111) slab of five layers was adopted, where one side was covered with the Si(Al) surface layer. The slabs were repeatedly separated by vacuum at a distance of 23 ˚A. Structural optimization was performed by minimizing the force upon each atom. At the bottom of the Al(111) slab, the coordinates of the atoms were fixed, while those of the other atoms were relaxed. The force convergence criterion was 1.0 × 10<sup>−</sup><sup>4</sup> Hartree/Bohr. After the convergence of energy and force was achieved, the total energy of the system was calculated to assess the system stability. To simulate the 2D slab, an effective screening medium method was applied in which a semi-infinite vacuum was inserted at the bottom and top of the system[\[40](#page-22-2)[–43\]](#page-22-3). ![FIG. 2. Schematic of the flat silicene model. Si atoms are labeled with numbers 1 to 8. A 3×3 unit cell is depicted.](path) #### III. RESULTS AND DISCUSSION #### A. Positron diffraction TRHEPD was conducted to investigate the structure of the Al(111)3×3-Si surface. TRHEPD rocking curves were measured at 13◦ off from the [11¯2] direction (one-beam condition), and along the [11¯2] and [1¯10] directions. Under the one-beam condition, the specular (00) spot intensities depended mainly on the surface-normal components (z) of atomic positions because simultaneous reflections in the surface-parallel plane were sufficiently suppressed[\[31\]](#page-21-7). In the TRHEPD analyses, a surface structure was determined by searching for a model that had agreements between simulated and experimental curves[\[11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1). A honeycomb lattice of Si atoms, shown in Fig. 2, was adopted as an initial model because it had previously described the Si layer (silicene) on Ag(111) and the Ge layer (germanene) on Al(111)[\[11,](#page-20-3) [23\]](#page-21-0). ![FIG. 3. Real parts of scattering potentials, Re[U00], averaged in the surface-parallel plane for the Si honeycomb lattice layer on Al(111) (red) without or (blue) with Al substitution. Bulk and surface layers are indicated by labels.](path) When performing the surface analysis via positron diffraction, it is important to note that Al and Si are located next to each other in the periodic table and thus have similar scattering potentials for a positron beam. This was demonstrated by the calculations shown in Fig. 3. The scattering event can be evaluated by the positron scattering potential. The scattering potential at the Si surface layer appeared nearly identical without or with Al substitution. This indicated that the surface structure model created uncertainty regarding embedded Al atoms. However, the structure parameters of the Si(Al) surface layer can be elaborated by considering only Si atoms. ![FIG. 4. (a) A contour plot of the R-factor with respect to z positions of one (z<sup>7</sup> in Fig. 2) and the other Si atoms (z1,z2,z3,z4,z5,z6,z<sup>8</sup> in Fig. 2) in the Si honeycomb lattice layer on Al(111). (b) Total-reflection high-energy positron diffraction rocking curves for the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam under the one-beam condition.](path) Figure 4(a) shows the 2D R-factor results for TRHEPD rocking curves under the onebeam condition [Fig. 4(b)] for the silicene model on Al(111), calculated via massively parallel data-analysis using 2DMAT[\[32\]](#page-21-5). In the model, there were eight atoms in the 3×3 unit cell and the height was z<sup>i</sup> for the i th atom (i = 1 ∼ 8). To examine deformation of the flat structure, the z<sup>i</sup> parameters were classified into two groups, one atom (z7) and the other containing atoms with the same height (z1=z2=z3=z4=z5=z6=z8). R-factor values were then plotted with respect to these heights [Fig. 4(a)]. In the 2D diagram, the minimum R-factor of 1.26 % was found at z<sup>7</sup> = 2.40 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.35 ˚A. The heights, z<sup>i</sup> 's, were essentially equal at each Si site. Further searches for optimum values around the minimum of R-factors were performed using Nelder-Mead methods with 2DMAT[\[32\]](#page-21-5). The optimum silicene atomic positions had nearly the same heights (z<sup>7</sup> = 2.38 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.37 ˚A). As shown in Fig. 4(b), the TRHEPD rocking curve calculated from the optimum heights was in good agreement with the experimental curve. Consequently, the analysis confirmed that the silicene layer was flat with no buckling structure. ![FIG. 5. Positron diffraction analysis with the Al-embedded silicene model. (a,b) Total-reflection high-energy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with an 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions. (c) Schematic of the model structure. A 3×3 unit cell is depicted.](path) Figure 5(a,b) is a set of TRHEPD rocking curves of the Al(111)3×3-Si surface, with comparisons to simulation curves (z1−<sup>8</sup> = 2.40 ˚A) of the flat Al-embedded silicene model that replaces one of the Si atoms with an Al atom [Fig. 5 (c)]. The results show good agreements that confirm the Al-Si mixed model. A R-factor value of the Al-embedded model was R = 2.12%, which was slightly smaller than R = 2.38% for the flat silicene model (Fig. 2). The R-factor values derived from Figs.4 and 5 were different because the former TRHEPD results were obtained with only a specular beam, while the latter were obtained with many diffracted beams. Previous reports proposed a "kagome-like silicene" model as the surface structure[\[21\]](#page-20-10). However, the TRHEPD analysis here was in disagreement, as shown in Fig. 12 in the Appendix. The first-principles calculation with OpenMX has also resulted in the flat structure for the Al-embedded silicene layer on Al(111) after the optimization. A standard derivation of the height (SDH) of the Al-embedded silicene layer on the Al(111) was only 0.06 ˚A. The result is consistent to the flat surface structure, as determined by the TRHEPD analyses. ## B. Core-level spectra Positron diffraction experiments revealed that the Al(111)3×3-Si surface structure could be described by a flat honeycomb lattice model. However, substitution of Al atoms in the Si overlayer was uncertain because of indistinguishable positron scattering. The Si atoms could be bonded with neighboring Si and/or Al atoms. This creates different chemical environments for these Si sites that should result in corresponding chemical shifts in Si core-level photoemission spectra[\[1\]](#page-19-0). Figure 6(a) shows Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface. There were at least five peaks, as reported previously[\[21\]](#page-20-10). Because a single chemical site for Si generates a doublet state, Si 2p3/<sup>2</sup> and 2p1/2, from spin-orbit splitting in the binding energy range, the spectrum indicated the existence of multiple chemical environments for Si atoms at the surface. To reveal the components, spectral curve-fittings were conducted using Doniach-Sunjic functions for various emission angles, θ's. The spin-orbit splitting and the branching ratio were fixed at 0.4 eV and 0.5, respectively. The Doniach-Sunjic width was fixed at 0.0425 eV to be consistent to the 0.085 eV reference Lorentzian width reported previously[\[44\]](#page-22-4). The curve-fit spectra are shown in Fig. 6(b-d), with parameters given in Table I. The results indicated four Si components, or four chemical sites, on the Al(111)3×3-Si surface. To analyze chemical environments at the Al(111)3×3-Si surface, the first-principles spectral simulations were calculated using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0). Figure 7(a) shows four prominent peaks in a simulated spectrum for the silicene layer on Al(111). Because of spin-orbit splitting, each Si 2p core-level appeared as a doublet state, and the simulations indicated two ![FIG. 6. Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface, taken at emission angles of (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with 150 eV photons. In (b-d), experimental spectra are dotted lines, while curve-fit spectra of the Si components are given by green and red curves, respectively. The curve-fit parameters are summarized in Table I.](path) TABLE I. Fitting parameters of the Si 2p core-level spectra for the Al(111)3×3-Si surface, shown in Fig. 6. The spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.6 eV, 0.0425 eV, and 0.5, respectively. Si components for the model structure. As depicted in the inset, one was attributed to Si atoms at atop Al sites and the other to bridge Al sites. While the calculations successfully linked the surface structure and the core-level spectra, the simulated results for the silicene layer were completely different from the experimental data (Fig. 6). Thus, it was necessity to consider a surface structure beyond a simple silicene layer for the Al(111)3×3-Si surface. ![FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦ , (c) θ = 30◦ , and (d) θ = 60◦ . In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b-d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.](path) Si sites in the Al-embedded silicene layer model (Fig. 8). As noted earlier, Si atoms in the surface layer can bond with different numbers of intralayer Si and/or Al atoms. For example, in Fig. 8(a), one Si atom used for the calculations was bonded with two Si atoms and one Al nearest neighbor (NN). Table II lists binding energies of the Si 2p3/<sup>2</sup> level for various structure models of silicene and Al-embedded silicene layers on Al(111). The core-level binding energies varied with respect to the number of NN Al atoms in the Si-Al bonding. Smaller NN values had higher binding energies, which was consistent with the silicene model results given in Table II. Moreover, there was a quantitative relationship between the binding energy and the NN value: 98.4±0.1eV (NN=3), 98.6±0.1eV (NN=2), 98.7±0.1eV (NN=1), and 98.8±0.1eV (NN=0). Because the model calculations in Fig. 8 considered only a single (atop) site on the Al substrate, differences in the substitution sites should be considered in TABLE III. Simulation parameters for the Si 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 7. The spectral components were given by Voigt functions with 0.085 eV Lorentzian widths and Gaussian widths given in the table. Spin-orbit splitting and a branching ratio were set at 0.6 eV and 0.5, respectively. The simulation was performed using OpenMX[\[34,](#page-21-8) [38\]](#page-22-0). On the basis of the calculated binding energies, the experimental spectra of the Al(111)3×3- Si surface were fairly well reproduced by five components, S1, S2, S2', S3 and S4, with the parameters in Table III. From the binding energies, S3 and S4 could be assigned to Si atoms that bonded only with NN Si atoms, while S1, S2, and S2' were also bonded to NN Al ![FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.](path) atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[46].	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. ![FIG. 1. Low-energy electron diffraction patterns acquired with 88.2 eV electrons. (a) A pristine Al(111) surface, and (b) the Al(111)3×3-Si surface[\[22\]](#page-20-9), .](path) # II. EXPERIMENTS AND SIMULATIONS ## A. Sample preparation The 3×3 phase was prepared via Si deposition on the Al(111) substrate in an ultrahigh vacuum chamber that had a base pressure of 2 × 10<sup>−</sup><sup>10</sup> mbar. A clean Al(111) surface was prepared via cycles of Ar<sup>+</sup> sputtering and annealing at 670 K. A clean and ordered surface was confirmed by elemental analyses via X-ray photoelectron spectroscopy and electron (positron) diffraction patterns, respectively. Here, various diffraction methods were used to characterize the sample surface: low-energy electron diffraction (LEED), reflection highenergy electron diffraction, and total-reflection high-energy positron diffraction (TRHEPD). Figure 1(a) shows a LEED pattern of a pristine Al(111) surface. Si was then deposited on this Al(111) substrate at 350 K. The Si source was based on sublimation of a Si wafer via Joule heating. Formation of the Al(111)3×3-Si phase was checked with a electron diffraction pattern, as shown in Fig. 1(b). The sample surface was then transferred in situ to perform positron diffraction and core-level spectroscopy. #### B. Positron diffraction The Al(111)3×3-Si structure was examined with TRHEPD [\[1,](#page-19-0) [11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1) performed at the Slow Positron Facility at KEK[\[28\]](#page-21-2). Details of the experimental setup were reported previously[\[29\]](#page-21-3). The energy of the incident positron beam was 10 keV, and rocking curves and spot intensities as a function of glancing angle of the beam were measured by rotating the sample holder up to 6◦ with step sizes of 0.1 ◦ . The surface structure was determined by searching for a model that had good agreement between experimental and simulated rocking curves. It was also quantitatively judged by a R-factor with the minimum value for the most appropriate structure model[\[11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1). The R-factor (R) was defined by: The data-driven analyses used TRHEPD results taken under the one-beam condition, and simulation results obtained with the 2DMAT software package[\[32\]](#page-21-5). Bayesian inference was performed via the massively parallel Monte Carlo method on the "Fugaku" supercomputer. #### C. Core-level photoemission spectroscopy Core-level photoemission spectroscopy measurements were performed at a vacuum ultraviolet photoemission beamline (Elettra, Trieste). Details were described previously[\[1,](#page-19-0) [12,](#page-20-4) [13\]](#page-20-11). The beamline covered a photon energy range of 20 eV to 750 eV, and the base pressure was 1 × 10<sup>−</sup><sup>11</sup> mbar. Simulations of the spectroscopy were performed with OpenMX[\[33–](#page-21-6)[38\]](#page-22-0). In the calculations, 3,3,2 optimized radial functions were allocated for Al s, p, and d orbitals with a cutoff radius of 7 Bohr. For Si atoms, 2, 2, 1 optimized radial functions were allocated for s, p, and d orbitals with a cutoff radius of 7 Bohr. A Perdew-Burke-Ernzerhof generalized gradient approximation was used as the exchange correlation functional[\[39\]](#page-22-1). In the spectral simulations, both atomic orbitals and pseudo-potentials included contributions from the 2p orbital. The energy cutoff and the energy convergence criterion were 220 Rydberg and 1.0× 10<sup>−</sup><sup>8</sup> Hartree, respectively. For the wavevector space, a 6×6×1 grid was used. In the calculations, an Al(111) slab of seven layers was used, where one side was covered with the Si(Al) surface layer. The slabs were repeatedly separated by a vacuum at a distance of 17 ˚A and facing the bare and overlayer surfaces. The binding energy of each core-level was obtained by calculating the total energy difference before and after creation of the core-hole state[\[38\]](#page-22-0). #### D. Electronic structure calculations Calculations of structural optimizations and band dispersion curves were performed using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0) with the same conditions used for the spectral simulations described above for optimized radial functions, a cutoff radius, and an exchange correlation functional. The energy cutoff was 300 Rydberg, while the energy convergence criterion was 1.0× 10<sup>−</sup><sup>8</sup> Hartree. A grid of 6×6×1 was used for the wavevector space. In the calculations, an Al(111) slab of five layers was adopted, where one side was covered with the Si(Al) surface layer. The slabs were repeatedly separated by vacuum at a distance of 23 ˚A. Structural optimization was performed by minimizing the force upon each atom. At the bottom of the Al(111) slab, the coordinates of the atoms were fixed, while those of the other atoms were relaxed. The force convergence criterion was 1.0 × 10<sup>−</sup><sup>4</sup> Hartree/Bohr. After the convergence of energy and force was achieved, the total energy of the system was calculated to assess the system stability. To simulate the 2D slab, an effective screening medium method was applied in which a semi-infinite vacuum was inserted at the bottom and top of the system[\[40](#page-22-2)[–43\]](#page-22-3). ![FIG. 2. Schematic of the flat silicene model. Si atoms are labeled with numbers 1 to 8. A 3×3 unit cell is depicted.](path) #### III. RESULTS AND DISCUSSION #### A. Positron diffraction TRHEPD was conducted to investigate the structure of the Al(111)3×3-Si surface. TRHEPD rocking curves were measured at 13◦ off from the [11¯2] direction (one-beam condition), and along the [11¯2] and [1¯10] directions. Under the one-beam condition, the specular (00) spot intensities depended mainly on the surface-normal components (z) of atomic positions because simultaneous reflections in the surface-parallel plane were sufficiently suppressed[\[31\]](#page-21-7). In the TRHEPD analyses, a surface structure was determined by searching for a model that had agreements between simulated and experimental curves[\[11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1). A honeycomb lattice of Si atoms, shown in Fig. 2, was adopted as an initial model because it had previously described the Si layer (silicene) on Ag(111) and the Ge layer (germanene) on Al(111)[\[11,](#page-20-3) [23\]](#page-21-0). ![FIG. 3. Real parts of scattering potentials, Re[U00], averaged in the surface-parallel plane for the Si honeycomb lattice layer on Al(111) (red) without or (blue) with Al substitution. Bulk and surface layers are indicated by labels.](path) When performing the surface analysis via positron diffraction, it is important to note that Al and Si are located next to each other in the periodic table and thus have similar scattering potentials for a positron beam. This was demonstrated by the calculations shown in Fig. 3. The scattering event can be evaluated by the positron scattering potential. The scattering potential at the Si surface layer appeared nearly identical without or with Al substitution. This indicated that the surface structure model created uncertainty regarding embedded Al atoms. However, the structure parameters of the Si(Al) surface layer can be elaborated by considering only Si atoms. ![FIG. 4. (a) A contour plot of the R-factor with respect to z positions of one (z<sup>7</sup> in Fig. 2) and the other Si atoms (z1,z2,z3,z4,z5,z6,z<sup>8</sup> in Fig. 2) in the Si honeycomb lattice layer on Al(111). (b) Total-reflection high-energy positron diffraction rocking curves for the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam under the one-beam condition.](path) Figure 4(a) shows the 2D R-factor results for TRHEPD rocking curves under the onebeam condition [Fig. 4(b)] for the silicene model on Al(111), calculated via massively parallel data-analysis using 2DMAT[\[32\]](#page-21-5). In the model, there were eight atoms in the 3×3 unit cell and the height was z<sup>i</sup> for the i th atom (i = 1 ∼ 8). To examine deformation of the flat structure, the z<sup>i</sup> parameters were classified into two groups, one atom (z7) and the other containing atoms with the same height (z1=z2=z3=z4=z5=z6=z8). R-factor values were then plotted with respect to these heights [Fig. 4(a)]. In the 2D diagram, the minimum R-factor of 1.26 % was found at z<sup>7</sup> = 2.40 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.35 ˚A. The heights, z<sup>i</sup> 's, were essentially equal at each Si site. Further searches for optimum values around the minimum of R-factors were performed using Nelder-Mead methods with 2DMAT[\[32\]](#page-21-5). The optimum silicene atomic positions had nearly the same heights (z<sup>7</sup> = 2.38 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.37 ˚A). As shown in Fig. 4(b), the TRHEPD rocking curve calculated from the optimum heights was in good agreement with the experimental curve. Consequently, the analysis confirmed that the silicene layer was flat with no buckling structure. ![FIG. 5. Positron diffraction analysis with the Al-embedded silicene model. (a,b) Total-reflection high-energy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with an 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions. (c) Schematic of the model structure. A 3×3 unit cell is depicted.](path) Figure 5(a,b) is a set of TRHEPD rocking curves of the Al(111)3×3-Si surface, with comparisons to simulation curves (z1−<sup>8</sup> = 2.40 ˚A) of the flat Al-embedded silicene model that replaces one of the Si atoms with an Al atom [Fig. 5 (c)]. The results show good agreements that confirm the Al-Si mixed model. A R-factor value of the Al-embedded model was R = 2.12%, which was slightly smaller than R = 2.38% for the flat silicene model (Fig. 2). The R-factor values derived from Figs.4 and 5 were different because the former TRHEPD results were obtained with only a specular beam, while the latter were obtained with many diffracted beams. Previous reports proposed a "kagome-like silicene" model as the surface structure[\[21\]](#page-20-10). However, the TRHEPD analysis here was in disagreement, as shown in Fig. 12 in the Appendix. The first-principles calculation with OpenMX has also resulted in the flat structure for the Al-embedded silicene layer on Al(111) after the optimization. A standard derivation of the height (SDH) of the Al-embedded silicene layer on the Al(111) was only 0.06 ˚A. The result is consistent to the flat surface structure, as determined by the TRHEPD analyses. ## B. Core-level spectra Positron diffraction experiments revealed that the Al(111)3×3-Si surface structure could be described by a flat honeycomb lattice model. However, substitution of Al atoms in the Si overlayer was uncertain because of indistinguishable positron scattering. The Si atoms could be bonded with neighboring Si and/or Al atoms. This creates different chemical environments for these Si sites that should result in corresponding chemical shifts in Si core-level photoemission spectra[\[1\]](#page-19-0). Figure 6(a) shows Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface. There were at least five peaks, as reported previously[\[21\]](#page-20-10). Because a single chemical site for Si generates a doublet state, Si 2p3/<sup>2</sup> and 2p1/2, from spin-orbit splitting in the binding energy range, the spectrum indicated the existence of multiple chemical environments for Si atoms at the surface. To reveal the components, spectral curve-fittings were conducted using Doniach-Sunjic functions for various emission angles, θ's. The spin-orbit splitting and the branching ratio were fixed at 0.4 eV and 0.5, respectively. The Doniach-Sunjic width was fixed at 0.0425 eV to be consistent to the 0.085 eV reference Lorentzian width reported previously[\[44\]](#page-22-4). The curve-fit spectra are shown in Fig. 6(b-d), with parameters given in Table I. The results indicated four Si components, or four chemical sites, on the Al(111)3×3-Si surface. To analyze chemical environments at the Al(111)3×3-Si surface, the first-principles spectral simulations were calculated using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0). Figure 7(a) shows four prominent peaks in a simulated spectrum for the silicene layer on Al(111). Because of spin-orbit splitting, each Si 2p core-level appeared as a doublet state, and the simulations indicated two ![FIG. 6. Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface, taken at emission angles of (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with 150 eV photons. In (b-d), experimental spectra are dotted lines, while curve-fit spectra of the Si components are given by green and red curves, respectively. The curve-fit parameters are summarized in Table I.](path) TABLE I. Fitting parameters of the Si 2p core-level spectra for the Al(111)3×3-Si surface, shown in Fig. 6. The spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.6 eV, 0.0425 eV, and 0.5, respectively. Si components for the model structure. As depicted in the inset, one was attributed to Si atoms at atop Al sites and the other to bridge Al sites. While the calculations successfully linked the surface structure and the core-level spectra, the simulated results for the silicene layer were completely different from the experimental data (Fig. 6). Thus, it was necessity to consider a surface structure beyond a simple silicene layer for the Al(111)3×3-Si surface. ![FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦ , (c) θ = 30◦ , and (d) θ = 60◦ . In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b-d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.](path) Si sites in the Al-embedded silicene layer model (Fig. 8). As noted earlier, Si atoms in the surface layer can bond with different numbers of intralayer Si and/or Al atoms. For example, in Fig. 8(a), one Si atom used for the calculations was bonded with two Si atoms and one Al nearest neighbor (NN). Table II lists binding energies of the Si 2p3/<sup>2</sup> level for various structure models of silicene and Al-embedded silicene layers on Al(111). The core-level binding energies varied with respect to the number of NN Al atoms in the Si-Al bonding. Smaller NN values had higher binding energies, which was consistent with the silicene model results given in Table II. Moreover, there was a quantitative relationship between the binding energy and the NN value: 98.4±0.1eV (NN=3), 98.6±0.1eV (NN=2), 98.7±0.1eV (NN=1), and 98.8±0.1eV (NN=0). Because the model calculations in Fig. 8 considered only a single (atop) site on the Al substrate, differences in the substitution sites should be considered in TABLE III. Simulation parameters for the Si 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 7. The spectral components were given by Voigt functions with 0.085 eV Lorentzian widths and Gaussian widths given in the table. Spin-orbit splitting and a branching ratio were set at 0.6 eV and 0.5, respectively. The simulation was performed using OpenMX[\[34,](#page-21-8) [38\]](#page-22-0). On the basis of the calculated binding energies, the experimental spectra of the Al(111)3×3- Si surface were fairly well reproduced by five components, S1, S2, S2', S3 and S4, with the parameters in Table III. From the binding energies, S3 and S4 could be assigned to Si atoms that bonded only with NN Si atoms, while S1, S2, and S2' were also bonded to NN Al ![FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.](path) atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦, (c) θ = 30◦, and (d) θ = 60◦. In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b–d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. ![FIG. 1. Low-energy electron diffraction patterns acquired with 88.2 eV electrons. (a) A pristine Al(111) surface, and (b) the Al(111)3×3-Si surface[\[22\]](#page-20-9), .](path) # II. EXPERIMENTS AND SIMULATIONS ## A. Sample preparation The 3×3 phase was prepared via Si deposition on the Al(111) substrate in an ultrahigh vacuum chamber that had a base pressure of 2 × 10<sup>−</sup><sup>10</sup> mbar. A clean Al(111) surface was prepared via cycles of Ar<sup>+</sup> sputtering and annealing at 670 K. A clean and ordered surface was confirmed by elemental analyses via X-ray photoelectron spectroscopy and electron (positron) diffraction patterns, respectively. Here, various diffraction methods were used to characterize the sample surface: low-energy electron diffraction (LEED), reflection highenergy electron diffraction, and total-reflection high-energy positron diffraction (TRHEPD). Figure 1(a) shows a LEED pattern of a pristine Al(111) surface. Si was then deposited on this Al(111) substrate at 350 K. The Si source was based on sublimation of a Si wafer via Joule heating. Formation of the Al(111)3×3-Si phase was checked with a electron diffraction pattern, as shown in Fig. 1(b). The sample surface was then transferred in situ to perform positron diffraction and core-level spectroscopy. #### B. Positron diffraction The Al(111)3×3-Si structure was examined with TRHEPD [\[1,](#page-19-0) [11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1) performed at the Slow Positron Facility at KEK[\[28\]](#page-21-2). Details of the experimental setup were reported previously[\[29\]](#page-21-3). The energy of the incident positron beam was 10 keV, and rocking curves and spot intensities as a function of glancing angle of the beam were measured by rotating the sample holder up to 6◦ with step sizes of 0.1 ◦ . The surface structure was determined by searching for a model that had good agreement between experimental and simulated rocking curves. It was also quantitatively judged by a R-factor with the minimum value for the most appropriate structure model[\[11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1). The R-factor (R) was defined by: The data-driven analyses used TRHEPD results taken under the one-beam condition, and simulation results obtained with the 2DMAT software package[\[32\]](#page-21-5). Bayesian inference was performed via the massively parallel Monte Carlo method on the "Fugaku" supercomputer. #### C. Core-level photoemission spectroscopy Core-level photoemission spectroscopy measurements were performed at a vacuum ultraviolet photoemission beamline (Elettra, Trieste). Details were described previously[\[1,](#page-19-0) [12,](#page-20-4) [13\]](#page-20-11). The beamline covered a photon energy range of 20 eV to 750 eV, and the base pressure was 1 × 10<sup>−</sup><sup>11</sup> mbar. Simulations of the spectroscopy were performed with OpenMX[\[33–](#page-21-6)[38\]](#page-22-0). In the calculations, 3,3,2 optimized radial functions were allocated for Al s, p, and d orbitals with a cutoff radius of 7 Bohr. For Si atoms, 2, 2, 1 optimized radial functions were allocated for s, p, and d orbitals with a cutoff radius of 7 Bohr. A Perdew-Burke-Ernzerhof generalized gradient approximation was used as the exchange correlation functional[\[39\]](#page-22-1). In the spectral simulations, both atomic orbitals and pseudo-potentials included contributions from the 2p orbital. The energy cutoff and the energy convergence criterion were 220 Rydberg and 1.0× 10<sup>−</sup><sup>8</sup> Hartree, respectively. For the wavevector space, a 6×6×1 grid was used. In the calculations, an Al(111) slab of seven layers was used, where one side was covered with the Si(Al) surface layer. The slabs were repeatedly separated by a vacuum at a distance of 17 ˚A and facing the bare and overlayer surfaces. The binding energy of each core-level was obtained by calculating the total energy difference before and after creation of the core-hole state[\[38\]](#page-22-0). #### D. Electronic structure calculations Calculations of structural optimizations and band dispersion curves were performed using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0) with the same conditions used for the spectral simulations described above for optimized radial functions, a cutoff radius, and an exchange correlation functional. The energy cutoff was 300 Rydberg, while the energy convergence criterion was 1.0× 10<sup>−</sup><sup>8</sup> Hartree. A grid of 6×6×1 was used for the wavevector space. In the calculations, an Al(111) slab of five layers was adopted, where one side was covered with the Si(Al) surface layer. The slabs were repeatedly separated by vacuum at a distance of 23 ˚A. Structural optimization was performed by minimizing the force upon each atom. At the bottom of the Al(111) slab, the coordinates of the atoms were fixed, while those of the other atoms were relaxed. The force convergence criterion was 1.0 × 10<sup>−</sup><sup>4</sup> Hartree/Bohr. After the convergence of energy and force was achieved, the total energy of the system was calculated to assess the system stability. To simulate the 2D slab, an effective screening medium method was applied in which a semi-infinite vacuum was inserted at the bottom and top of the system[\[40](#page-22-2)[–43\]](#page-22-3). ![FIG. 2. Schematic of the flat silicene model. Si atoms are labeled with numbers 1 to 8. A 3×3 unit cell is depicted.](path) #### III. RESULTS AND DISCUSSION #### A. Positron diffraction TRHEPD was conducted to investigate the structure of the Al(111)3×3-Si surface. TRHEPD rocking curves were measured at 13◦ off from the [11¯2] direction (one-beam condition), and along the [11¯2] and [1¯10] directions. Under the one-beam condition, the specular (00) spot intensities depended mainly on the surface-normal components (z) of atomic positions because simultaneous reflections in the surface-parallel plane were sufficiently suppressed[\[31\]](#page-21-7). In the TRHEPD analyses, a surface structure was determined by searching for a model that had agreements between simulated and experimental curves[\[11,](#page-20-3) [23–](#page-21-0)[27\]](#page-21-1). A honeycomb lattice of Si atoms, shown in Fig. 2, was adopted as an initial model because it had previously described the Si layer (silicene) on Ag(111) and the Ge layer (germanene) on Al(111)[\[11,](#page-20-3) [23\]](#page-21-0). ![FIG. 3. Real parts of scattering potentials, Re[U00], averaged in the surface-parallel plane for the Si honeycomb lattice layer on Al(111) (red) without or (blue) with Al substitution. Bulk and surface layers are indicated by labels.](path) When performing the surface analysis via positron diffraction, it is important to note that Al and Si are located next to each other in the periodic table and thus have similar scattering potentials for a positron beam. This was demonstrated by the calculations shown in Fig. 3. The scattering event can be evaluated by the positron scattering potential. The scattering potential at the Si surface layer appeared nearly identical without or with Al substitution. This indicated that the surface structure model created uncertainty regarding embedded Al atoms. However, the structure parameters of the Si(Al) surface layer can be elaborated by considering only Si atoms. ![FIG. 4. (a) A contour plot of the R-factor with respect to z positions of one (z<sup>7</sup> in Fig. 2) and the other Si atoms (z1,z2,z3,z4,z5,z6,z<sup>8</sup> in Fig. 2) in the Si honeycomb lattice layer on Al(111). (b) Total-reflection high-energy positron diffraction rocking curves for the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam under the one-beam condition.](path) Figure 4(a) shows the 2D R-factor results for TRHEPD rocking curves under the onebeam condition [Fig. 4(b)] for the silicene model on Al(111), calculated via massively parallel data-analysis using 2DMAT[\[32\]](#page-21-5). In the model, there were eight atoms in the 3×3 unit cell and the height was z<sup>i</sup> for the i th atom (i = 1 ∼ 8). To examine deformation of the flat structure, the z<sup>i</sup> parameters were classified into two groups, one atom (z7) and the other containing atoms with the same height (z1=z2=z3=z4=z5=z6=z8). R-factor values were then plotted with respect to these heights [Fig. 4(a)]. In the 2D diagram, the minimum R-factor of 1.26 % was found at z<sup>7</sup> = 2.40 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.35 ˚A. The heights, z<sup>i</sup> 's, were essentially equal at each Si site. Further searches for optimum values around the minimum of R-factors were performed using Nelder-Mead methods with 2DMAT[\[32\]](#page-21-5). The optimum silicene atomic positions had nearly the same heights (z<sup>7</sup> = 2.38 ˚A and z1=z2=z3=z4=z5=z6=z<sup>8</sup> = 2.37 ˚A). As shown in Fig. 4(b), the TRHEPD rocking curve calculated from the optimum heights was in good agreement with the experimental curve. Consequently, the analysis confirmed that the silicene layer was flat with no buckling structure. ![FIG. 5. Positron diffraction analysis with the Al-embedded silicene model. (a,b) Total-reflection high-energy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with an 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions. (c) Schematic of the model structure. A 3×3 unit cell is depicted.](path) Figure 5(a,b) is a set of TRHEPD rocking curves of the Al(111)3×3-Si surface, with comparisons to simulation curves (z1−<sup>8</sup> = 2.40 ˚A) of the flat Al-embedded silicene model that replaces one of the Si atoms with an Al atom [Fig. 5 (c)]. The results show good agreements that confirm the Al-Si mixed model. A R-factor value of the Al-embedded model was R = 2.12%, which was slightly smaller than R = 2.38% for the flat silicene model (Fig. 2). The R-factor values derived from Figs.4 and 5 were different because the former TRHEPD results were obtained with only a specular beam, while the latter were obtained with many diffracted beams. Previous reports proposed a "kagome-like silicene" model as the surface structure[\[21\]](#page-20-10). However, the TRHEPD analysis here was in disagreement, as shown in Fig. 12 in the Appendix. The first-principles calculation with OpenMX has also resulted in the flat structure for the Al-embedded silicene layer on Al(111) after the optimization. A standard derivation of the height (SDH) of the Al-embedded silicene layer on the Al(111) was only 0.06 ˚A. The result is consistent to the flat surface structure, as determined by the TRHEPD analyses. ## B. Core-level spectra Positron diffraction experiments revealed that the Al(111)3×3-Si surface structure could be described by a flat honeycomb lattice model. However, substitution of Al atoms in the Si overlayer was uncertain because of indistinguishable positron scattering. The Si atoms could be bonded with neighboring Si and/or Al atoms. This creates different chemical environments for these Si sites that should result in corresponding chemical shifts in Si core-level photoemission spectra[\[1\]](#page-19-0). Figure 6(a) shows Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface. There were at least five peaks, as reported previously[\[21\]](#page-20-10). Because a single chemical site for Si generates a doublet state, Si 2p3/<sup>2</sup> and 2p1/2, from spin-orbit splitting in the binding energy range, the spectrum indicated the existence of multiple chemical environments for Si atoms at the surface. To reveal the components, spectral curve-fittings were conducted using Doniach-Sunjic functions for various emission angles, θ's. The spin-orbit splitting and the branching ratio were fixed at 0.4 eV and 0.5, respectively. The Doniach-Sunjic width was fixed at 0.0425 eV to be consistent to the 0.085 eV reference Lorentzian width reported previously[\[44\]](#page-22-4). The curve-fit spectra are shown in Fig. 6(b-d), with parameters given in Table I. The results indicated four Si components, or four chemical sites, on the Al(111)3×3-Si surface. To analyze chemical environments at the Al(111)3×3-Si surface, the first-principles spectral simulations were calculated using OpenMX[\[33](#page-21-6)[–38\]](#page-22-0). Figure 7(a) shows four prominent peaks in a simulated spectrum for the silicene layer on Al(111). Because of spin-orbit splitting, each Si 2p core-level appeared as a doublet state, and the simulations indicated two ![FIG. 6. Si 2p core-level photoemission spectra of the Al(111)3×3-Si surface, taken at emission angles of (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with 150 eV photons. In (b-d), experimental spectra are dotted lines, while curve-fit spectra of the Si components are given by green and red curves, respectively. The curve-fit parameters are summarized in Table I.](path) TABLE I. Fitting parameters of the Si 2p core-level spectra for the Al(111)3×3-Si surface, shown in Fig. 6. The spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.6 eV, 0.0425 eV, and 0.5, respectively. Si components for the model structure. As depicted in the inset, one was attributed to Si atoms at atop Al sites and the other to bridge Al sites. While the calculations successfully linked the surface structure and the core-level spectra, the simulated results for the silicene layer were completely different from the experimental data (Fig. 6). Thus, it was necessity to consider a surface structure beyond a simple silicene layer for the Al(111)3×3-Si surface. ![FIG. 7. Simulations of Si 2p core-level photoemission spectra of surface layers on Al(111). (a) The flat silicene model, and the Al-embedded silicene model for (b) θ = 0◦ , (c) θ = 30◦ , and (d) θ = 60◦ . In (a), the simulated component spectrum is depicted in black (red). The model structure is depicted in the inset. In (b-d), experimental spectra are shown by dotted lines, while simulated spectra and the components are given by green and red curves, respectively. The simulation parameters are summarized in Table III.](path) Si sites in the Al-embedded silicene layer model (Fig. 8). As noted earlier, Si atoms in the surface layer can bond with different numbers of intralayer Si and/or Al atoms. For example, in Fig. 8(a), one Si atom used for the calculations was bonded with two Si atoms and one Al nearest neighbor (NN). Table II lists binding energies of the Si 2p3/<sup>2</sup> level for various structure models of silicene and Al-embedded silicene layers on Al(111). The core-level binding energies varied with respect to the number of NN Al atoms in the Si-Al bonding. Smaller NN values had higher binding energies, which was consistent with the silicene model results given in Table II. Moreover, there was a quantitative relationship between the binding energy and the NN value: 98.4±0.1eV (NN=3), 98.6±0.1eV (NN=2), 98.7±0.1eV (NN=1), and 98.8±0.1eV (NN=0). Because the model calculations in Fig. 8 considered only a single (atop) site on the Al substrate, differences in the substitution sites should be considered in TABLE III. Simulation parameters for the Si 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 7. The spectral components were given by Voigt functions with 0.085 eV Lorentzian widths and Gaussian widths given in the table. Spin-orbit splitting and a branching ratio were set at 0.6 eV and 0.5, respectively. The simulation was performed using OpenMX[\[34,](#page-21-8) [38\]](#page-22-0). On the basis of the calculated binding energies, the experimental spectra of the Al(111)3×3- Si surface were fairly well reproduced by five components, S1, S2, S2', S3 and S4, with the parameters in Table III. From the binding energies, S3 and S4 could be assigned to Si atoms that bonded only with NN Si atoms, while S1, S2, and S2' were also bonded to NN Al ![FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.](path) atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
FIG. 8. Structural models of Al-embedded silicene layers with Si coverages of (a) 7/9 ML, (b-d) 6/9 ML, and (e-h) 5/9 ML. Black circles indicate Si atoms used for calculating the Si 2p3/<sup>2</sup> binding energies, while the red circles in (a) indicate example positions of nearest-neighbor atoms in the surface layer.	# Surface structure of the 3×3-Si phase on Al(111), studied by the multiple usages of positron diffraction and core-level photoemission spectroscopy Chi-Cheng Lee,<sup>5</sup> Kazuyoshi Yoshimi,<sup>1</sup> Taisuke Ozaki,<sup>1</sup> Takeru Nakashima,<sup>6</sup> Yasunobu Ando,<sup>6</sup> Hiroaki Aoyama,<sup>7</sup> Tadashi Abukawa,<sup>7</sup> Yuki Tsujikawa,<sup>1</sup> # Abstract The structure of an Al(111)3×3-Si surface was examined by combining data from positron diffraction and core-level photoemission spectroscopy. Analysis of the diffraction rocking curves indicated that the overlayer had a flat honeycomb lattice structure. Simulations of Si core-level spectra calculated via the first-principles indicated that one of the Si atoms in the unit cell was replaced by an Al atom. The surface superstructure was thus a two-dimensional layer of Al-embedded silicene on Al(111). #### I. INTRODUCTION Two-dimensional (2D) materials have been a central issue in materials science because of the emergence of physical or chemical phenomena that are specific to its dimensionality, with promising applications in nanotechnology[\[1\]](#page-19-0). A Xene is a 2D material named after the elemental atomic sheet[\[2\]](#page-19-1), such as graphene (C)[\[3](#page-19-2)[–5\]](#page-19-3), silicene (Si)[\[6,](#page-19-4) [7\]](#page-19-5), or borophene (B)[\[8\]](#page-20-0). Diversity in Xene research was triggered by the discovery of silicene on the Ag(111) surface[\[9,](#page-20-1) [10\]](#page-20-2). The surface structure is fundamental information for a material, and that of the silicene layer was determined by positron diffraction. It formed a honeycomb lattice with out-of-plane buckling[\[11\]](#page-20-3). The electronic structure was expected to show Dirac cones at the K point in the 2D Brillouin zone, but the actual band mappings via photoemission spectroscopy revealed different positions in momentum (wavenumber) space[\[12–](#page-20-4)[15\]](#page-20-5). These investigations have indicated that Xene properties differ between free-standing layers and those on surfaces. Such external effects have been reported for various combinations of overlayers and substrates[\[16,](#page-20-6) [17\]](#page-20-7). Detailed analyses of Xene structures and electronic states are still required for specific surfaces before a universal understanding is achieved. On Al(111) surfaces, it has been reported that Si deposition induces the formation of a long-range ordered 2D phase, or the Al(111)3×3-Si surface[\[18–](#page-20-8)[22\]](#page-20-9). The ordered phase was found at a Si coverage of 0.4 ∼ 0.8 ML[\[18](#page-20-8)[–21\]](#page-20-10), where 1 ML corresponds to the surface atomic density of Al(111), which is 1.42 × 10<sup>19</sup>atoms/cm<sup>2</sup> . The surface band structure was probed via angle-resolved photoemission spectroscopy and consistently matched the band diagram calculated for a honeycomb lattice Si layer (silicene) on Al(111)[\[22\]](#page-20-9). However, a chemical analysis of the surface via Si 2p core-level photoemission spectroscopy indicated Si environments that were much more complex than a simple silicene layer, and the "kagomelike silicene" structure model was proposed [\[21\]](#page-20-10). These experiments have implied that this Xene system has unique properties, but no surface structure analysis has been reported. Here, we formed the Al(111)3×3-Si surface and conducted structure analyses via multiple experiments using positron diffraction and core-level photoemission spectroscopy. Datadriven simulations and the first-principles calculations indicated that it had a flat honeycomb lattice arrangement with Al atoms embedded in the overlayer. The Al-embedded silicene model described the experimental results when the Si coverage was 7/9 ML. The band calculations also reproduced the photoemission band diagram reported previously[\[22\]](#page-20-9). The Al(111)3×3-Si surface is a unique silicene system where the layer has partially embedded substrate atoms, showing an intriguing external effect of the surface. This work thus suggests possible doping of Xenes by external atoms, which may lead to novel functional layers. atoms. A ratio of these two types of components was used to evaluate the extent of Al substitutions in the Al-embedded silicene layer. Those ratios, RAl/Si's, reflecting numbers of Si atoms bonding with Al atoms (S1, S2, S2' components) and those without Al bonding (S3, S4 components), were evaluated from Table III. Specifically, RAl/Si = 0.7, 0.83, and 1 for spectra acquired at θ = 0◦ , 30◦ , and 60◦ , respectively, in Figs. 6 and 7. According to previous reports on Al(111)3×3-Si, the 2D phase was found at Si coverages from 4/9 ML to 7/9 ML[\[18](#page-20-8)[–21\]](#page-20-10). RAl/Si ≤ 1 could only be satisfied at 7/9 ML coverage with RAl/Si = 0.75. For the eight sites in the honeycomb lattice in a 3×3 unit cell, this result corresponded to one Al-substitution. The Al atom bonded with three Si atoms at NN sites (S1, S2, S2' components) and the remaining four Si atoms bonded only with NN Si atoms (S3, S4 components). The Al-embedded silicene model corresponds to that in Fig. 5(c), and it reproduced the TRHEPD curves, as shown in Fig. 5(a,b). There were slight deviations between experiments and simulations for photoemission binding energies and for intensities in positron diffraction rocking curves. This likely indicated that embedded Al atom positions were not specific but rather random in the eight sites in the 3×3 unit cell. Thus, the structure of the Al(111)3×3-Si surface could be described as a layer of Al-embedded silicene with a Si coverage of 7/9 ML. To examine the chemical environment at the Al site, a set of Al 2p core-level spectra for the pristine Al(111) and the Al(111)3×3-Si surfaces were acquired, as shown in Fig. 9(a). The Al core-level states generated a doublet structure via spin-orbit splitting, and a pair of the 2p3/<sup>2</sup> and 2p1/<sup>2</sup> peaks shows broadening at higher binding energies upon formation of the 3×3 surface superstructure. The spectral broadening was confirmed in the data taken ![FIG. 9. Al 2p core-level photoemission spectra of the Al(111)3×3-Si surface, acquired at emission angles (a,b) θ = 0◦ , (c)θ = 30◦ , and (d)θ = 60◦ . The spectra were acquired at room temperature with an incident photon energy of hν=123 eV. In (a,c,d), red and black solid curves correspond to the Al(111)3×3-Si and the Al(111) surfaces, respectively. In (b), an experimental spectrum of (a) is shown by a dotted line, while the curve-fit spectrum and its components are given by green and red curves, respectively. The fitting parameters are summarized in Table IV.](path) TABLE IV. Fitting parameters for the Al 2p core-level spectra of the Al(111)3×3-Si surface, shown in Fig. 2(b). Spin-orbit splitting, the Doniach-Sunjic width, and the branching ratio were set at 0.4 eV, 0.015 eV, and 0.5, respectively. at different emission angles, as shown in Fig. 9(c,d). To examine the spectral features in detail, curve-fitting was performed for the Al(111)3×3-Si spectrum, which revealed two components at binding energies of 72.73 eV (A1) and 72.83 eV (A2), as shown in Fig. 9(b). The fit used parameters that are summarized in Table IV. Because the A<sup>1</sup> component was also observed on the pristine Al(111) surface, it could be assigned to bulk Al. However, the A<sup>2</sup> component was assigned to surface Al atoms, and likely confirmed the presence of an Al atom in the Al-embedded silicene layer. Similar spectral broadening was reported for Ag 3d core-level spectra of the Ag(111)4×4-Si surface that had a silicene overlayer[\[45\]](#page-22-5). The A<sup>2</sup> peak may contain contributions from Al atoms in the Al-embedded silicene layer and those in the subsurface. ![FIG. 10. Calculated band structure of the Al-embedded silicene model on Al(111). (a) Total band diagrams, including both surface and substrate contributions. (b) Band diagram of the Al(111) substrate. (c) Band diagram of a Al-embedded silicene surface layer at a Si coverage of 7/9 ML. Both band structures were unfolded into the Al(111)1×1 surface Brillouin zone with labeled high symmetry points, Γ, M, and K. The color scale corresponds to the spectral weights of the electronic states.](path) #### C. Surface band structure The structure model was determined to be the Al-embedded silicene layer on Al(111), as shown in Fig. 5(c), and therefore electronic band structures were calculated for the Al(111)3×3-Si surface. Figure 10 shows the calculated band diagrams with different weights for the surface system: (a) the total system, (b) the Al(111) substrate, and (c) the Alembedded silicene surface layer. The band structure appeared complex at the Fermi level (EF) because of hybridizations between the substrate and surface electronic states. At the Γ point, there are electronic states at E − E<sup>F</sup> ∼ - 2 eV in Fig. 10 that show dispersion ![FIG. 11. Electronic structure of a free-standing layer of Al-embedded silicene. Optimized structure model: (a) top and (b) side views. Silicon and aluminum atoms are shown as red and gray circles, respectively. A unit cell for the calculations is depicted in the figure. (c) Calculated band structure of the free-standing atomic layers: (blue, red) Al-embedded silicene and (black) silicene. The red and blue color scale corresponds to the spectral weights of electronic states in the Si-π and Si-σ bands, respectively. Band-unfolding was applied to compare band diagrams of the layers in the same Brillouin zone[\[46\]](#page-22-6).](path) curves at higher binding energies along the Γ-M and Γ-K directions. The electronic features were found in the experimental band diagram reported previously via angle-resolved photoemission spectroscopy[\[22\]](#page-20-9). These agreements confirmed the present structure model. The calculated band diagram was similar to that reported for a pristine silicene layer on Al(111)[\[22\]](#page-20-9). Information from the band structure was not sufficient to distinguish structure models of silicene and Al-embedded silicene overlayers. The electronic structure of the Al-embedded silicene layer on Al(111) appeared complex because of interactions between the overlayer and the substrate, as shown in Fig. 10. Thus, it was not suitable to use the system to understand electronic variations of a silicene layer attributed to Al substitution. Instead, we conducted band calculations of a free-standing Al-embedded silicene layer in vacuum, as shown in Fig. 11. After structural optimization, the free-standing layer exhibited a buckled structure with a 0.29 ˚A in SDH. In contrast, the OpenMX first-principles calculation indicated that the layer was flat (0.06 ˚A in SDH) on the Al(111) substrate. For reference, the SDH of the pure silicene on the Al(111) was 0.02 ˚A, while the SDH of a pure silicene layer in vacuum was 0.36 ˚A. Concerning the band structure, the Si-π band formed a gap at the K point for the surface, as opposed to the Dirac cones in the silicene layer shown for comparison. At E<sup>F</sup> , the Si-σ bands dispersed at higher energies and similar curves were found in the surface system [Fig. 10(c)]. The conservation was sharply in contrast to the Si-π band at the surface. The difference was likely attributed to layer-substrate interactions along the surface normal, which was sensitive (insensitive) to the Si-π (Si-σ) bands. Such electronic properties were also reported for various surface systems[\[17\]](#page-20-7). The Al atom has been reported to be an acceptor-type donor in bulk (diamond) silicon. In the Al-embedded silicene layer, the Mulliken charge of the Si atoms was 3.89 ∼ 3.94 e <sup>−</sup>, while that of the Al atoms was 3.01 ∼ 3.06 e <sup>−</sup>, where neutral Si and Al atoms in the calculations have electrons of 4 and 3 e −, respectively. A substituted Al atom in the silicene layer was essentially neutral and was not suitable as an acceptor-type donor. #### IV. CONCLUSION The surface structure of a long-range ordered 3×3 phase, prepared by Si deposition on Al(111), was subjected to a multi-beam investigation using positron diffraction and Xray photoemission spectroscopy. Total reflection high-energy positron diffraction analyses revealed that the surface layer had a flat honeycomb lattice structure. Core-level photoemission spectroscopy indicated that eight sites in the 3×3 unit cell were occupied by one Al atom and seven Si atoms. Hence, the superstructure could be described as a Al-embedded silicene layer on the Al(111) surface. This work thus revealed a silicene layer that contained a substrate atom. This could provide a future approach to dope external atoms in functional Xene surface layers. #### ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI grants (Grants No. JP21H05012, No. JP19H04398, and No. JP18H03874), by a Grant-in-Aid for JSPS Fellows (Grant No. 21J21993), and by JST, CREST Grant No. JPMJCR21O4, Japan. The preliminary experiment was performed at facilities of the Synchrotron Radiation Research Organization, the University of Tokyo. The parts of this work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2014S2-004). The numerical computation was carried out partially on the supercomputers of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. The numerical computation was carried out partially, also, on the Fugaku supercomputer through the HPCI projects (hp210228, hp210267, hp220248, hp230304) and the Wisteria-Odyssey supercomputer as part of the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba. We acknowledge Elettra Sincrotrone Trieste for providing access to its synchrotron radiation facilities and we thank Dr. Paolo Moras, Dr. Polina M. Sheverdyaeva, and Dr. Asish Kumar Kundu for assistance in using the VUV-Photoemission beamline (beamtime nr. 20185157). A. K. K. acknowledges the project EUROFEL-ROADMAP ESFRI of the Italian Ministry of University and Research. #### Appendix: TRHEPD analysis of the kagome-like silicene model Figure 12(a,b) is a set of TRHEPD rocking curves that are compared with a simulation of the kagome-like silicene model[\[21\]](#page-20-10) depicted in Fig. 12(c). The structure contained various types of Si atoms that generate the kagome network on the Al(111) substrate and behaved as adatoms. The model was proposed to qualitatively explain the complex Si 2p core-level spectra of the surface that indicated existence of various Si sites[\[21\]](#page-20-10). As shown in Fig. 12(a,b), the correlation between experimental and simulated curves was poor (R = 3.65%) relative to that for the Al-embedded silicene model (R = 2.12%) in Fig. 5. Thus, the Al-embedded silicene model was more likely than the kagome-like silicene model for the structure of the Al(111)3×3-Si surface. ![FIG. 12. Positron diffraction analysis of the kagome silicene model. (a,b) Total-reflection highenergy positron diffraction rocking curves of the Al(111)3×3-Si surface, acquired at room temperature with a 10 keV positron beam incident along the (a) [11¯2] and (b) [1¯10] directions.(c) Schematic of the structure model.](path) - <span id="page-20-6"></span>[16] B. Feng, B. Fu, S. Kasamatsu, S. Ito, P. Cheng, C.-C. Liu, S. K. Mahatha, P. Sheverdyaeva, P. Moras, M. Arita, O. Sugino, T.-C. Chiang, K. Wu, L. Chen, Y. Yao, and I. Matsuda, Nat. Comm. 8, 1007 (2017). - <span id="page-20-7"></span>[17] M. Cameau, R. Yukawa, C.-H. Chen, A. Huang, S. Ito, R. Ishibiki, K. Horiba, Y. Obata, T. Kondo, H. Kumigashira, H.-T. Jeng, M. D'angelo, I. Matsuda, Phys. Rev. Materials 3, 044004 (2019). - [21] Y. Sassa, F. O. L. Johansson, A. Lindblad, M. G. Yazdi, K. Simonov, J. Weissenrieder, M. Muntwiler, F. Iyikanat, H. Sahin, T. Angot, E. Salomon, and G. L. Lay, Appl. Surf. Sci. 530, 1471 (2020). - <span id="page-20-9"></span>[22] Y. Sato, Y. Fukaya, M. Cameau, A. K. Kundu, D. Shiga, R. Yukawa, K. Horiba, C.-H. Chen, A. Huang, H.-T. Jeng, T. Ozaki, H. Kumigashira, M. Niibe, I. Matsuda, Phys. Rev. Materials 4, 064005 (2020). - [37] K. Lejaeghere, G. Bihlmayer, T. Bj¨orkman, P. Blaha, S. Bl¨ugel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dulak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Gr˚an¨as, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. K¨uc¨ukbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordstr¨om, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunstr¨om, A. Tkatchenko, M. Torrent, - [45] J. T. K¨uchle, A. Baklanov, A. P. Seitsonen, P. T. P. Ryan, P. Feulner, P. Pendem, T.-L. Lee, M. Muntwiler, M. Schwarz, F. Haag, J. V. Barth, W. Auw¨arter, D. A. Duncan, F. Allegretti, 2D Mater. 9, 045021 (2022).
Figure 1.5 Flexoelectric voltage generated in the cantilever around its resonance frequency.	# Measuring direct flexoelectricity at the nanoscale #### 1.1 Abstract Flexoelectricity is a property of all dielectric materials, where inhomogeneous strain induces electrical polarization. is effect becomes particularly prominent at the nanoscale where larger strain gradients can be obtained. While flexoelectric charges have been measured in mm-scale systems, direct measurements in nanoscale-thickness materials have not yet been achieved. Given that one of the most prominent applications of flexoelectricity is in nanoelectro-mechanical systems (NEMS), confirming the presence and magnitude of the effect at these scales is essential. is study presents the first-ever measurements of flexoelectricgenerated charges (direct effect) in nanoscale-thickness materials, using cantilevers with a 50 nm hafnium oxide layer. We confirm that the estimated flexoelectric coefficient from said measurements aligns with the values obtained from complementary experiments using the flexoelectric inverse effect. Additionally, by changing the cantilever geometry (modifying the width of the cantilevers), we demonstrate a 40% increase in the effective flexoelectric coefficient, explained by the interplay of different flexoelectric tensor components. These findings not only validate the presence of flexoelectric effects at the nanoscale but also open the possibility for full flexoelectric transduction of the motion in NEMS/MEMS devices. ### 1.2 Introduction Flexoelectricity is a property of dielectric materials where an inhomogeneous mechanical deformation, such as bending, induces electrical polarization (direct effect) [1]. Conversely, an applied electric field induces mechanical deformation (inverse effect) [2]. Unlike piezoelectricity, which requires non-centrosymmetric materials, flexoelectricity can occur in all dielectric materials [3]. is effect is particularly significant at the nanoscale, where large strain gradients are easier to achieve [4]. e simplest expression of this relationship is captured in Equation [1.1,](#page-1-0) where the flexoelectric coefficient is represented by , the material strain gradient is ⁄ and the generated polarization is [5]. A conceptual visualization of flexoelectricity can be illustrated using a simple ionic lattice [\(Figure 1.1a](#page-1-1)). When the material bends, the strain gradient causes a relative displacement between positively and negatively charged ions, leading to a local redistribution of charges and inducing a net polarization [\(Figure 1.1c](#page-1-1)). In materials where ionic movement is minimal, flexoelectricity can also be understood as a redistribution of electron density under strain gradients [6], enabling the effect in any type of dielectrics. ![<span id="page-1-1"></span>![](_page_1_Figure_3.jpeg)](path) Direct flexoelectricity in a mechanical structure [\(Figure 1.1b](#page-1-1)) generates surface charges across the beam when it bends along its length. To facilitate the reading of small charges, studies in the literature tend to use samples of large lateral dimensions (millimeters to centimeters). Consequently, to avoid breaking the sample when applying the load, the samples have thicknesses that range from 0.5 to 3 mm. Typically these measurements employ a three- or four-point bending tester, connected to an amplifier [7], [8], [9], [10], [11]. Other authors produce the strain in the material with a loudspeaker and measure charges with a lock-in amplifier. In these cases, the displacement of the loudspeaker is measured using a Displacement Voltage Ratio Transformer (DVRT) [1], [12], [13], [14] or an optical sensor [15], [16]. Additionally, some approaches utilize a piezoelectric shaker to flex the sample, with the displacement controlled by a Laser Doppler Vibrometer (LDV). In these cases, charge measurements are conducted using an amplifier [17], [18]. To date, no method reported in the literature has successfully measured direct flexoelectric effect in materials with nanoscale thicknesses. is is primarily due to the challenges in achieving sufficient sensitivity and accurately controlling strain gradients at such small scales [2]. Given that one of the most promising applications of flexoelectricity lies in Nano-Electro-Mechanical Systems (NEMS), where device dimensions are on the nanometer scale, confirming the presence and magnitude of the flexoelectric effect at these scales is crucial. is could enable the development of novel sensors, actuators, and energy-harvesting devices that use nanoscale flexoelectricity [19]. # 1.3 Methodology Our approach for measuring flexoelectric charges at the nanoscale starts by fabricating cantilevers with a thin (50 nm) layer of hafnium oxide, sandwiched between two platinum electrodes (20 nm). Hafnium oxide is selected due to its high dielectric constant (within microelectronic compatible materials), well-characterized coefficient [20], [21], and its wide availability in cleanroom facilities. e cantilevers measure 20 µm in length and vary in width from 5 µm to 50 µm [\(Figure 1.2\)](#page-2-0). Even though we favor narrower devices, we include a range of widths in this study to facilitate charge detection. ![<span id="page-2-0"></span>Figure 1.2 SEM images of fabricated cantilevers. a) Cantilever with a length of 20 µm and a width of 5 µm. b) Cantilever with a length of 20 µm and a width of 50 µm.](path) To maximize the flexoelectric response, we excite the cantilevers at their resonance frequency using a piezoelectric shaker under the cantilevers chip. Operating at resonance amplifies the mechanical displacement, improving the signal-to-noise ratio. Measurements are conducted under vacuum (3 · 10−4 ) to reduce air damping, further enhancing displacement amplitude. e generated flexoelectric charges are collected by platinum electrodes and amplified using a low-noise voltage amplifier (Sierra Amps [22]) placed in close proximity to minimize the load capacitance and signal loss. e experimental setup [\(Figure 1.3\)](#page-3-0) includes a Laser Doppler Vibrometer to monitor cantilever displacement. All signals are processed through a lock-in amplifier (Zurich Instruments UHFLI), which controls the shaker, reads the electrical charges, and monitors the cantilever's displacement. Importantly, the electrical charges and the vibrometer signal are not monitored simultaneously, but subsequently. ![<span id="page-3-0"></span>Figure 1.3 Schematic of the experimental setup to measure flexoelectric charges in nanoscale-thickness materials. e cantilever is actuated by a piezoelectric shaker at its resonance frequency under vacuum. A low-noise amplifier amplifies the generated flexoelectric charges () and parasitic charges ( ). A Laser Doppler Vibrometer monitors the displacement of the cantilever. All signals are processed in a lockin amplifier.](path) e first measurement quantifies the cantilever's displacement for a certain shaker actuation. By sweeping the excitation frequency around the cantilever's resonance frequency, we measure the displacement at the laser's measurement point [\(Figure 1.4\)](#page-4-0). However, our primary interest is the displacement at the cantilever's tip, which is essential for calculating the strain gradient (beam curvature). To extrapolate the tip displacement from the measured point, we analyze the thermomechanical noise and use a standard cantilever beam model. is method allows us to translate the measurements at any point along the cantilever to the tip [23], [24]. ![<span id="page-4-0"></span>Figure 1.4 Laser Doppler Vibrometer measurements of cantilever displacement around the resonance frequency. e displacement amplitude is linearly proportional to the actuation voltage applied to the shaker. ermomechanical noise measurements, conducted without actuation, are used to verify the laser position on the cantilever and translate the amplitude measurements to cantilever's tip displacement.](path) ![<span id="page-4-1"></span>Figure 1.5 Flexoelectric voltage generated in the cantilever around its resonance frequency. a) Normalized output voltage for a cantilever with a width of 40 µm and a length of 20 µm. We observe the flexoelectric charges superimposed on the background signal. b) Lorentzian signal corresponding to the cantilever's motion, obtained by subtracting the background signal from the output signal.](path) e second measurement evaluates the electrical charges generated by the flexoelectric effect. We perform the measurements without the laser, as its illumination can induce photoelectric effects and interfere with the flexoelectric charges [25]. [Figure 1.5](#page-4-1) shows the electrical measurements obtained from a cantilever with dimensions of 40 µm in width and 20 µm in length. In [Figure 1.5a](#page-4-1), the normalized transfer function is plotted as a function of frequency, revealing the cantilever's resonance peak superimposed on a background signal. is background arises from parasitic capacitance pathways between the shaker's actuation and the cantilever. To isolate the flexoelectric signal, we perform a linear baseline subtraction for both components in phase and in quadrature with the drive, to remove the background contribution, resulting in the cleaned data presented in [Figure 1.5b](#page-4-1). e extracted signal corresponds to the flexoelectric charges generated due to the cantilever's motion, which are linearly dependent on the shaker's actuation. By combining the measured electrical amplitudes () with the cantilever displacements at the tip () at various shaker actuations, we calculate the effective flexoelectric coefficient () using Equation [1.2.](#page-5-0) is calculation also involves the cantilever's width (), the capacitance of the measurement path leading to the low-noise amplifier ( ) and the derivative of the normalized n-mode shape (′ ()). For the cantilever's first mode, 1 ′ () = − 1.3765⁄, where L represents the length of the cantilever. Detailed derivation is provided in the supplementary material. #### 1.4 Results and discussion [Figure 1.6](#page-6-0) presents the effective flexoelectric coefficient for cantilevers of varying widths. We observe that increases with increasing width, reaching a saturation value for widths larger than 40 µm. is trend is attributed to a regime shift, as the cantilevers transition from the < condition to > . is change affects how the flexoelectric tensor components 11 and 12 [4] interact to determine the generated charges. [Table 1.1](#page-6-1) describes the theoretical asymptotic limits of the effective flexoelectric coefficient for both narrow and wide cantilevers (compared to the length). Detailed derivations are provided in the supplementary material. By making the ratio between the limits we obtain ,⁄, = 1⁄(1 − ), which depends solely on the material's Poisson's ratio (), and not on the flexoelectric tensor terms. For hafnium oxide, with = 0.3, the ratio evaluates to approximately 1.42. is theoretical prediction aligns well with the data i[n Figure 1.6,](#page-6-0) where increasing the width results in a 42% increase in . <span id="page-6-1"></span>Table 1.1 Asymptotical effective flexoelectric coefficients. Asymptotical limits of the effective flexoelectric coefficient for narrow and wide cantilevers. e ratio between these limits is independent of the flexoelectric tensor components, depending only on Poisson's ratio of the material. (t = Cantilever's thickness). ![<span id="page-6-0"></span>Figure 1.6 Effective flexoelectric coefficient for cantilevers with different widths. e growth in the effective flexoelectric coefficient is theoretically expected, as the interplay between the tensorial components depend on the width of the cantilever. e figure shows the asymptotical limits for the extreme cases of the width.](path) e direct flexoelectric measurements presented in this work [\(Figure 1.6\)](#page-6-0) align with the published coefficients we previously obtained from inverse flexoelectric measurements [20]. In that study, we report a value of = 105 ± 10 / for narrow cantilevers, which aligns closely with the direct flexoelectric measurements. # 1.5 Conclusion is study presents the first flexoelectric charge measurements in nanometrically thin materials. We employ microcantilevers with a 50 nm layer of hafnium oxide between platinum electrodes. Electrical signals are captured using a low-noise amplifier and they are compared with the cantilever displacement which is measured with a Laser Doppler Vibrometer. e results align with hafnium oxide coefficient values obtained through inverse methods. Additionally, we demonstrate that the effective flexoelectric coefficient increases with cantilever width (for a fixed length), showing the dependence on cantilever geometry. ese findings could help in the fabrication of next-generation NEMS devices. ## References - [8] P. Zubko, G. Catalan, A. Buckley, P. R. L. Welche, and J. F. Scott, "Strain-Gradient-Induced Polarization in SrTiO 3 Single Crystals," Phys. Rev. Lett., vol. 99, no. 16, p. 167601, Oct. 2007, doi: 10.1103/PhysRevLett.99.167601. - [10] J. Narvaez, S. Saremi, J. Hong, M. Stengel, and G. Catalan, "Large Flexoelectric Anisotropy in Paraelectric Barium Titanate," Phys. Rev. Lett., vol. 115, no. 3, p. 037601, Jul. 2015, doi: 10.1103/PhysRevLett.115.037601. - [12] W. Ma and L. E. Cross, "Flexoelectric polarization of barium strontium titanate in the paraelectric state," Applied Physics Letters, vol. 81, no. 18, pp. 3440–3442, Oct. 2002, doi: 10.1063/1.1518559. - [16] Y. Li, L. Shu, W. Huang, X. Jiang, and H. Wang, "Giant flexoelectricity in Ba0.6Sr0.4TiO3/Ni0.8Zn0.2Fe2O4 composite," Applied Physics Letters, vol. 105, no. 16, p. 162906, Oct. 2014, doi: 10.1063/1.4899060. - [17] W. Huang, K. Kim, S. Zhang, F.-G. Yuan, and X. Jiang, "Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers: Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers," Phys. Status Solidi RRL, vol. 5, no. 9, pp. 350–352, Sep. 2011, doi: 10.1002/pssr.201105326. - [18] S. Huang, T. Kim, D. Hou, D. Cann, J. L. Jones, and X. Jiang, "Flexoelectric characterization of BaTiO3-0.08Bi(Zn1/2Ti1/2)O3," Applied Physics Letters, vol. 110, no. 22, p. 222904, May 2017, doi: 10.1063/1.4984212. - [19] Q. Deng, M. Kammoun, A. Erturk, and P. Sharma, "Nanoscale flexoelectric energy harvesting," International Journal of Solids and Structures, vol. 51, no. 18, pp. 3218– 3225, Sep. 2014, doi: 10.1016/j.ijsolstr.2014.05.018. - [23] L. G. Villanueva, R. B. Karabalin, M. H. Matheny, D. Chi, J. E. Sader, and M. L. Roukes, "Nonlinearity in nanomechanical cantilevers," Phys. Rev. B, vol. 87, no. 2, p. 024304, Jan. 2013, doi: 10.1103/PhysRevB.87.024304. # 1.6 Supplementary Material #### 1.6.1 Flexoelectric formula derivation Derivation of the formula to convert the read voltage (after background subtraction) and cantilever tip displacement into effective flexoelectric coefficient. ![Figure 1.7 Equivalent circuit of the electronic setup. e cantilever is represented as a current source ( ), which has an equivalent parallel resistance () and capacitance ( ). e capacitance of the connector is represented by ( ), the input lownoise amplifier resistance ( ) and capacitance ( ).](path) $$\phi\_n(\mathbf{x}) = \frac{1}{2} \left[ \cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} (\sin(\beta\_n \mathbf{x}) - \sinh(\beta\_n \mathbf{x})) \right] \tag{1.12}$$ $$\phi\_n'(\mathbf{x}) = \frac{1}{2} \left[ -\beta\_n \cdot \sin(\beta\_n \mathbf{x}) - \beta\_n \cdot \sinh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} \cdot \beta\_n \cdot (\cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x})) \right] \tag{1.13}$$ #### 1.6.2 Measurement uncertainty evaluation Extended formula for the direct measurement of flexoelectricity, including all dependencies. $$\mu\_{direct} = \frac{\left(\mathcal{C}\_{device} + \mathcal{C}\_{connect} + \mathcal{C}\_{amplitude}\right) \cdot L}{1.3765 \cdot W} \cdot \frac{V\_{initial}}{Amp\_{sweep} \cdot \sqrt{\frac{4 \cdot Q \cdot K\_B T}{1 \cdot L \cdot W \cdot \Sigma \rho\_l t\_l \cdot (2\pi f\_R)^3 \cdot P \text{SD}/2}}} \tag{1.15}$$ ## 1.6.3 Flexoelectric tensor contributions to the effective flexoelectric coefficient ![∂z](path) ## 1.6.4 Equivalent flexoelectric charges for the measured voltages Here, just for curiosity, we calculate what are the equivalent flexoelectric charges for the measured flexoelectric voltages . Our methodology does not measure flexoelectric charges directly, we measure currents, which is an integration of the charges at a relatively high frequency (200 kHz). We found these calculations interesting to have an estimated electron charge produced by the flexoelectric effect for hafnium oxide.
Figure 1.6 Effective flexoelectric coefficient for cantilevers with different widths. e growth in the effective flexoelectric coefficient is theoretically expected, as the interplay between the tensorial components depend on the width of the cantilever. e figure shows the asymptotical limits for the extreme cases of the width.	# Measuring direct flexoelectricity at the nanoscale #### 1.1 Abstract Flexoelectricity is a property of all dielectric materials, where inhomogeneous strain induces electrical polarization. is effect becomes particularly prominent at the nanoscale where larger strain gradients can be obtained. While flexoelectric charges have been measured in mm-scale systems, direct measurements in nanoscale-thickness materials have not yet been achieved. Given that one of the most prominent applications of flexoelectricity is in nanoelectro-mechanical systems (NEMS), confirming the presence and magnitude of the effect at these scales is essential. is study presents the first-ever measurements of flexoelectricgenerated charges (direct effect) in nanoscale-thickness materials, using cantilevers with a 50 nm hafnium oxide layer. We confirm that the estimated flexoelectric coefficient from said measurements aligns with the values obtained from complementary experiments using the flexoelectric inverse effect. Additionally, by changing the cantilever geometry (modifying the width of the cantilevers), we demonstrate a 40% increase in the effective flexoelectric coefficient, explained by the interplay of different flexoelectric tensor components. These findings not only validate the presence of flexoelectric effects at the nanoscale but also open the possibility for full flexoelectric transduction of the motion in NEMS/MEMS devices. ### 1.2 Introduction Flexoelectricity is a property of dielectric materials where an inhomogeneous mechanical deformation, such as bending, induces electrical polarization (direct effect) [1]. Conversely, an applied electric field induces mechanical deformation (inverse effect) [2]. Unlike piezoelectricity, which requires non-centrosymmetric materials, flexoelectricity can occur in all dielectric materials [3]. is effect is particularly significant at the nanoscale, where large strain gradients are easier to achieve [4]. e simplest expression of this relationship is captured in Equation [1.1,](#page-1-0) where the flexoelectric coefficient is represented by , the material strain gradient is ⁄ and the generated polarization is [5]. A conceptual visualization of flexoelectricity can be illustrated using a simple ionic lattice [\(Figure 1.1a](#page-1-1)). When the material bends, the strain gradient causes a relative displacement between positively and negatively charged ions, leading to a local redistribution of charges and inducing a net polarization [\(Figure 1.1c](#page-1-1)). In materials where ionic movement is minimal, flexoelectricity can also be understood as a redistribution of electron density under strain gradients [6], enabling the effect in any type of dielectrics. ![<span id="page-1-1"></span>![](_page_1_Figure_3.jpeg)](path) Direct flexoelectricity in a mechanical structure [\(Figure 1.1b](#page-1-1)) generates surface charges across the beam when it bends along its length. To facilitate the reading of small charges, studies in the literature tend to use samples of large lateral dimensions (millimeters to centimeters). Consequently, to avoid breaking the sample when applying the load, the samples have thicknesses that range from 0.5 to 3 mm. Typically these measurements employ a three- or four-point bending tester, connected to an amplifier [7], [8], [9], [10], [11]. Other authors produce the strain in the material with a loudspeaker and measure charges with a lock-in amplifier. In these cases, the displacement of the loudspeaker is measured using a Displacement Voltage Ratio Transformer (DVRT) [1], [12], [13], [14] or an optical sensor [15], [16]. Additionally, some approaches utilize a piezoelectric shaker to flex the sample, with the displacement controlled by a Laser Doppler Vibrometer (LDV). In these cases, charge measurements are conducted using an amplifier [17], [18]. To date, no method reported in the literature has successfully measured direct flexoelectric effect in materials with nanoscale thicknesses. is is primarily due to the challenges in achieving sufficient sensitivity and accurately controlling strain gradients at such small scales [2]. Given that one of the most promising applications of flexoelectricity lies in Nano-Electro-Mechanical Systems (NEMS), where device dimensions are on the nanometer scale, confirming the presence and magnitude of the flexoelectric effect at these scales is crucial. is could enable the development of novel sensors, actuators, and energy-harvesting devices that use nanoscale flexoelectricity [19]. # 1.3 Methodology Our approach for measuring flexoelectric charges at the nanoscale starts by fabricating cantilevers with a thin (50 nm) layer of hafnium oxide, sandwiched between two platinum electrodes (20 nm). Hafnium oxide is selected due to its high dielectric constant (within microelectronic compatible materials), well-characterized coefficient [20], [21], and its wide availability in cleanroom facilities. e cantilevers measure 20 µm in length and vary in width from 5 µm to 50 µm [\(Figure 1.2\)](#page-2-0). Even though we favor narrower devices, we include a range of widths in this study to facilitate charge detection. ![<span id="page-2-0"></span>Figure 1.2 SEM images of fabricated cantilevers. a) Cantilever with a length of 20 µm and a width of 5 µm. b) Cantilever with a length of 20 µm and a width of 50 µm.](path) To maximize the flexoelectric response, we excite the cantilevers at their resonance frequency using a piezoelectric shaker under the cantilevers chip. Operating at resonance amplifies the mechanical displacement, improving the signal-to-noise ratio. Measurements are conducted under vacuum (3 · 10−4 ) to reduce air damping, further enhancing displacement amplitude. e generated flexoelectric charges are collected by platinum electrodes and amplified using a low-noise voltage amplifier (Sierra Amps [22]) placed in close proximity to minimize the load capacitance and signal loss. e experimental setup [\(Figure 1.3\)](#page-3-0) includes a Laser Doppler Vibrometer to monitor cantilever displacement. All signals are processed through a lock-in amplifier (Zurich Instruments UHFLI), which controls the shaker, reads the electrical charges, and monitors the cantilever's displacement. Importantly, the electrical charges and the vibrometer signal are not monitored simultaneously, but subsequently. ![<span id="page-3-0"></span>Figure 1.3 Schematic of the experimental setup to measure flexoelectric charges in nanoscale-thickness materials. e cantilever is actuated by a piezoelectric shaker at its resonance frequency under vacuum. A low-noise amplifier amplifies the generated flexoelectric charges () and parasitic charges ( ). A Laser Doppler Vibrometer monitors the displacement of the cantilever. All signals are processed in a lockin amplifier.](path) e first measurement quantifies the cantilever's displacement for a certain shaker actuation. By sweeping the excitation frequency around the cantilever's resonance frequency, we measure the displacement at the laser's measurement point [\(Figure 1.4\)](#page-4-0). However, our primary interest is the displacement at the cantilever's tip, which is essential for calculating the strain gradient (beam curvature). To extrapolate the tip displacement from the measured point, we analyze the thermomechanical noise and use a standard cantilever beam model. is method allows us to translate the measurements at any point along the cantilever to the tip [23], [24]. ![<span id="page-4-0"></span>Figure 1.4 Laser Doppler Vibrometer measurements of cantilever displacement around the resonance frequency. e displacement amplitude is linearly proportional to the actuation voltage applied to the shaker. ermomechanical noise measurements, conducted without actuation, are used to verify the laser position on the cantilever and translate the amplitude measurements to cantilever's tip displacement.](path) ![<span id="page-4-1"></span>Figure 1.5 Flexoelectric voltage generated in the cantilever around its resonance frequency. a) Normalized output voltage for a cantilever with a width of 40 µm and a length of 20 µm. We observe the flexoelectric charges superimposed on the background signal. b) Lorentzian signal corresponding to the cantilever's motion, obtained by subtracting the background signal from the output signal.](path) e second measurement evaluates the electrical charges generated by the flexoelectric effect. We perform the measurements without the laser, as its illumination can induce photoelectric effects and interfere with the flexoelectric charges [25]. [Figure 1.5](#page-4-1) shows the electrical measurements obtained from a cantilever with dimensions of 40 µm in width and 20 µm in length. In [Figure 1.5a](#page-4-1), the normalized transfer function is plotted as a function of frequency, revealing the cantilever's resonance peak superimposed on a background signal. is background arises from parasitic capacitance pathways between the shaker's actuation and the cantilever. To isolate the flexoelectric signal, we perform a linear baseline subtraction for both components in phase and in quadrature with the drive, to remove the background contribution, resulting in the cleaned data presented in [Figure 1.5b](#page-4-1). e extracted signal corresponds to the flexoelectric charges generated due to the cantilever's motion, which are linearly dependent on the shaker's actuation. By combining the measured electrical amplitudes () with the cantilever displacements at the tip () at various shaker actuations, we calculate the effective flexoelectric coefficient () using Equation [1.2.](#page-5-0) is calculation also involves the cantilever's width (), the capacitance of the measurement path leading to the low-noise amplifier ( ) and the derivative of the normalized n-mode shape (′ ()). For the cantilever's first mode, 1 ′ () = − 1.3765⁄, where L represents the length of the cantilever. Detailed derivation is provided in the supplementary material. #### 1.4 Results and discussion [Figure 1.6](#page-6-0) presents the effective flexoelectric coefficient for cantilevers of varying widths. We observe that increases with increasing width, reaching a saturation value for widths larger than 40 µm. is trend is attributed to a regime shift, as the cantilevers transition from the < condition to > . is change affects how the flexoelectric tensor components 11 and 12 [4] interact to determine the generated charges. [Table 1.1](#page-6-1) describes the theoretical asymptotic limits of the effective flexoelectric coefficient for both narrow and wide cantilevers (compared to the length). Detailed derivations are provided in the supplementary material. By making the ratio between the limits we obtain ,⁄, = 1⁄(1 − ), which depends solely on the material's Poisson's ratio (), and not on the flexoelectric tensor terms. For hafnium oxide, with = 0.3, the ratio evaluates to approximately 1.42. is theoretical prediction aligns well with the data i[n Figure 1.6,](#page-6-0) where increasing the width results in a 42% increase in . <span id="page-6-1"></span>Table 1.1 Asymptotical effective flexoelectric coefficients. Asymptotical limits of the effective flexoelectric coefficient for narrow and wide cantilevers. e ratio between these limits is independent of the flexoelectric tensor components, depending only on Poisson's ratio of the material. (t = Cantilever's thickness). ![<span id="page-6-0"></span>Figure 1.6 Effective flexoelectric coefficient for cantilevers with different widths. e growth in the effective flexoelectric coefficient is theoretically expected, as the interplay between the tensorial components depend on the width of the cantilever. e figure shows the asymptotical limits for the extreme cases of the width.](path) e direct flexoelectric measurements presented in this work [\(Figure 1.6\)](#page-6-0) align with the published coefficients we previously obtained from inverse flexoelectric measurements [20]. In that study, we report a value of = 105 ± 10 / for narrow cantilevers, which aligns closely with the direct flexoelectric measurements. # 1.5 Conclusion is study presents the first flexoelectric charge measurements in nanometrically thin materials. We employ microcantilevers with a 50 nm layer of hafnium oxide between platinum electrodes. Electrical signals are captured using a low-noise amplifier and they are compared with the cantilever displacement which is measured with a Laser Doppler Vibrometer. e results align with hafnium oxide coefficient values obtained through inverse methods. Additionally, we demonstrate that the effective flexoelectric coefficient increases with cantilever width (for a fixed length), showing the dependence on cantilever geometry. ese findings could help in the fabrication of next-generation NEMS devices. ## References - [8] P. Zubko, G. Catalan, A. Buckley, P. R. L. Welche, and J. F. Scott, "Strain-Gradient-Induced Polarization in SrTiO 3 Single Crystals," Phys. Rev. Lett., vol. 99, no. 16, p. 167601, Oct. 2007, doi: 10.1103/PhysRevLett.99.167601. - [10] J. Narvaez, S. Saremi, J. Hong, M. Stengel, and G. Catalan, "Large Flexoelectric Anisotropy in Paraelectric Barium Titanate," Phys. Rev. Lett., vol. 115, no. 3, p. 037601, Jul. 2015, doi: 10.1103/PhysRevLett.115.037601. - [12] W. Ma and L. E. Cross, "Flexoelectric polarization of barium strontium titanate in the paraelectric state," Applied Physics Letters, vol. 81, no. 18, pp. 3440–3442, Oct. 2002, doi: 10.1063/1.1518559. - [16] Y. Li, L. Shu, W. Huang, X. Jiang, and H. Wang, "Giant flexoelectricity in Ba0.6Sr0.4TiO3/Ni0.8Zn0.2Fe2O4 composite," Applied Physics Letters, vol. 105, no. 16, p. 162906, Oct. 2014, doi: 10.1063/1.4899060. - [17] W. Huang, K. Kim, S. Zhang, F.-G. Yuan, and X. Jiang, "Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers: Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers," Phys. Status Solidi RRL, vol. 5, no. 9, pp. 350–352, Sep. 2011, doi: 10.1002/pssr.201105326. - [18] S. Huang, T. Kim, D. Hou, D. Cann, J. L. Jones, and X. Jiang, "Flexoelectric characterization of BaTiO3-0.08Bi(Zn1/2Ti1/2)O3," Applied Physics Letters, vol. 110, no. 22, p. 222904, May 2017, doi: 10.1063/1.4984212. - [19] Q. Deng, M. Kammoun, A. Erturk, and P. Sharma, "Nanoscale flexoelectric energy harvesting," International Journal of Solids and Structures, vol. 51, no. 18, pp. 3218– 3225, Sep. 2014, doi: 10.1016/j.ijsolstr.2014.05.018. - [23] L. G. Villanueva, R. B. Karabalin, M. H. Matheny, D. Chi, J. E. Sader, and M. L. Roukes, "Nonlinearity in nanomechanical cantilevers," Phys. Rev. B, vol. 87, no. 2, p. 024304, Jan. 2013, doi: 10.1103/PhysRevB.87.024304. # 1.6 Supplementary Material #### 1.6.1 Flexoelectric formula derivation Derivation of the formula to convert the read voltage (after background subtraction) and cantilever tip displacement into effective flexoelectric coefficient. ![Figure 1.7 Equivalent circuit of the electronic setup. e cantilever is represented as a current source ( ), which has an equivalent parallel resistance () and capacitance ( ). e capacitance of the connector is represented by ( ), the input lownoise amplifier resistance ( ) and capacitance ( ).](path) $$\phi\_n(\mathbf{x}) = \frac{1}{2} \left[ \cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} (\sin(\beta\_n \mathbf{x}) - \sinh(\beta\_n \mathbf{x})) \right] \tag{1.12}$$ $$\phi\_n'(\mathbf{x}) = \frac{1}{2} \left[ -\beta\_n \cdot \sin(\beta\_n \mathbf{x}) - \beta\_n \cdot \sinh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} \cdot \beta\_n \cdot (\cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x})) \right] \tag{1.13}$$ #### 1.6.2 Measurement uncertainty evaluation Extended formula for the direct measurement of flexoelectricity, including all dependencies. $$\mu\_{direct} = \frac{\left(\mathcal{C}\_{device} + \mathcal{C}\_{connect} + \mathcal{C}\_{amplitude}\right) \cdot L}{1.3765 \cdot W} \cdot \frac{V\_{initial}}{Amp\_{sweep} \cdot \sqrt{\frac{4 \cdot Q \cdot K\_B T}{1 \cdot L \cdot W \cdot \Sigma \rho\_l t\_l \cdot (2\pi f\_R)^3 \cdot P \text{SD}/2}}} \tag{1.15}$$ ## 1.6.3 Flexoelectric tensor contributions to the effective flexoelectric coefficient ![∂z](path) ## 1.6.4 Equivalent flexoelectric charges for the measured voltages Here, just for curiosity, we calculate what are the equivalent flexoelectric charges for the measured flexoelectric voltages . Our methodology does not measure flexoelectric charges directly, we measure currents, which is an integration of the charges at a relatively high frequency (200 kHz). We found these calculations interesting to have an estimated electron charge produced by the flexoelectric effect for hafnium oxide.
∂z μeff,<sup>n</sup> = −ν · μ11 + μ12(1 − ν) 1 − ν μeff,<sup>w</sup> <sup>=</sup> μ11 · <sup>ν</sup> ∂z ν − <sup>1</sup> <sup>+</sup>μ12	# Measuring direct flexoelectricity at the nanoscale #### 1.1 Abstract Flexoelectricity is a property of all dielectric materials, where inhomogeneous strain induces electrical polarization. is effect becomes particularly prominent at the nanoscale where larger strain gradients can be obtained. While flexoelectric charges have been measured in mm-scale systems, direct measurements in nanoscale-thickness materials have not yet been achieved. Given that one of the most prominent applications of flexoelectricity is in nanoelectro-mechanical systems (NEMS), confirming the presence and magnitude of the effect at these scales is essential. is study presents the first-ever measurements of flexoelectricgenerated charges (direct effect) in nanoscale-thickness materials, using cantilevers with a 50 nm hafnium oxide layer. We confirm that the estimated flexoelectric coefficient from said measurements aligns with the values obtained from complementary experiments using the flexoelectric inverse effect. Additionally, by changing the cantilever geometry (modifying the width of the cantilevers), we demonstrate a 40% increase in the effective flexoelectric coefficient, explained by the interplay of different flexoelectric tensor components. These findings not only validate the presence of flexoelectric effects at the nanoscale but also open the possibility for full flexoelectric transduction of the motion in NEMS/MEMS devices. ### 1.2 Introduction Flexoelectricity is a property of dielectric materials where an inhomogeneous mechanical deformation, such as bending, induces electrical polarization (direct effect) [1]. Conversely, an applied electric field induces mechanical deformation (inverse effect) [2]. Unlike piezoelectricity, which requires non-centrosymmetric materials, flexoelectricity can occur in all dielectric materials [3]. is effect is particularly significant at the nanoscale, where large strain gradients are easier to achieve [4]. e simplest expression of this relationship is captured in Equation [1.1,](#page-1-0) where the flexoelectric coefficient is represented by , the material strain gradient is ⁄ and the generated polarization is [5]. A conceptual visualization of flexoelectricity can be illustrated using a simple ionic lattice [\(Figure 1.1a](#page-1-1)). When the material bends, the strain gradient causes a relative displacement between positively and negatively charged ions, leading to a local redistribution of charges and inducing a net polarization [\(Figure 1.1c](#page-1-1)). In materials where ionic movement is minimal, flexoelectricity can also be understood as a redistribution of electron density under strain gradients [6], enabling the effect in any type of dielectrics. ## 1.6.4 Equivalent flexoelectric charges for the measured voltages Here, just for curiosity, we calculate what are the equivalent flexoelectric charges for the measured flexoelectric voltages . Our methodology does not measure flexoelectric charges directly, we measure currents, which is an integration of the charges at a relatively high frequency (200 kHz). We found these calculations interesting to have an estimated electron charge produced by the flexoelectric effect for hafnium oxide.
Figure 1.2 SEM images of fabricated cantilevers. a) Cantilever with a length of 20 µm and a width of 5 µm. b) Cantilever with a length of 20 µm and a width of 50 µm.	# Measuring direct flexoelectricity at the nanoscale #### 1.1 Abstract Flexoelectricity is a property of all dielectric materials, where inhomogeneous strain induces electrical polarization. is effect becomes particularly prominent at the nanoscale where larger strain gradients can be obtained. While flexoelectric charges have been measured in mm-scale systems, direct measurements in nanoscale-thickness materials have not yet been achieved. Given that one of the most prominent applications of flexoelectricity is in nanoelectro-mechanical systems (NEMS), confirming the presence and magnitude of the effect at these scales is essential. is study presents the first-ever measurements of flexoelectricgenerated charges (direct effect) in nanoscale-thickness materials, using cantilevers with a 50 nm hafnium oxide layer. We confirm that the estimated flexoelectric coefficient from said measurements aligns with the values obtained from complementary experiments using the flexoelectric inverse effect. Additionally, by changing the cantilever geometry (modifying the width of the cantilevers), we demonstrate a 40% increase in the effective flexoelectric coefficient, explained by the interplay of different flexoelectric tensor components. These findings not only validate the presence of flexoelectric effects at the nanoscale but also open the possibility for full flexoelectric transduction of the motion in NEMS/MEMS devices. ### 1.2 Introduction Flexoelectricity is a property of dielectric materials where an inhomogeneous mechanical deformation, such as bending, induces electrical polarization (direct effect) [1]. Conversely, an applied electric field induces mechanical deformation (inverse effect) [2]. Unlike piezoelectricity, which requires non-centrosymmetric materials, flexoelectricity can occur in all dielectric materials [3]. is effect is particularly significant at the nanoscale, where large strain gradients are easier to achieve [4]. e simplest expression of this relationship is captured in Equation [1.1,](#page-1-0) where the flexoelectric coefficient is represented by , the material strain gradient is ⁄ and the generated polarization is [5]. A conceptual visualization of flexoelectricity can be illustrated using a simple ionic lattice [\(Figure 1.1a](#page-1-1)). When the material bends, the strain gradient causes a relative displacement between positively and negatively charged ions, leading to a local redistribution of charges and inducing a net polarization [\(Figure 1.1c](#page-1-1)). In materials where ionic movement is minimal, flexoelectricity can also be understood as a redistribution of electron density under strain gradients [6], enabling the effect in any type of dielectrics. ![<span id="page-1-1"></span>![](_page_1_Figure_3.jpeg)](path) Direct flexoelectricity in a mechanical structure [\(Figure 1.1b](#page-1-1)) generates surface charges across the beam when it bends along its length. To facilitate the reading of small charges, studies in the literature tend to use samples of large lateral dimensions (millimeters to centimeters). Consequently, to avoid breaking the sample when applying the load, the samples have thicknesses that range from 0.5 to 3 mm. Typically these measurements employ a three- or four-point bending tester, connected to an amplifier [7], [8], [9], [10], [11]. Other authors produce the strain in the material with a loudspeaker and measure charges with a lock-in amplifier. In these cases, the displacement of the loudspeaker is measured using a Displacement Voltage Ratio Transformer (DVRT) [1], [12], [13], [14] or an optical sensor [15], [16]. Additionally, some approaches utilize a piezoelectric shaker to flex the sample, with the displacement controlled by a Laser Doppler Vibrometer (LDV). In these cases, charge measurements are conducted using an amplifier [17], [18]. To date, no method reported in the literature has successfully measured direct flexoelectric effect in materials with nanoscale thicknesses. is is primarily due to the challenges in achieving sufficient sensitivity and accurately controlling strain gradients at such small scales [2]. Given that one of the most promising applications of flexoelectricity lies in Nano-Electro-Mechanical Systems (NEMS), where device dimensions are on the nanometer scale, confirming the presence and magnitude of the flexoelectric effect at these scales is crucial. is could enable the development of novel sensors, actuators, and energy-harvesting devices that use nanoscale flexoelectricity [19]. # 1.3 Methodology Our approach for measuring flexoelectric charges at the nanoscale starts by fabricating cantilevers with a thin (50 nm) layer of hafnium oxide, sandwiched between two platinum electrodes (20 nm). Hafnium oxide is selected due to its high dielectric constant (within microelectronic compatible materials), well-characterized coefficient [20], [21], and its wide availability in cleanroom facilities. e cantilevers measure 20 µm in length and vary in width from 5 µm to 50 µm [\(Figure 1.2\)](#page-2-0). Even though we favor narrower devices, we include a range of widths in this study to facilitate charge detection. ![<span id="page-2-0"></span>Figure 1.2 SEM images of fabricated cantilevers. a) Cantilever with a length of 20 µm and a width of 5 µm. b) Cantilever with a length of 20 µm and a width of 50 µm.](path) To maximize the flexoelectric response, we excite the cantilevers at their resonance frequency using a piezoelectric shaker under the cantilevers chip. Operating at resonance amplifies the mechanical displacement, improving the signal-to-noise ratio. Measurements are conducted under vacuum (3 · 10−4 ) to reduce air damping, further enhancing displacement amplitude. e generated flexoelectric charges are collected by platinum electrodes and amplified using a low-noise voltage amplifier (Sierra Amps [22]) placed in close proximity to minimize the load capacitance and signal loss. e experimental setup [\(Figure 1.3\)](#page-3-0) includes a Laser Doppler Vibrometer to monitor cantilever displacement. All signals are processed through a lock-in amplifier (Zurich Instruments UHFLI), which controls the shaker, reads the electrical charges, and monitors the cantilever's displacement. Importantly, the electrical charges and the vibrometer signal are not monitored simultaneously, but subsequently. ![<span id="page-3-0"></span>Figure 1.3 Schematic of the experimental setup to measure flexoelectric charges in nanoscale-thickness materials. e cantilever is actuated by a piezoelectric shaker at its resonance frequency under vacuum. A low-noise amplifier amplifies the generated flexoelectric charges () and parasitic charges ( ). A Laser Doppler Vibrometer monitors the displacement of the cantilever. All signals are processed in a lockin amplifier.](path) e first measurement quantifies the cantilever's displacement for a certain shaker actuation. By sweeping the excitation frequency around the cantilever's resonance frequency, we measure the displacement at the laser's measurement point [\(Figure 1.4\)](#page-4-0). However, our primary interest is the displacement at the cantilever's tip, which is essential for calculating the strain gradient (beam curvature). To extrapolate the tip displacement from the measured point, we analyze the thermomechanical noise and use a standard cantilever beam model. is method allows us to translate the measurements at any point along the cantilever to the tip [23], [24]. ![<span id="page-4-0"></span>Figure 1.4 Laser Doppler Vibrometer measurements of cantilever displacement around the resonance frequency. e displacement amplitude is linearly proportional to the actuation voltage applied to the shaker. ermomechanical noise measurements, conducted without actuation, are used to verify the laser position on the cantilever and translate the amplitude measurements to cantilever's tip displacement.](path) ![<span id="page-4-1"></span>Figure 1.5 Flexoelectric voltage generated in the cantilever around its resonance frequency. a) Normalized output voltage for a cantilever with a width of 40 µm and a length of 20 µm. We observe the flexoelectric charges superimposed on the background signal. b) Lorentzian signal corresponding to the cantilever's motion, obtained by subtracting the background signal from the output signal.](path) e second measurement evaluates the electrical charges generated by the flexoelectric effect. We perform the measurements without the laser, as its illumination can induce photoelectric effects and interfere with the flexoelectric charges [25]. [Figure 1.5](#page-4-1) shows the electrical measurements obtained from a cantilever with dimensions of 40 µm in width and 20 µm in length. In [Figure 1.5a](#page-4-1), the normalized transfer function is plotted as a function of frequency, revealing the cantilever's resonance peak superimposed on a background signal. is background arises from parasitic capacitance pathways between the shaker's actuation and the cantilever. To isolate the flexoelectric signal, we perform a linear baseline subtraction for both components in phase and in quadrature with the drive, to remove the background contribution, resulting in the cleaned data presented in [Figure 1.5b](#page-4-1). e extracted signal corresponds to the flexoelectric charges generated due to the cantilever's motion, which are linearly dependent on the shaker's actuation. By combining the measured electrical amplitudes () with the cantilever displacements at the tip () at various shaker actuations, we calculate the effective flexoelectric coefficient () using Equation [1.2.](#page-5-0) is calculation also involves the cantilever's width (), the capacitance of the measurement path leading to the low-noise amplifier ( ) and the derivative of the normalized n-mode shape (′ ()). For the cantilever's first mode, 1 ′ () = − 1.3765⁄, where L represents the length of the cantilever. Detailed derivation is provided in the supplementary material. #### 1.4 Results and discussion [Figure 1.6](#page-6-0) presents the effective flexoelectric coefficient for cantilevers of varying widths. We observe that increases with increasing width, reaching a saturation value for widths larger than 40 µm. is trend is attributed to a regime shift, as the cantilevers transition from the < condition to > . is change affects how the flexoelectric tensor components 11 and 12 [4] interact to determine the generated charges. [Table 1.1](#page-6-1) describes the theoretical asymptotic limits of the effective flexoelectric coefficient for both narrow and wide cantilevers (compared to the length). Detailed derivations are provided in the supplementary material. By making the ratio between the limits we obtain ,⁄, = 1⁄(1 − ), which depends solely on the material's Poisson's ratio (), and not on the flexoelectric tensor terms. For hafnium oxide, with = 0.3, the ratio evaluates to approximately 1.42. is theoretical prediction aligns well with the data i[n Figure 1.6,](#page-6-0) where increasing the width results in a 42% increase in . <span id="page-6-1"></span>Table 1.1 Asymptotical effective flexoelectric coefficients. Asymptotical limits of the effective flexoelectric coefficient for narrow and wide cantilevers. e ratio between these limits is independent of the flexoelectric tensor components, depending only on Poisson's ratio of the material. (t = Cantilever's thickness). ![<span id="page-6-0"></span>Figure 1.6 Effective flexoelectric coefficient for cantilevers with different widths. e growth in the effective flexoelectric coefficient is theoretically expected, as the interplay between the tensorial components depend on the width of the cantilever. e figure shows the asymptotical limits for the extreme cases of the width.](path) e direct flexoelectric measurements presented in this work [\(Figure 1.6\)](#page-6-0) align with the published coefficients we previously obtained from inverse flexoelectric measurements [20]. In that study, we report a value of = 105 ± 10 / for narrow cantilevers, which aligns closely with the direct flexoelectric measurements. # 1.5 Conclusion is study presents the first flexoelectric charge measurements in nanometrically thin materials. We employ microcantilevers with a 50 nm layer of hafnium oxide between platinum electrodes. Electrical signals are captured using a low-noise amplifier and they are compared with the cantilever displacement which is measured with a Laser Doppler Vibrometer. e results align with hafnium oxide coefficient values obtained through inverse methods. Additionally, we demonstrate that the effective flexoelectric coefficient increases with cantilever width (for a fixed length), showing the dependence on cantilever geometry. ese findings could help in the fabrication of next-generation NEMS devices. ## References - [8] P. Zubko, G. Catalan, A. Buckley, P. R. L. Welche, and J. F. Scott, "Strain-Gradient-Induced Polarization in SrTiO 3 Single Crystals," Phys. Rev. Lett., vol. 99, no. 16, p. 167601, Oct. 2007, doi: 10.1103/PhysRevLett.99.167601. - [10] J. Narvaez, S. Saremi, J. Hong, M. Stengel, and G. Catalan, "Large Flexoelectric Anisotropy in Paraelectric Barium Titanate," Phys. Rev. Lett., vol. 115, no. 3, p. 037601, Jul. 2015, doi: 10.1103/PhysRevLett.115.037601. - [12] W. Ma and L. E. Cross, "Flexoelectric polarization of barium strontium titanate in the paraelectric state," Applied Physics Letters, vol. 81, no. 18, pp. 3440–3442, Oct. 2002, doi: 10.1063/1.1518559. - [16] Y. Li, L. Shu, W. Huang, X. Jiang, and H. Wang, "Giant flexoelectricity in Ba0.6Sr0.4TiO3/Ni0.8Zn0.2Fe2O4 composite," Applied Physics Letters, vol. 105, no. 16, p. 162906, Oct. 2014, doi: 10.1063/1.4899060. - [17] W. Huang, K. Kim, S. Zhang, F.-G. Yuan, and X. Jiang, "Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers: Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers," Phys. Status Solidi RRL, vol. 5, no. 9, pp. 350–352, Sep. 2011, doi: 10.1002/pssr.201105326. - [18] S. Huang, T. Kim, D. Hou, D. Cann, J. L. Jones, and X. Jiang, "Flexoelectric characterization of BaTiO3-0.08Bi(Zn1/2Ti1/2)O3," Applied Physics Letters, vol. 110, no. 22, p. 222904, May 2017, doi: 10.1063/1.4984212. - [19] Q. Deng, M. Kammoun, A. Erturk, and P. Sharma, "Nanoscale flexoelectric energy harvesting," International Journal of Solids and Structures, vol. 51, no. 18, pp. 3218– 3225, Sep. 2014, doi: 10.1016/j.ijsolstr.2014.05.018. - [23] L. G. Villanueva, R. B. Karabalin, M. H. Matheny, D. Chi, J. E. Sader, and M. L. Roukes, "Nonlinearity in nanomechanical cantilevers," Phys. Rev. B, vol. 87, no. 2, p. 024304, Jan. 2013, doi: 10.1103/PhysRevB.87.024304. # 1.6 Supplementary Material #### 1.6.1 Flexoelectric formula derivation Derivation of the formula to convert the read voltage (after background subtraction) and cantilever tip displacement into effective flexoelectric coefficient. ![Figure 1.7 Equivalent circuit of the electronic setup. e cantilever is represented as a current source ( ), which has an equivalent parallel resistance () and capacitance ( ). e capacitance of the connector is represented by ( ), the input lownoise amplifier resistance ( ) and capacitance ( ).](path) $$\phi\_n(\mathbf{x}) = \frac{1}{2} \left[ \cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} (\sin(\beta\_n \mathbf{x}) - \sinh(\beta\_n \mathbf{x})) \right] \tag{1.12}$$ $$\phi\_n'(\mathbf{x}) = \frac{1}{2} \left[ -\beta\_n \cdot \sin(\beta\_n \mathbf{x}) - \beta\_n \cdot \sinh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} \cdot \beta\_n \cdot (\cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x})) \right] \tag{1.13}$$ #### 1.6.2 Measurement uncertainty evaluation Extended formula for the direct measurement of flexoelectricity, including all dependencies. $$\mu\_{direct} = \frac{\left(\mathcal{C}\_{device} + \mathcal{C}\_{connect} + \mathcal{C}\_{amplitude}\right) \cdot L}{1.3765 \cdot W} \cdot \frac{V\_{initial}}{Amp\_{sweep} \cdot \sqrt{\frac{4 \cdot Q \cdot K\_B T}{1 \cdot L \cdot W \cdot \Sigma \rho\_l t\_l \cdot (2\pi f\_R)^3 \cdot P \text{SD}/2}}} \tag{1.15}$$ ## 1.6.3 Flexoelectric tensor contributions to the effective flexoelectric coefficient ![∂z](path) ## 1.6.4 Equivalent flexoelectric charges for the measured voltages Here, just for curiosity, we calculate what are the equivalent flexoelectric charges for the measured flexoelectric voltages . Our methodology does not measure flexoelectric charges directly, we measure currents, which is an integration of the charges at a relatively high frequency (200 kHz). We found these calculations interesting to have an estimated electron charge produced by the flexoelectric effect for hafnium oxide.
Direct flexoelectricity in a mechanical structure [\(Figure 1.1b](#page-1-1))	# Measuring direct flexoelectricity at the nanoscale #### 1.1 Abstract Flexoelectricity is a property of all dielectric materials, where inhomogeneous strain induces electrical polarization. is effect becomes particularly prominent at the nanoscale where larger strain gradients can be obtained. While flexoelectric charges have been measured in mm-scale systems, direct measurements in nanoscale-thickness materials have not yet been achieved. Given that one of the most prominent applications of flexoelectricity is in nanoelectro-mechanical systems (NEMS), confirming the presence and magnitude of the effect at these scales is essential. is study presents the first-ever measurements of flexoelectricgenerated charges (direct effect) in nanoscale-thickness materials, using cantilevers with a 50 nm hafnium oxide layer. We confirm that the estimated flexoelectric coefficient from said measurements aligns with the values obtained from complementary experiments using the flexoelectric inverse effect. Additionally, by changing the cantilever geometry (modifying the width of the cantilevers), we demonstrate a 40% increase in the effective flexoelectric coefficient, explained by the interplay of different flexoelectric tensor components. These findings not only validate the presence of flexoelectric effects at the nanoscale but also open the possibility for full flexoelectric transduction of the motion in NEMS/MEMS devices. ### 1.2 Introduction Flexoelectricity is a property of dielectric materials where an inhomogeneous mechanical deformation, such as bending, induces electrical polarization (direct effect) [1]. Conversely, an applied electric field induces mechanical deformation (inverse effect) [2]. Unlike piezoelectricity, which requires non-centrosymmetric materials, flexoelectricity can occur in all dielectric materials [3]. is effect is particularly significant at the nanoscale, where large strain gradients are easier to achieve [4]. e simplest expression of this relationship is captured in Equation [1.1,](#page-1-0) where the flexoelectric coefficient is represented by , the material strain gradient is ⁄ and the generated polarization is [5]. A conceptual visualization of flexoelectricity can be illustrated using a simple ionic lattice [\(Figure 1.1a](#page-1-1)). When the material bends, the strain gradient causes a relative displacement between positively and negatively charged ions, leading to a local redistribution of charges and inducing a net polarization [\(Figure 1.1c](#page-1-1)). In materials where ionic movement is minimal, flexoelectricity can also be understood as a redistribution of electron density under strain gradients [6], enabling the effect in any type of dielectrics. generates surface charges across the beam when it bends along its length. To facilitate the reading of small charges, studies in the literature tend to use samples of large lateral dimensions (millimeters to centimeters). Consequently, to avoid breaking the sample when applying the load, the samples have thicknesses that range from 0.5 to 3 mm. Typically these measurements employ a three- or four-point bending tester, connected to an amplifier [7], [8], [9], [10], [11]. Other authors produce the strain in the material with a loudspeaker and measure charges with a lock-in amplifier. In these cases, the displacement of the loudspeaker is measured using a Displacement Voltage Ratio Transformer (DVRT) [1], [12], [13], [14] or an optical sensor [15], [16]. Additionally, some approaches utilize a piezoelectric shaker to flex the sample, with the displacement controlled by a Laser Doppler Vibrometer (LDV). In these cases, charge measurements are conducted using an amplifier [17], [18]. To date, no method reported in the literature has successfully measured direct flexoelectric effect in materials with nanoscale thicknesses. is is primarily due to the challenges in achieving sufficient sensitivity and accurately controlling strain gradients at such small scales [2]. Given that one of the most promising applications of flexoelectricity lies in Nano-Electro-Mechanical Systems (NEMS), where device dimensions are on the nanometer scale, confirming the presence and magnitude of the flexoelectric effect at these scales is crucial. is could enable the development of novel sensors, actuators, and energy-harvesting devices that use nanoscale flexoelectricity [19]. # 1.3 Methodology Our approach for measuring flexoelectric charges at the nanoscale starts by fabricating cantilevers with a thin (50 nm) layer of hafnium oxide, sandwiched between two platinum electrodes (20 nm). Hafnium oxide is selected due to its high dielectric constant (within microelectronic compatible materials), well-characterized coefficient [20], [21], and its wide availability in cleanroom facilities. e cantilevers measure 20 µm in length and vary in width from 5 µm to 50 µm [\(Figure 1.2\)](#page-2-0). Even though we favor narrower devices, we include a range of widths in this study to facilitate charge detection. ![<span id="page-2-0"></span>Figure 1.2 SEM images of fabricated cantilevers. a) Cantilever with a length of 20 µm and a width of 5 µm. b) Cantilever with a length of 20 µm and a width of 50 µm.](path) To maximize the flexoelectric response, we excite the cantilevers at their resonance frequency using a piezoelectric shaker under the cantilevers chip. Operating at resonance amplifies the mechanical displacement, improving the signal-to-noise ratio. Measurements are conducted under vacuum (3 · 10−4 ) to reduce air damping, further enhancing displacement amplitude. e generated flexoelectric charges are collected by platinum electrodes and amplified using a low-noise voltage amplifier (Sierra Amps [22]) placed in close proximity to minimize the load capacitance and signal loss. e experimental setup [\(Figure 1.3\)](#page-3-0) includes a Laser Doppler Vibrometer to monitor cantilever displacement. All signals are processed through a lock-in amplifier (Zurich Instruments UHFLI), which controls the shaker, reads the electrical charges, and monitors the cantilever's displacement. Importantly, the electrical charges and the vibrometer signal are not monitored simultaneously, but subsequently. ![<span id="page-3-0"></span>Figure 1.3 Schematic of the experimental setup to measure flexoelectric charges in nanoscale-thickness materials. e cantilever is actuated by a piezoelectric shaker at its resonance frequency under vacuum. A low-noise amplifier amplifies the generated flexoelectric charges () and parasitic charges ( ). A Laser Doppler Vibrometer monitors the displacement of the cantilever. All signals are processed in a lockin amplifier.](path) e first measurement quantifies the cantilever's displacement for a certain shaker actuation. By sweeping the excitation frequency around the cantilever's resonance frequency, we measure the displacement at the laser's measurement point [\(Figure 1.4\)](#page-4-0). However, our primary interest is the displacement at the cantilever's tip, which is essential for calculating the strain gradient (beam curvature). To extrapolate the tip displacement from the measured point, we analyze the thermomechanical noise and use a standard cantilever beam model. is method allows us to translate the measurements at any point along the cantilever to the tip [23], [24]. ![<span id="page-4-0"></span>Figure 1.4 Laser Doppler Vibrometer measurements of cantilever displacement around the resonance frequency. e displacement amplitude is linearly proportional to the actuation voltage applied to the shaker. ermomechanical noise measurements, conducted without actuation, are used to verify the laser position on the cantilever and translate the amplitude measurements to cantilever's tip displacement.](path) ![<span id="page-4-1"></span>Figure 1.5 Flexoelectric voltage generated in the cantilever around its resonance frequency. a) Normalized output voltage for a cantilever with a width of 40 µm and a length of 20 µm. We observe the flexoelectric charges superimposed on the background signal. b) Lorentzian signal corresponding to the cantilever's motion, obtained by subtracting the background signal from the output signal.](path) e second measurement evaluates the electrical charges generated by the flexoelectric effect. We perform the measurements without the laser, as its illumination can induce photoelectric effects and interfere with the flexoelectric charges [25]. [Figure 1.5](#page-4-1) shows the electrical measurements obtained from a cantilever with dimensions of 40 µm in width and 20 µm in length. In [Figure 1.5a](#page-4-1), the normalized transfer function is plotted as a function of frequency, revealing the cantilever's resonance peak superimposed on a background signal. is background arises from parasitic capacitance pathways between the shaker's actuation and the cantilever. To isolate the flexoelectric signal, we perform a linear baseline subtraction for both components in phase and in quadrature with the drive, to remove the background contribution, resulting in the cleaned data presented in [Figure 1.5b](#page-4-1). e extracted signal corresponds to the flexoelectric charges generated due to the cantilever's motion, which are linearly dependent on the shaker's actuation. By combining the measured electrical amplitudes () with the cantilever displacements at the tip () at various shaker actuations, we calculate the effective flexoelectric coefficient () using Equation [1.2.](#page-5-0) is calculation also involves the cantilever's width (), the capacitance of the measurement path leading to the low-noise amplifier ( ) and the derivative of the normalized n-mode shape (′ ()). For the cantilever's first mode, 1 ′ () = − 1.3765⁄, where L represents the length of the cantilever. Detailed derivation is provided in the supplementary material. #### 1.4 Results and discussion [Figure 1.6](#page-6-0) presents the effective flexoelectric coefficient for cantilevers of varying widths. We observe that increases with increasing width, reaching a saturation value for widths larger than 40 µm. is trend is attributed to a regime shift, as the cantilevers transition from the < condition to > . is change affects how the flexoelectric tensor components 11 and 12 [4] interact to determine the generated charges. [Table 1.1](#page-6-1) describes the theoretical asymptotic limits of the effective flexoelectric coefficient for both narrow and wide cantilevers (compared to the length). Detailed derivations are provided in the supplementary material. By making the ratio between the limits we obtain ,⁄, = 1⁄(1 − ), which depends solely on the material's Poisson's ratio (), and not on the flexoelectric tensor terms. For hafnium oxide, with = 0.3, the ratio evaluates to approximately 1.42. is theoretical prediction aligns well with the data i[n Figure 1.6,](#page-6-0) where increasing the width results in a 42% increase in . <span id="page-6-1"></span>Table 1.1 Asymptotical effective flexoelectric coefficients. Asymptotical limits of the effective flexoelectric coefficient for narrow and wide cantilevers. e ratio between these limits is independent of the flexoelectric tensor components, depending only on Poisson's ratio of the material. (t = Cantilever's thickness). ![<span id="page-6-0"></span>Figure 1.6 Effective flexoelectric coefficient for cantilevers with different widths. e growth in the effective flexoelectric coefficient is theoretically expected, as the interplay between the tensorial components depend on the width of the cantilever. e figure shows the asymptotical limits for the extreme cases of the width.](path) e direct flexoelectric measurements presented in this work [\(Figure 1.6\)](#page-6-0) align with the published coefficients we previously obtained from inverse flexoelectric measurements [20]. In that study, we report a value of = 105 ± 10 / for narrow cantilevers, which aligns closely with the direct flexoelectric measurements. # 1.5 Conclusion is study presents the first flexoelectric charge measurements in nanometrically thin materials. We employ microcantilevers with a 50 nm layer of hafnium oxide between platinum electrodes. Electrical signals are captured using a low-noise amplifier and they are compared with the cantilever displacement which is measured with a Laser Doppler Vibrometer. e results align with hafnium oxide coefficient values obtained through inverse methods. Additionally, we demonstrate that the effective flexoelectric coefficient increases with cantilever width (for a fixed length), showing the dependence on cantilever geometry. ese findings could help in the fabrication of next-generation NEMS devices. ## References - [8] P. Zubko, G. Catalan, A. Buckley, P. R. L. Welche, and J. F. Scott, "Strain-Gradient-Induced Polarization in SrTiO 3 Single Crystals," Phys. Rev. Lett., vol. 99, no. 16, p. 167601, Oct. 2007, doi: 10.1103/PhysRevLett.99.167601. - [10] J. Narvaez, S. Saremi, J. Hong, M. Stengel, and G. Catalan, "Large Flexoelectric Anisotropy in Paraelectric Barium Titanate," Phys. Rev. Lett., vol. 115, no. 3, p. 037601, Jul. 2015, doi: 10.1103/PhysRevLett.115.037601. - [12] W. Ma and L. E. Cross, "Flexoelectric polarization of barium strontium titanate in the paraelectric state," Applied Physics Letters, vol. 81, no. 18, pp. 3440–3442, Oct. 2002, doi: 10.1063/1.1518559. - [16] Y. Li, L. Shu, W. Huang, X. Jiang, and H. Wang, "Giant flexoelectricity in Ba0.6Sr0.4TiO3/Ni0.8Zn0.2Fe2O4 composite," Applied Physics Letters, vol. 105, no. 16, p. 162906, Oct. 2014, doi: 10.1063/1.4899060. - [17] W. Huang, K. Kim, S. Zhang, F.-G. Yuan, and X. Jiang, "Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers: Scaling effect of flexoelectric (Ba,Sr)TiO 3 microcantilevers," Phys. Status Solidi RRL, vol. 5, no. 9, pp. 350–352, Sep. 2011, doi: 10.1002/pssr.201105326. - [18] S. Huang, T. Kim, D. Hou, D. Cann, J. L. Jones, and X. Jiang, "Flexoelectric characterization of BaTiO3-0.08Bi(Zn1/2Ti1/2)O3," Applied Physics Letters, vol. 110, no. 22, p. 222904, May 2017, doi: 10.1063/1.4984212. - [19] Q. Deng, M. Kammoun, A. Erturk, and P. Sharma, "Nanoscale flexoelectric energy harvesting," International Journal of Solids and Structures, vol. 51, no. 18, pp. 3218– 3225, Sep. 2014, doi: 10.1016/j.ijsolstr.2014.05.018. - [23] L. G. Villanueva, R. B. Karabalin, M. H. Matheny, D. Chi, J. E. Sader, and M. L. Roukes, "Nonlinearity in nanomechanical cantilevers," Phys. Rev. B, vol. 87, no. 2, p. 024304, Jan. 2013, doi: 10.1103/PhysRevB.87.024304. # 1.6 Supplementary Material #### 1.6.1 Flexoelectric formula derivation Derivation of the formula to convert the read voltage (after background subtraction) and cantilever tip displacement into effective flexoelectric coefficient. ![Figure 1.7 Equivalent circuit of the electronic setup. e cantilever is represented as a current source ( ), which has an equivalent parallel resistance () and capacitance ( ). e capacitance of the connector is represented by ( ), the input lownoise amplifier resistance ( ) and capacitance ( ).](path) $$\phi\_n(\mathbf{x}) = \frac{1}{2} \left[ \cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} (\sin(\beta\_n \mathbf{x}) - \sinh(\beta\_n \mathbf{x})) \right] \tag{1.12}$$ $$\phi\_n'(\mathbf{x}) = \frac{1}{2} \left[ -\beta\_n \cdot \sin(\beta\_n \mathbf{x}) - \beta\_n \cdot \sinh(\beta\_n \mathbf{x}) - \frac{\cos(\beta\_n L) + \cosh(\beta\_n L)}{\sin(\beta\_n L) + \sinh(\beta\_n L)} \cdot \beta\_n \cdot (\cos(\beta\_n \mathbf{x}) - \cosh(\beta\_n \mathbf{x})) \right] \tag{1.13}$$ #### 1.6.2 Measurement uncertainty evaluation Extended formula for the direct measurement of flexoelectricity, including all dependencies. $$\mu\_{direct} = \frac{\left(\mathcal{C}\_{device} + \mathcal{C}\_{connect} + \mathcal{C}\_{amplitude}\right) \cdot L}{1.3765 \cdot W} \cdot \frac{V\_{initial}}{Amp\_{sweep} \cdot \sqrt{\frac{4 \cdot Q \cdot K\_B T}{1 \cdot L \cdot W \cdot \Sigma \rho\_l t\_l \cdot (2\pi f\_R)^3 \cdot P \text{SD}/2}}} \tag{1.15}$$ ## 1.6.3 Flexoelectric tensor contributions to the effective flexoelectric coefficient ![∂z](path) ## 1.6.4 Equivalent flexoelectric charges for the measured voltages Here, just for curiosity, we calculate what are the equivalent flexoelectric charges for the measured flexoelectric voltages . Our methodology does not measure flexoelectric charges directly, we measure currents, which is an integration of the charges at a relatively high frequency (200 kHz). We found these calculations interesting to have an estimated electron charge produced by the flexoelectric effect for hafnium oxide.
Figure D.2: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm² scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm² measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ∥ = 40% and ϵ⊥ = 40%, respectively. In each of the maps, the coating direction goes from bottom left to top right.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [\(II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5\)](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [\(1\)](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [\(5\)](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [\(3\)](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [\(B1\)](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm² scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm² measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ∥ = 40% and ϵ⊥ = 40%, respectively. In each of the maps, the coating direction goes from bottom left to top right.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [\(II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5\)](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [\(1\)](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [\(5\)](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [\(3\)](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [\(B1\)](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [\(II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5\)](#page-8-1). ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [\(1\)](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [\(5\)](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [\(3\)](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [\(B1\)](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [\(II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5\)](#page-8-1). Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [\(5\)](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [\(3\)](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [\(B1\)](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵmax = 40 %), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2.3]). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [\(II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5\)](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [\(1\)](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [\(5\)](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [\(3\)](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [\(B1\)](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure 4: Electrical four‐point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB‐silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R∥ and R⊥ (multiplied by film thickness to compensate geometric variations, see eq. [\(1\)](#page-3-2) in Section [II.2.1](#page-3-0)), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R⊥ were calculated from all local pairs of R∥ and R⊥ of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [\(II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5\)](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [\(1\)](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [\(5\)](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [\(3\)](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [\(B1\)](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm² per material state). Scans of the unstretched 7 vol% CB‐film and one scan on the 7 vol% CB‐film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB‐ and matrix-dominated regions.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [\(II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5\)](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [\(1\)](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [\(5\)](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [\(3\)](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [\(B1\)](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure D.1: Visualization of PeakForce QNM signals as derived from force‐distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real‐time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [\(II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5\)](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [\(1\)](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [\(5\)](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [\(3\)](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [\(B1\)](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [\(II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5\)](#page-8-1). Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.
Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10⁻² s⁻¹ strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.	# Flow-induced anisotropy in a carbon black-filled silicone elastomer: electromechanical properties and structure # Abstract Carbon black (CB)-elastomers can serve as low-cost, highly deformable sensor materials, but hardly any work exists on their structure-property relationships. We report on flow-induced anisotropy, considering CB-silicone films generated via doctor blade coating. Cured films showed slight electrical anisotropy, with conductivity parallel to the coating direction being lower than perpendicular to it. Furthermore, piezoresistive sensitivity was much larger for stretch perpendicular to the coating direction than for parallel stretch. Structural analysis for length scales up to the CB agglomerate level yielded only weak evidence of anisotropy. Based on this evidence and insight from CB network simulations, we hypothesize that shear flow during coating fragments the CB network and then induces a preferential aggregate alignment, as well as increased inter-particle distances, parallel to the coating direction. As a practical conclusion, already weak anisotropic structuration suffices to cause significant electric anisotropy. #### <span id="page-1-0"></span>I. INTRODUCTION Carbon black (CB) is a common filler used to tune mechanical properties of rubbers [\[1–](#page-32-0)[5\]](#page-32-1). In addition, it introduces electrical conductivity when its concentration exceeds a critical value (percolation threshold) needed to form a network within the polymeric matrix [\[6–](#page-32-2)[9\]](#page-32-3). CB-filled elastomers can thus be used to create mechanically robust electronic devices such as flexible electrodes and highly deformable piezoresistive sensors [\[3,](#page-32-4) [10–](#page-33-0)[14\]](#page-33-1). Compared to conductive fillers like carbon nanotubes (CNTs) [\[15–](#page-33-2)[17\]](#page-33-3), graphene [\[18,](#page-33-4) [19\]](#page-33-5), and silver nanoparticles [\[20,](#page-33-6) [21\]](#page-33-7), CB is an attractive alternative in that it is much less expensive and more readily available [\[1\]](#page-32-0). To optimize CB elastomers for electronic applications, the correlation between structure (esp. CB dispersion state) and electromechanical properties must be understood. Literature on CB composites reports on numerous relevant factors, e.g. CB primary structure and surface chemistry [\[4,](#page-32-5) [22](#page-33-8)[–24\]](#page-34-0), CB aggregate size distribution [\[25,](#page-34-1) [26\]](#page-34-2), CB-matrix interactions [\[27\]](#page-34-3), additives (e.g. ionic liquids [\[28,](#page-34-4) [29\]](#page-34-5), inorganic salts [\[26\]](#page-34-2), non-ionic plasticizer [\[30\]](#page-34-6)), matrix viscosity during processing [\[31\]](#page-34-7), and curing temperature [\[31\]](#page-34-7). In addition, the processing method has a decisive impact [\[1–](#page-32-0) [3,](#page-32-4) [6,](#page-32-2) [7,](#page-32-6) [32–](#page-34-8)[36\]](#page-34-9). Depending on the flow history of the precursor (mixture of CB and the yet liquid matrix), different CB morphologies can form, incl. anisotropic ones. The latter result from material flow in preferential directions, e.g. during injection molding [\[35\]](#page-34-10), compression molding [\[2,](#page-32-7) [6\]](#page-32-2), meltcasting [\[37\]](#page-34-11), and extrusion [\[36\]](#page-34-9). While extensive research on flow-induced anisotropy has been done on CB suspended in low viscosity organic liquids [\[38–](#page-34-12)[41\]](#page-35-0) and CB-filled thermoplasts [\[6,](#page-32-2) [35–](#page-34-10)[37\]](#page-34-11), literature on the microstructuredependent electromechanical properties of CB elastomers is scarce. We did not find any work for CB elastomers with chemically crosslinked matrices (e.g. silicone rubber) and only two publications [\[2,](#page-32-7) [37\]](#page-34-11) for CB-filled thermoplastic elastomers (thermoplastic matrix with elastomeric properties). Flandin et al. [\[2\]](#page-32-7) briefly discussed the possible role of compression molding in the electrical anisotropy of a CB-filled ethylene-octene elastomer in the undeformed state, but due to the absence of structural analysis, the structure-property relationship was not clarified. Ehrburger-Dolle et al. [\[37\]](#page-34-11) reported anisotropic X-ray scattering patterns of ethylene propylene rubber with CB contents slightly above the percolation threshold. The anisotropy, which presumably originates from the liquid composite being sheared during melt-casting, is correlated with different degrees of interpenetration of CB aggregates in two principal orientations. Yet, no link between the principal orientations in the patterns and the flow conditions is established, and the consequences for electrical conductivity are not addressed. In light of this lack, this work explores flow-induced anisotropy of a CB-filled silicone elastomer in terms of electromechanical properties and structure. For this, we exposed the yet liquid CBsilicone mixtures to shear flow by means of doctor blade coating at two gap heights (60 µm, 350 µm) and various coating speeds. The resulting films were cured at elevated temperature to retain the process-induced microstructure. Electrical resistance of cured, unstrained films of three CB concentrations above the percolation threshold was measured both parallel and perpendicular to the coating direction, and the influence of process parameters (blade speed, film thickness) was investigated (Section [III.1\)](#page-8-0). In addition, piezoresistivity (electrical resistance change under mechanical deformation) was studied via uniaxial tensile tests with in-situ electrical resistance measurement along the stretch axis (parallel or perpendicular to the coating direction, Section [III.2\)](#page-11-0). Shearing the uncured CB-silicone mixtures lead to significant electrical and piezoresistive anisotropy in the cured material, whereas mechanical anisotropy was negligible. We discuss implications of this (piezo-)electric anisotropy for industrial practice and explore its structural origin via characterization of unstretched and stretched states by small-angle X-ray scattering (SAXS) as well as nanomechanical mapping (PeakForce QNM) combined with segmentation of the signal maps (Section [III.3\)](#page-17-0). In addition, we present simulations of the fractal filler network, namely on the impact of shear flow on particle alignment and consequences for electrical anisotropy (Section [III.4\)](#page-23-0). A structural hypothesis for the observed phenomenology is given (Section [III.5\)](#page-29-0) which, due to the rather indirect nature of the structural evidence, needs to be validated in future work. ### II. EXPERIMENTAL #### <span id="page-2-0"></span>II.1. Film fabrication and sample preparation CB-filled elastomeric films of at least 5x7 cm<sup>2</sup> were fabricated in ambient air by dispersing CB (7/9/11 vol% CB, above the percolation threshold of ∼5 vol% [\[25\]](#page-34-1)) in a crosslinking silicone matrix (Sylgard® 184, Dow® ), doctor blade coating the resulting reactive mixtures (two gap heights, hgap = 60 µm and 350 µm), and curing for 2 h at 100 °C. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . More details on film fabrication are found in Appendix [A.](#page-38-0) Cured, unstrained films were characterized electrically by the four-point method (Section [II.2.1\)](#page-3-0) without any further sample preparation. Samples characterized by further methods (tensile test with in situ electrical two-point measurement, PeakForce QNM and SAXS, films coated at hgap = 350 µm, 20 mm/s) were cut from near the edges of the films as exemplified in Fig. [1a](#page-3-1) (left/right side for stretch parallel to the coating direction, start/end position for stretch perpendicular to the coating direction). Exact sample dimensions are given in the corresponding sections [\(II.2.2,](#page-4-0) [II.2.3](#page-5-0) and [II.2.5\)](#page-8-1). <span id="page-3-1"></span>![Figure 1: To study anisotropy of CB-filled silicone films generated by doctor blade coating, samples were prepared and characterized parallel and perpendicular to the coating direction. a) film with tensile specimens cut from near the edges and example of measuring grid (spacing 3 – 4 cm) for electrical four-point measurements indicating 'start', 'middle', and 'end' positions, b) probe arrangement and polarity of four-point measurements in the square setup, c) definition of directions and planes with respect to the coating direction.](path) ## <span id="page-3-3"></span>II.2. Characterization #### <span id="page-3-0"></span>II.2.1. Four-point probe measurement Electrical resistance of undeformed films was measured in ambient air (22°C ± 1 K, 26 ± 19 % r.h.) with a Keithley instrument (2450 Interactive SourceMeter®, current sweep from -10 µA to +10 µA, R obtained from linear fits of the ohmic voltage-current curves) using a customized four-point probe setup in the square arrangement (Fig. [1b](#page-3-1)). With the used probe distance of 2 mm, all films can be approximated as thin films, giving the following analytical expressions for resistance, R, in two orthogonal directions x and y [\[42\]](#page-35-1): To measure resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R⊥, the films were arranged such that the latter was parallel or perpendicular to Rx, respectively (see Fig. [1b](#page-3-1)). When oriented in parallel, R<sup>x</sup> equals R∥; when oriented perpendicular, it equals R⊥. Spatial variations were analyzed by choosing three to four measuring spots at the 'start', 'middle' and 'end' sections of each film, as schematically depicted in Fig. [1a](#page-3-1). The respective mean and maximal error of R<sup>∥</sup> and R<sup>⊥</sup> were then calculated from the three to four spots per section (one measurement per spot for each orientation). Note that the spots at a given grid position were not exactly the same for the parallel and perpendicular direction since the measuring direction was switched by rotating the film (rather than the probe polarity). According to eq. [\(1\)](#page-3-2), the resistivities, ρ<sup>x</sup> and ρy, cannot be determined separately from measuring R<sup>x</sup> and Ry. Despite this analytical limitation, the quotient R∥/R<sup>⊥</sup> is a suitable measure for material anisotropy since it grows monotonically with the ratio of the intrinsic resistivities, ρx/ρ<sup>y</sup> [\[42\]](#page-35-1): ## <span id="page-4-0"></span>II.2.2. Uniaxial tensile test with electrical two-point probe measurement To study piezoresistivity, uniaxial tensile tests (universal testing machine Zwick 1446, sample stiffness negligible compared to stiffness of the machine) with in-situ electrical two-point measurements (DAQ6510 by Keithley, constant test current in 'auto range' mode, = 10 µA for the measured samples) were performed in ambient air (22°C ± 1 K, 30 ± 15 % r.h.). As samples, rectangular strips of 4x55 mm<sup>2</sup> were cut from 'thick' films (hgap = 350 µm) coated at 20 mm/s, both for stretch in the parallel (engineering strain ϵ = ϵ∥) and the perpendicular direction (ϵ = ϵ⊥), respectively (Fig. [1a](#page-3-1)). For in-situ resistance measurement along the respective stretch axis, R∥(ϵ∥) and R⊥(ϵ⊥), aluminum strips were glued to the top and bottom of the samples with conductive silver glue (Elektrodag 1415 M by Plano) and fixated with copper foil. The resulting probe distance is identical to the gauge length for straining and equals L<sup>0</sup> = 35 mm in the unstretched state. Samples mounted in the testing machine were contacted electrically with crocodile clamps. After mounting, the tensile force was zeroed and the sample stretched to a pre-load of 0.05 N. The tensile test started after both force and electrical resistance had stabilized (roughly after 20 – 25 min), with the force zeroed at the start of the loading phase. The testing procedure consisted of 4 load-unload cycles between 0 % strain and maximal strains of ϵmax = 10/20/30/40 %, with a strain rate of 10−<sup>2</sup> s −1 (controlled via the position of the crosshead). After each loading/unloading, the material was allowed to relax for 20 min at the given strain plateau. From these relaxation phases, relaxed resistance values for each strain plateau were derived as explained in Appendix [B.](#page-39-0) To differentiate between the impact of doctor blade coating on the silicone matrix vs. on the CB network, neat Sylgard (0 vol% CB, same blade speed, slightly higher gap height of 400 µm resulting in a similar film thickness, see Appendix [A\)](#page-38-0) was characterized in addition to the CB composites (7/9/11 vol% CB). Mechanical testing was identical to the CB-filled samples except for a lower Resistance measured along the stretch axis, Rx, is related to intrinsic electrical resistivity, ρx, and the geometrical contribution (sample length, L, and cross-sectional area, A) according to the well-known equation where the x-orientation is either parallel or perpendicular to the coating direction in our tests. Thus, in addition to intrinsic resistivity, the geometrical part can lead to piezoresistive anisotropy: Mechanical anisotropy in the form of direction-dependent transverse contraction (compressibility) leads to different cross-sectional areas for a given stretch. ### <span id="page-5-0"></span>II.2.3. PeakForce QNM (quantitative nanomechanics) PeakForce QNM™ by Bruker is a mode of scanning force microscopy (SFM) for quantitative nanomechanical mapping. It outputs sample topography ('height' signal) and, thanks to real-time analysis of force-distance curves for each pixel on the sample surface, local material properties (mirrored by the signals 'deformation', 'dissipation', 'modulus' and 'adhesion'). For more information on PeakForce QNM, the reader is referred to [\[43](#page-35-2)[–45\]](#page-35-3) and Appendix [D.](#page-44-0) Measurements were performed with Bruker's Dimension Icon in ambient air (22.5°C ± 0.5 K, 34 ± 5 % r.h.). The most important methodological details are compiled in Table [I.](#page-6-0) Calibration for quantitative measurements (deflection sensitivity, sync distance, PFT amplitude sensitivity) was done on sapphire via the 'touch calibration' feature of the software (NanoScope 9.30). For characterization in unstrained and strained (40 % parallel/perpendicular to the coating direction) states, rectangular strips were cut from the 350 µm-films coated at 20 mm/s as shown in Section [II.1.](#page-2-0) Sample width (∼1 cm) and length (∼5 cm) were chosen big enough to ensure a center region suitable for scanning (no edge effects) and to allow fixation in the stretched state. Glass slides glued to the SFM stage using white-out served as a substrate. In the unstrained state, sample adhesion to the glass was sufficient for stable scanning. For the strained state, the strips were manually stretched to 40 % strain, fixated with tape, and allowed to relax before characterization (at least 30 min). Samples were scanned on their bottom surface, i.e., the surface generated by the contact to the substrate foil during coating. In contrast to the top surfaces generated by the doctor blade, the bottom surfaces of all investigated compositions (7/9/11 vol% CB) have a topography which allows high quality measurements (no serious artifacts from topological features; rms-roughness = 2 – 5 nm when unstretched). To test for drift during scanning, measuring spots were scanned at least twice. All data presented and discussed here is devoid of artifacts from sample drift. To minimize bias from the control of the vertical position of the cantilever (z), the scanning angle was set to 45° with respect to the coating direction. The parallel and perpendicular stretch axes in the resulting images are indicated in Fig. [2c](#page-7-0). Any artifacts from the z-control during scanning (shape distortion of carbon black in particular) are thus equal for the parallel and the perpendicular orientation. Probe and measuring parameters were chosen to give clean force-distance curves on the soft areas dominated by the silicone matrix as well as on the much stiffer CB-rich regions (see Fig. [2a](#page-7-0) for examples). This involves sufficient indentation on stiff regions (for high quality force-distance curves) and minimal indentation on soft regions (corresponding to maximal lateral resolution). According to the frequency density of the deformation signal (see Fig. [2b](#page-7-0) as well as Appendix [D](#page-44-0) for further explanation), the indentation depth varied between 0 nm and 70 nm for all examined CB contents (7/9/11 vol%). Together with the tip end radius of 30 nm, this corresponds to contact radius values in the same range (a few 10<sup>1</sup> nm). The chosen pixel density (256 per line) resulted in a pixel size suitable for these values (20/40/80 nm for scans of 5x5/10x10/20x20 µm<sup>2</sup> ). Note that the sample volume contributing to the material response exceeds the contact region and indentation depth. As a rule of thumb, it extends to some multiples of the contact radius (downward from the sample surface + laterally from the rotational axis of the tip, see e.g. [\[46\]](#page-35-4) for calculations), which in our case equates to a probed depth of some 10<sup>1</sup> – 10<sup>2</sup> nm. As indicated by the shaded area in Fig. [2b](#page-7-0) (see Appendix [D](#page-44-0) for discussion), regions dominated by CB are much less deformable than the ones dominated by the silicone matrix, with indentation depths of a few nm to about 20 nm. As evidenced by our results presented in Section [III.3.1,](#page-18-0) the resulting lateral resolution suffices to discriminate CB aggregates. <span id="page-7-0"></span>![Figure 2: PeakForce QNM revealed the morphology of near-surface CB. Suitable probe and measuring parameters ensured clean and reproducible data: a) Examples of force-distance curves (adapted from screenshots of real-time display in NanoScope), b) frequency distributions of deformation signal (exported from Gwyddion 2.62) of unstretched films with 7/9/11 vol% CB (grey shaded area: range dominated by CB particles, rest: range dominated by the silicone matrix), c) visualization of the 45° scanning angle which minimizes bias from the z-control of the cantilever (smearing of features in to the scanning direction) parallel and perpendicular to the coating direction.](path) ## <span id="page-7-1"></span>II.2.4. Segmentation and statistical analysis of PeakForce QNM data A particle analysis was carried out with ImageJ and Fiji 2.15.8, respectively [\[47,](#page-35-5) [48\]](#page-35-6). For this purpose, the dissipation signals of the PeakForce QNM measurements were first converted into 8-bit grayscale images using Gwyddion (2.64, Delayed Drifter). The images were then segmented using numerical gray value thresholding. The threshold was set using the MaxEntropy algorithm, implemented in Fiji. The filter uses the entropy of the gray value histogram derived on the basis of information theory to determine a threshold value [\[49\]](#page-35-7). The basic challenge is that the images show both details of particles that are exposed on the surface (black, also visible in the adhesion image) and of particles that are covered by matrix elastomer (dark to light gray). The volume fraction V<sup>V</sup> of the CB particles is evaluated using the approach V<sup>V</sup> = AA, where A<sup>A</sup> denotes the fraction of particles in a perfect two-dimensional surface section through the bulk sample [\[50\]](#page-35-8), assuming a uniform density of CB particles in the film. In overview scans to 20x20 µm<sup>2</sup> , the threshold determination with the MaxEntropy filter proved successful, as it gave a good approximation of the volume fraction of samples with 7/9/11 vol% CB, with a tendency to slight overestimation. The volume fractions of the CB particles measured in 10x10 µm<sup>2</sup> images of the 7 vol% film are indicated in Fig. [3.](#page-9-0) This is followed by erosion and dilation by one pixel each to achieve a better discrimination of CB aggregates, as SFM scans tend to smear the edges of raised features in the scan direction due to control delay. The volume fraction is then determined again. This is followed by the actual particle analysis: First, the angle for each particle in which the maximum Feret diameter lies with respect to the horizontal direction is determined and this is plotted as a histogram over all particles from 3 scans each for the same parameter. (The coating direction is at 45◦ with respect to the horizontal direction). The average particle diameter 2p area/π, the particle size distribution and the average particle distance as well as the circularity of the particles as <sup>4</sup>πarea perimeter<sup>2</sup> are determined as well, see Fig. [8.](#page-21-0) The circularity approaches 1.0 for perfectly round particles and 0.0 for elongated particles. Particles that are cut by the edges of the image are excluded from the evaluation. ## <span id="page-8-1"></span>II.2.5. SAXS SAXS measurements were performed on a laboratory-scale Xeuss 2.0 instrument (Xenocs SA, Grenoble, France). The X-ray beam from a copper K<sup>α</sup> source (wavelength 1.54 ˚A) was focused on the sample with a spot size of 0.25 mm<sup>2</sup> . The samples were located at a sample-detector distance (SDD) of 2500 mm, calibrated using a silver behenate standard. The resulting measurable momentum transfer, q, ranges from 5 · 10−<sup>3</sup>˚A −1 to 2 · 10−<sup>1</sup>˚A −1 , with q being defined as q = 4πsin(θ/2)/λ and θ the scattering angle. 2D scattering patterns were obtained using a Pilatus 300K detector (Dectris, Baden, Switzerland) with a pixel size of 0.172 × 0.172 mm<sup>2</sup> and an acquisition time of 1 h for each sample. The samples (0 vol% CB coated at 400 µm, 20 mm/s and 9 vol% CB coated at 350 µm, 20 mm/s, ∼7x50 mm<sup>2</sup> -strips cut from near the film edges parallel and perpendicular to the coating direction, respectively) were placed directly in the beam, without the need of using a sample container. Sample strain is induced by manually stretching and fixing the samples at 140 % of their original length (nominal strain of 40 %). To obtain I(q) parallel and perpendicular to the direction of strain, the 2D scattering patterns were azimuthally averaged within two angle ranges subtending 20 deg parallel and perpendicular to direction of strain, respectively. # III. RESULTS AND DISCUSSION #### <span id="page-8-0"></span>III.1. Electrical anisotropy of unstrained films Electrical resistances of unstrained films in the two main orientations relative to the coating direction, R<sup>∥</sup> and R⊥, along with corresponding values of the electrical anisotropy ratio, R∥/R⊥, are illustrated in Fig. [4.](#page-10-0) Resistance values are multiplied by the corresponding film thickness to eliminate geometrical effects and facilitate comparison (see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0). Symbols indicate the start, middle and end sections of film coatings. We found no relevant variation of <span id="page-9-0"></span>![Figure 3: Dissipation data from PeakForce QNM measurements for specimens with 7 vol% CB in the unstretched state and at 40 % strain (parallel and perpendicular to the coating direction) are converted to 8-bit grayscale images and segmented by gray value thresholding using the MaxEntropy algorithm implemented in Fiji. The images below illustrate the masks for particle detection. The inset shows the determined area and thus volume fractions of the particles, whereby 3 data sets were evaluated in each case. The enlarged value is the one that belongs to the data set shown.](path) Resistance values parallel to the coating direction were systematically higher than perpendicular to it, resulting in R∥/R<sup>⊥</sup> > 1 (mean: 1.1 – 1.4) for all films. Doctor blade coating the uncured <span id="page-10-0"></span>![Figure 4: Electrical four-point probe measurements (22°C ± 1 K, 26 ± 19 % r.h.) reveal that cured, unstrained CB-silicone films produced by doctor blade coating (7/9/11 vol% CB, gaps 60/350 µm, blade speeds 5 – 400 mm/s) are slightly less conductive in the coating direction that perpendicular to it. Shown in the figure are the respective resistances parallel and perpendicular to the coating direction, R<sup>∥</sup> and R<sup>⊥</sup> (multiplied by film thickness to compensate geometric variations, see eq. [\(1\)](#page-3-2) in Section [II.2.1\)](#page-3-0), measured at the start, middle and end position of the films (mean and maximal error from 3 – 4 points on each position), and the resulting anisotropy ratio, R∥/R⊥. The mean and maximal error of R∥/R<sup>⊥</sup> were calculated from all local pairs of R<sup>∥</sup> and R<sup>⊥</sup> of a given film (9 – 12 measuring spots, see Section [II.2.1](#page-3-0) for visualization of the measuring grid).](path) CB-silicone mixtures apparently introduced electrical anisotropy, decreasing conductivity in the coating direction relative to the perpendicular direction. Within experimental error, the degree of electrical anisotropy depended neither on CB concentration nor on film thickness or blade speed. This is surprising in that all these parameters have a big impact on hydrodynamics: Increasing CB content from 7 vol% to 11 vol% leads to much more viscous mixtures, as evidenced by handling them during the coating process, and the variations of film thickness and blade speed imply a large range of global shear rates of approximately 50 – 3080 s-1 (see Fig. [4\)](#page-10-0). Flow-induced electrical anisotropy in carbon filler-polymer composites has been reported previously. For example, CB-filled thermoplasts [\[6,](#page-32-2) [35,](#page-34-10) [36\]](#page-34-9) and a CNT-filled silicone [\[51\]](#page-35-9) all had larger electrical conductivity in the direction of melt flow than perpendicular to it, which was explained by preferential alignment. In light of this, it is unusual to find that conductivity is larger in the direction normal to shear, as we did here. An important difference to previous studies is the role of the matrix: In two of the three referenced sources on CB-filled thermoplasts, the alignment refers to the polymer chain segments rather than to CB [\[35,](#page-34-10) [36\]](#page-34-9); electrical anisotropy originated from an anisotropic distribution of CB particles (vs. preferential rotational state of anisometric aggregates and agglomerates) which was imposed by the packing and orientation of the matrix chains during processing (e.g. by CB adsorption to aligned PET fibers [\[36\]](#page-34-9)). These examples show that it is incorrect to generally equate flow-induced anisotropy with an alignment of CB particles along the shearing direction. In fact, work on low molecular weight CB suspensions documents anisotropic CB structures aligned perpendicular to the shearing direction [\[38](#page-34-12)[–41,](#page-35-0) [52\]](#page-35-10). The mechanism leading to flow-induced electrical anisotropy appears to have similar efficiency across the given range of flow kinetics and shear forces. Even the lowest shear rate of 50 s-1 is sufficient to induce electrical anisotropy. This is consistent with the fragmentation of a weak network of CB agglomerates upon exceeding a critical shear rate, which is commonly evidenced as shear-thinning behavior [\[53,](#page-35-11) [54\]](#page-35-12). Since the latter has been reported for a mixture very similar to our uncured CB-silicone mixtures in the concerned range of shear rates (see [\[28\]](#page-34-4): 9 vol% CB in silicone resin, steady-state viscosity at 50 s-1 ≈ 70 Pa·s, vs. ∼20 kPa·s at 0.1 s-1), we propose that the CB network gets disrupted by the shear forces, becomes anisotropic as a result of shear flow, and reforms rapidly upon cessation of shear. ## <span id="page-11-0"></span>III.2. Piezoresistive anisotropy Since conductive CB elastomers can be used in sensing applications, we not only consider how flow-induced anisotropy affects electrical conductivity of undeformed films, but how it affects piezoresistivity, i.e., strain-induced resistance changes. For this, we compare the electromechanical response (electrical resistance, mechanical stress) to uniaxial strain parallel vs. perpendicular to the coating direction. Since mechanical anisotropy can lead to piezoresistive anisotropy (see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0), we first discuss the mechanical stress response to strain (Section [III.2.1\)](#page-11-1). In a second step, the anisotropy of piezoresistive sensitivity is examined (Section [III.2.2\)](#page-12-0). ## <span id="page-11-1"></span>III.2.1. Mechanical anisotropy The stress-strain curves in Fig. [5](#page-12-1) show that increasing CB concentration reinforced the composites. This is indicative of strong filler-matrix adhesion and a fine dispersion of CB in the matrix (see Section [III.3.1](#page-18-0) for confirmation). Furthermore, the load cycles for unfilled and CB-filled silicone (Fig. [5a](#page-12-1)-b, 0 vol% vs. 9/11 vol% CB) indicate that CB introduced strain softening: The composites became less stiff with increasing cycle number (evident for ≥ 10 % strain), and the effect was stronger for higher CB contents (maximal for 11 vol% CB, stress reduction by up to ∼15 %). This is common in filled rubbers and referred to as the Mullins effect [\[55\]](#page-35-13). Microstructural hypotheses for this behavior (see [\[33\]](#page-34-13) and sources therein, for example) include the rupture of the filler network and the damage of polymer-filler interphases. Irrespective of the mechanism(s) in our specific <span id="page-12-1"></span>![Figure 5: The stress response to uniaxial strain (strain rate 10-2 s -1, 22°C ± 1 K, 30 ± 15 % r.h.) of CB-silicone films is isotropic within the limits of accuracy. Shown are the stress-strain curves (nominal stress, σ, vs. nominal strain, ϵ) of: a) all load cycles for stretch parallel to the coating direction, b) all load cycles for stretch perpendicular to the coating direction, c) load phase of cycle 4 for all samples and stretching directions. In graphs a) + b), the curves for 9 vol% CB have been omitted for clarity; they lie just below the ones for 11 vol% CB and have the same shape.](path) The tensile tests brought no signs of mechanical anisotropy whatsoever. Figure [5](#page-12-1) illustrates the same stress response for the two stretch axes, σ∥(ϵ∥) and σ⊥(ϵ⊥), within experimental scatter of 3 – 5 % (maximal difference between σ<sup>∥</sup> and σ<sup>⊥</sup> relative to σ∥: 5/5/-6/3 % for 0/7/9/11 vol% CB). In contrast, electrical conductivity was clearly anisotropic, both in the unstrained (Section [III.1\)](#page-8-0) and strained state (Section [III.2.2\)](#page-12-0). We conclude that the flow-induced structuration responsible for electrical anisotropy is either irrelevant for the mechanical stress response or its mechanical effect is too weak to be resolved. In the following, we consider the anisotropy of piezoresistance that links mechanical deformation and electrical resistance change. ## <span id="page-12-0"></span>III.2.2. Anisotropy of piezoresistive sensitivity Electrical resistances always increased when applying uniaxial strain parallel and perpendicular to the coating direction, respectively. As qualitatively represented by the data for 9 vol% CB (Fig. [6a](#page-13-0)), the resistance increase was much larger for stretch perpendicular to the coating direction than for stretch parallel to it. In conclusion, doctor blade coating the liquid CB-silicone mixtures induced anisotropy in the piezoresistive response, with sensitivity being higher perpendicular to the coating direction (dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥). <span id="page-13-0"></span>![Figure 6: Electrical resistance of CB-silicone films (7/9/11 vol% CB) increased more strongly for strain perpendicular to the coating direction than parallel to it. a) Example of resistance data (9 vol% CB, change along stretch axis relative to initial value before stretching, R0) during load-unload cycles to ϵmax = 0/10/20/30/40 % (22°C ± 1 K, 30 ± 15 % r.h.). For quantitative analysis of piezoresistive anisotropy, relaxed resistances are compared for parallel and perpendicular stretch (mean and maximal error from two samples each): b) – d) overall resistance change relative to R0, along with irreversible (permanent increase after unloading) and reversible (overall minus irreversible increase) parts (lines = trend), e) irreversible parts (lines = trend), f) first derivatives of trend curves in panels b-d as a measure of piezoresistive sensitivity.](path) In general, both intrinsic electrical resistivity and sample geometry contribute to resistance. As explained in Section [II.2.2,](#page-4-0) we cannot discriminate them due to the unknown (possibly anisotropic) strain-dependence of the cross-sectional area. In light of the continuity of the resistance increase (Fig. [6b](#page-13-0)-d), we assume that both contributions grow upon stretching, at least up to our moderate maximal strains of 40 %. Concerning intrinsic resistivity, a strain-induced increase is commonly observed for CB elastomers at small to moderate stretches up to 25 - 30 % [\[2,](#page-32-7) [10,](#page-33-0) [11,](#page-33-9) [36,](#page-34-9) [56\]](#page-35-14). The increase is attributed to growing CB interparticle distances, i.e., conductive pathways become more resistive and fewer in number when tunneling distances increase and electrical contacts are lost as a result of stretching [\[8,](#page-32-8) [11,](#page-33-9) [13,](#page-33-10) [36,](#page-34-9) [56\]](#page-35-14). This agrees with structural evidence for a CB elastomer presented in [\[37\]](#page-34-11) where the degree of interpenetration of CB aggregates diminishes parallel to the stretch axis. Concerning the structural origin of piezoresistive anisotropy, we have not found any work in literature. In light of the isotropic stress response, we assume that the geometrical contribution to R(ϵ) is irrelevant for piezoresistive anisotropy, and that the latter is solely based on anisotropic changes in intrinsic resistivity. We point out that a net increase of interparticle distance in the stretching direction is not necessarily the governing mechanism at small to moderate strains: The strain-induced translation and rotation of filler particles generally leads to both the breakdown and formation of conductive pathways, and either of the two may dominate during stretching. For example, Flandin et al. [\[2\]](#page-32-7) report that for a high structure CB-silicone composite, resistance decreases for up to 5 % strain and attribute this to the net creation of new conduction paths or improvement of existing ones. In addition to increasing interparticle distances, particle alignment along the stretch axis is reported to become significant at moderate strains for various conductive CB elastomers (e.g. for strains > 30 % [\[56\]](#page-35-14), > 30 % [\[4\]](#page-32-5), > 25 % [\[8\]](#page-32-8)). We will discuss this mechanism in Section [III.3](#page-17-0) for our films. To discriminate reversible and irreversible effects, Fig. [6b](#page-13-0)-e displays the relaxed resistance increase at the probed strain plateaus (derived from R(t) as explained in Appendix [B\)](#page-39-0) relative to resistance of unstretched samples, R0, along with its irreversible (resistance change after unloading from a given strain plateau and relaxing) and reversible (overall minus irreversible increase) parts. As seen in Fig. [6e](#page-13-0), the irreversible part increases quasi-linearly with strain. It reaches values of ≥ 1/10 R<sup>0</sup> at the smallest strain plateau (ϵmax = 10 %), and up to 1/2 to 100 % of R<sup>0</sup> at the biggest (ϵmax = 40 %). We mainly attribute the irreversible resistance increase to intrinsic resistivity changes (i.e., permanent modifications of the CB network) rather than geometric ones (i.e., plastic deformation at the macroscale). As explained in Appendix [B,](#page-39-0) plastic strains amount to 4 %, which would account for a resistance increase by the same, small percentage (via the sample length L, see eq. [\(4\)](#page-5-1) in Section [II.2.2\)](#page-4-0). A significant permanent decrease in the cross-sectional area is also excluded as this would seriously raise the nominal stress values (see Section [III.2.1,](#page-11-1) Fig. [5\)](#page-12-1). As a cause for permanent resistivity increase, literature frequently considers the damage of filler-matrix interphases (also termed 'debonding' or 'slippage' at the filler-matrix interface) [\[8,](#page-32-8) [56,](#page-35-14) [57\]](#page-35-15). Mechanical failure at the interphases is plausible here since the stress response is also indicative of it (strain softening for strains ≥ 10 %, see Section [III.2.1\)](#page-11-1); however, the exact role of filler-matrix delamination for electrical conductivity remains obscure. For example, Yamaguchi et al. [\[56\]](#page-35-14) argue that the effect of filler-matrix delamination on the spatial arrangement of filler particles should be rather small, whereas Knite et al. [\[57\]](#page-35-15) hypothesize the irreversible formation of isolated filler clusters. Irrespective of the exact mechanism, we conclude that straining provokes microstructural changes in the composite that result in a different set of conductive pathways of the CB network after unloading. Interestingly, the irreversible increase of resistance along the stretch axis is isotropic (Fig. [6b](#page-13-0)-e), as opposed to electrical resistance in the unstrained state (Section [III.1\)](#page-8-0) and the reversible resistance increase (Fig. [6b](#page-13-0)-d). In conclusion, only the reversible part of the resistance response to strain caused piezoresistive anisotropy. As a reminder, the stress response is isotropic within an experimental scatter of 3 – 5 % (Section [III.2.1\)](#page-11-1). In contrast, anisotropy in R(ϵ) is clearly significant, even within the more pronounced scatter of ≤ 17 % (e.g. relative difference in resistance increases for stretch parallel vs. perpendicular to the coating direction, ∆R(ϵ∥)/R<sup>0</sup> and ∆R(ϵ⊥)/R<sup>0</sup> at 40 % strain: 243/225/198 % for 7/9/11 vol% CB). Apparently, electrical properties of the composite are much more strongly affected by flowinduced anisotropy than mechanical properties. A basic reason for this is that electrical conductivity necessitates the existence of percolated paths of CB, whereas the reinforcing effect of CB does not. Concomitantly, small changes of the CB network can account for significant changes in conductivity (via loss/gain of conductive pathways) without (or barely) affecting mechanical reinforcement – a phenomenology that has also been reported in [\[56\]](#page-35-14). We now consider the effect of CB concentration on piezoresistive sensitivity and its anisotropy, as mirrored in Fig. [6b](#page-13-0)-f. The reversible (and irreversible) part of ∆R(ϵ) gets stronger upon lowering CB concentration from 11 vol% to 7 vol%, i.e., piezoresistive sensitivity increases upon approaching the percolation threshold (∼5 vol% [\[28\]](#page-34-4)). The trend is commonly noted for deformation-induced conductivity changes in CB composites [\[3,](#page-32-4) [4,](#page-32-5) [8,](#page-32-8) [11\]](#page-33-9). As a general explanation in terms of percolation, conductivity is most sensitive to microstructural changes when the filler concentration barely suffices to form a few conductive paths, and the loss of these paths cannot be compensated by a recombination with particles from the vicinity. This picture is supported by our data since the resistance increase as a function of strain transitions is convex close to the percolation threshold (7 vol% CB): A convex dependence is typical of only a few single conductive paths which can be modeled as conductive elements connected in series. In contrast, the dependence is concave for higher CB concentrations (9/11 vol% CB), characteristic of particles in a denser network which are increasingly connected in parallel. Concerning the anisotropy of piezoresistive sensitivity, it is also strongest when closest to the percolation threshold (compare relative differences between parallel and perpendicular stretch for the three CB contents in Fig. [6f](#page-13-0)). This is analogous to observations on anisotropy in carbon-filled polymers, e.g., for flow-induced electrical anisotropy of CB thermoplasts [\[6,](#page-32-2) [36\]](#page-34-9), electrical anisotropy of magnetically aligned graphite fibers in an epoxy [\[58\]](#page-36-0), and anisotropic X-ray scattering of the CB network in ethylene propylene rubber [\[37\]](#page-34-11). The trend of electrical anisotropy being maximal just above percolation is also seen for a simple model for anisometric conductive particles aligned in a dielectric continuum [\[58\]](#page-36-0), i.e., it can be derived from geometrical considerations in terms of preferential filler orientation. It is open to question whether this simple structural idea is adequate in the case of our films and other anisotropic CB polymers, already because there are anisotropic CB structures that are not governed by aggregate alignment (see review in Section [III.3\)](#page-17-0). In addition, the impact of filler concentration on hydrodynamics during processing must be taken into account (Section [III.1\)](#page-8-0): Structuration mechanisms leading to anisotropy may become less efficient with increasing CB concentration as the material gets more viscous and resistant to flow. The above phenomenology has important implications for the application of CB elastomers and other composites with conductive fillers. Our results show that electrical anisotropy from shear flow during film fabrication may be insignificant in the undeformed state but become pronounced upon deformation. As an example from our data, initial anisotropy is fairly weak with mean values of R∥/R<sup>⊥</sup> = 1.1 – 1.4, see Section [III.1\)](#page-8-0), whereas at 40 % strain, R∥/R<sup>⊥</sup> is as low as 0.5, i.e., the material is twice as resistant along the stretch axis when stretched perpendicular to the coating direction as when stretched parallel to it. Thus, in cases where (piezo-)electric anisotropy is not desired, one should test for it not only in the undeformed state but also along at least two stretch axes at strains relevant in practice. As a further practical measure, our data indicate that CB concentration should be chosen well above the percolation threshold since anisotropic effects in conductivity and piezoelectric sensitivity are then weaker. One could, however, also think of applications where (piezo-)electric anisotropy is favorable, and tune CB elastomers toward maximal piezoelectric anisotropy. For the interested reader, we therefore report on the anisotropic sensing performance of our CB-silicone films in Appendix [C.](#page-41-0) ### <span id="page-17-0"></span>III.3. Structural analysis Before presenting our results from structural characterization, we compile some general considerations relevant for identifying the mechanisms that lead to flow-induced anisotropic CB structures during doctor blade coating. As mentioned in Section [I,](#page-1-0) there seems to be no systematic work on flow-induced anisotropy in CB-filled polymers, as opposed to CB suspensions in low viscosity organic liquids [\[38](#page-34-12)[–41\]](#page-35-0) and other colloidal systems with aggregating particles (e.g. [\[59–](#page-36-1)[61\]](#page-36-2)). - As a basic requirement for flow-induced anisotropy, shear forces have to be strong enough to fragment the CB network. The latter manifests as shear-thinning flow in rheometry [\[53,](#page-35-11) [54\]](#page-35-12). Since the global shear rates of our coating process fall deep into the shear-thinning regime of a similar CB-silicone mixture (see [\[28\]](#page-34-4) and Section [III.1\)](#page-8-0), we expect doctor blade coating to be highly effective in breaking up the CB network and allowing for translation and rotation of CB agglomerates and primary aggregates. - Work on CB in low viscosity suspending media reports that for high shear rates (10<sup>2</sup> 10<sup>3</sup> s −1 for CB in tetradecane and CB in mineral oil [\[39](#page-35-16)[–41\]](#page-35-0)), primary aggregates are broken, resulting in reduced sizes and manifesting as shear-thickening flow. Even though our global shear rates (50 s −1 to 3000 s −1 ) cover this regime and would equate to even higher shear forces (due to the higher matrix viscosity), we do not expect a significant break-up of primary aggregates in light of the shear-thinning of a similar CB-silicone mixture in the concerned range of shear rates (see preceding point). To verify, we derive an aggregate size distribution from PeakForce QNM maps and compare it with reference data published previously (Section [III.3.2\)](#page-20-0). - A peculiarity of polymeric matrices (vs. low molecular weight matrices) is that they themselves can become anisotropic as a result of shear. As reviewed in Section 3.1, electrical anisotropy can stem from an anisotropic CB dispersion imposed by the packing and orientation of matrix chains during processing [\[35,](#page-34-10) [36\]](#page-34-9). The publications refer to semicrystalline thermoplasts, i.e., polymers where preferential chain orientation seems more likely than for our silicone matrix which is chemically crosslinked (steric hindrance to chain orientation) as well as amorphous during processing (coating at room temperature, cure at 100 °C) and characterization at room temperature [\[62\]](#page-36-3) (no tendency for alignment via crystallization). Yet, since we cannot generally exclude anisotropic effects, structural analysis must check for matrix anisotropy. This is done via SAXS (Section [III.3.3\)](#page-21-1). - Despite the complexity and diversity of flow-induced CB morphologies, one anisotropic version is reported both for CB suspensions [34-37] and short-range attraction colloidal suspensions of other particles (e.g. [\[59,](#page-36-1) [60\]](#page-36-4)): Under certain shear flow conditions (e.g. shear rate, filler concentration and size of the shear gap), vorticity-aligned structures (rods or sheets) form parallel to the compressional and neutral axes of shear flow. As the underlying mechanism, literature suggests the break-up of agglomerates and subsequent separation of aggregates/agglomerates along the extensional axis of shear flow, in combination with the densification/interpenetration of CB aggregates along the compressional axis [\[39,](#page-35-16) [41,](#page-35-0) [59,](#page-36-1) [60,](#page-36-4) [63\]](#page-36-5). In light of their generic appearance in the referenced systems, we shall consider that CB aggregates may become separated along the axis of maximal extension of shear flow / parallel to the coating direction. Concerning the undeformed state, this would readily explain why parallel resistance is higher than perpendicular resistance. # <span id="page-18-0"></span>III.3.1. Nanomechanical mapping (PeakForce QNM) The dispersion state of near-surface CB in unstretched samples is illustrated by the 20x20 µm<sup>2</sup> dissipation maps in Fig. [7a](#page-19-0)-c (for explanation and qualitative discussion of the Peak Force QNM signal maps, see Appendix [D\)](#page-44-0). The area density of near-surface CB markedly grows with CB concentration, and CB is finely dispersed in the matrix for all compositions (aggregate and agglomerate sizes of some 100 nm, see Fig. [7d](#page-19-0)-e as well as Section [III.3.2](#page-20-0) for segmental analysis). Upon assuming that both observations hold true also for the bulk, we have proven the structural basis for the mechanical stiffening by CB (Section [III.2.1\)](#page-11-1): The rather homogeneous dispersion of 100 nm-sized aggregates provides a sufficient portion of reinforcing CB-matrix interphases, and the latter become more prominent in the composite with increasing CB content. Note that the maps (Fig. [7,](#page-19-0) plus more examples in Fig. [3,](#page-9-0) Section [II.2.4](#page-7-1) as well as Fig. [D.2](#page-46-0) and Fig. [D.3,](#page-48-0) Appendix [D\)](#page-44-0) show hardly any agglomerates, and that they appear to be no bigger than 1 µm. In contrast to primary aggregates, agglomerates may not be fully captured as their size may exceed the information depth (some 10<sup>1</sup> nm to 10<sup>2</sup> nm). As a result, PeakForce QNM probably underestimates agglomerate size. Concerning anisotropy and strain-induced changes, the 20x20 µm<sup>2</sup> scans do not provide sufficient resolution to discern possible preferential orientation of CB aggregates and agglomerates. Instead, we turn to the more highly resolved 5x5 µm<sup>2</sup> maps in Fig. [7d](#page-19-0)-f. Shown are different scan areas for both unstrained and strained states of the 7 vol% CB film, i.e., the composition with the strongest piezoelectric effects (see Section [III.2.2\)](#page-12-0). The signal maps indicate that a significant portion of CB aggregates (visible as coherent dark spots) and agglomerates (clusters of aggregates separated by thin yellowish lines in the maps) is anisometric, i.e., the basic condition for their preferential orientation is met. Yet, the exemplary maps for the unstrained state (Fig. [7d](#page-19-0)) and perpendicular strain (Fig. [7f](#page-19-0)) do not readily evidence preferential alignment. Some smaller aggregates appear oriented horizontally, but this effect is not reliable as we cannot exclude scanning bias from the height control of the SFM cantilever. (It is for this reason that we have chosen a scanning angle of 45°, i.e., the horizontal bias will have equal effects in the two stretching directions.) The maps for the state strained parallel to the coating direction give an ambivalent picture with regard to particle alignment: In some regions, aggregates appear preferentially oriented along the axis of coating/stretching (left part of Fig. [7e](#page-19-0)), while they tend to be aligned perpendicular to the axis in <span id="page-19-0"></span>![Figure 7: PeakForce QNM maps of the CB-silicone films reveal that the near-surface CB morphology is uniform for all examined CB contents (7/9/11 vol% CB, unstretched, 20x20 µm<sup>2</sup> scans, panels a-c). The impact of uniaxial strain on CB particle distance and orientation is not readily discernible via qualitative inspection of the data, as seen by the deformation and dissipation maps of 5x5 µm<sup>2</sup> measuring spots of 7 vol% CB samples in the d) unstrained state vs. e-f) strained, relaxed states at ϵ<sup>∥</sup> = 40 % and ϵ<sup>⊥</sup> = 40 %, respectively. In each of the maps, the coating direction goes from bottom left to top right.](path) In conclusion, visual inspection of PeakForce QNM data does not allow statements on changes in CB morphology induced by doctor blade coating or stretching. We therefore opt for a more precise and comprehensive means of data interpretation, i.e., segmentation of a bigger overall data set for the three material states. Corresponding results are presented in the next section. We point out that concerning strain-induced damage, the lateral scan resolution does not suffice to detect voids smaller than 10<sup>1</sup> – 10<sup>2</sup> nm. We can, however, exclude the formation of bigger, microscopic pores, since the maps do not indicate any qualitative differences in material contrast between unstretched and stretched states (see Fig. [7](#page-19-0) as well as Fig. [3](#page-9-0) in Section [II.2.4\)](#page-7-1). The only hint of defects are a few brighter spots of some 10<sup>1</sup> nm in the matrix-dominated regions of the deformation signal, which could reflect pores in the silicone. However, they are also visible in the unstrained state, and the evidence is too weak overall for an unambiguous assignment. ## <span id="page-20-0"></span>III.3.2. Segmentation and statistical analysis of nanomechanical data The CB particle distribution was assessed after segmentation of the 10 x 10 µm<sup>2</sup> dissipation maps. Three maps were evaluated for each material state (unstrained, strained to 40 % parallel and perpendicular to the coating direction). The resulting histograms and average values of the particle diameter, circularity as well as the angle of the major feret axis to the horizontal direction are shown in Fig. [8.](#page-21-0) The CB particle diameter roughly follows a log-normal distribution with values in the 10<sup>2</sup> nmrange, confirming the size range derived from visual inspection of the PF-QNM maps in the previous section. In addition, the distribution agrees very well with aggregate size measurements for CB suspended in toluene [\[25\]](#page-34-1), indicating that (as expected) primary aggregates are not modified / broken down by shear during the coating process. (Note that the histograms provide no means to differentiate between primary aggregates and agglomerates; the same was found for USAXS measurements on bulk composites (3/5/7/9 vol% CB) where scattering by both species strongly overlapped [\[25\]](#page-34-1).) The circularity parameter is significantly smaller than 1 (means: 0.7 - 0.8), confirming the visual particle assessment in the previous section, i.e., CB aggregates are anisometric and can, as a result, exhibit preferential orientation. The histogram of the feret angle for the unstrained film does indeed show a trend to two maxima at roughly 20° and 170° relative to the horizontal (scanning) direction. This can be an indication of a slight preferential orientation; however, it is superimposed by a SFM scan artifact (smearing of particles in the horizontal/ 0°-direction) which prevents a definite conlusion. Since the coating direction is at 45° in the scans, preferential particle alignment from the coating process or straining would rather be expected at the 45° or 135° direction. With straining, the maxima of the histogram appear to shift from the extremes at approx. 20° and 170° towards the coating direction and perpendicular to it. This indicates a slight alignment of the CB particles due to the straining, but just like for the unstrained state, the evidence is too weak for an unambiguous interpretation. <span id="page-21-0"></span>![Figure 8: Assessment of CB particle diameter, circularity and angle of the major feret axis to the horizontal axis of the PeakForce QNM scans by histograms obtained from segmentation analysis of 10x10 µm<sup>2</sup> dissipation maps. Particles are anisometric (circularity < 1), and their size roughly follows a log-normal distribution in the range of about 40 nm (primary diameter) up to about 1 µm. The feret angle reflects weak evidence of two preferential orientations which tend to align with the coating direction (∥-sign in the histograms) and perpendicular to the coating direction (⊥-sign in the histograms) upon straining.](path) ## <span id="page-21-1"></span>III.3.3. SAXS Small-angle X-ray scattering (SAXS) measurements were performed to study the bulk structure of unstretched and stretched silicone films, both neat and filled with CB at a concentration of 9 vol%. Fig. [9a](#page-24-0) presents 1D scattering patterns of the neat silicone film in its unstretched state, obtained by azimuthally averaging the 2D scattering patterns in a narrow angular range, both in the coating direction and perpendicular to it (see experimental section for more information). Both curves feature a shoulder at ∼0.04 ˚A−<sup>1</sup> , assigned to scattering of the silicone network, and weak forward scattering due to large-scale structures. The latter may be assigned to frozen-in elastic forces inside the gel, leading to large-scale composition fluctuations [\[64,](#page-36-6) [65\]](#page-36-7). To quantify the involved length scales, the scattering curves were modeled using a contribution accounting for large-scale structures, ILS(q), and a contribution accounting for composition fluctuations, Ifluct(q), following the equation with K<sup>P</sup> the Porod amplitude and m the Porod exponent. The Ornstein-Zernike structure factor accounts for scattering by composition fluctuations on the level of single chains. It is given by ξ, which is proportional to the mesh size of the silicone network and thus a measure of crosslink density, equals 2.43 ± 0.03 nm parallel to the coating direction and 2.22 ± 0.03 nm perpendicular to the coating direction. Thus, the coating process introduced a notable structural anisotropy at the nanoscale of the silicone network, presumably via increased flow of chain segments along the extensional axis of shear (see Fig. [1c](#page-3-1)). In addition, the scattering at large-scale structures (q < 10−<sup>2</sup> ) is significantly stronger perpendicular to the coating direction, implying that they are larger in size and/or number. Data for yet lower q-values would be necessary to draw conclusions on possible anisotropy at these larger length scales of some 10<sup>2</sup> nm. Strain significantly changed the nanoscale structure of the silicone network, as is shown in Fig. [9\)](#page-24-0)b for 40 % uniaxial strain applied parallel to the coating direction and Fig. [9\)](#page-24-0)c for 40 % uniaxial strain applied perpendicular to the coating direction. In the direction of strain, the mesh size is significantly larger than perpendicular to the direction of strain (see ξ-values in Fig. [9b](#page-24-0)-c), reflecting transverse contraction. To quantify the corresponding compressibility, the Poisson ratio for finite strains according to Hencky, ν = −ln(λtrans)/ln(λaxial), was applied to the ξ-values from the SAXS fits. The extensional ratios in the transverse and axial directions, λtrans and λaxial, are thus each derived as the quotient of the ξ-values for the strained vs. unstrained state. The resulting values for stretch parallel vs. perpendicular to the coating direction, ν<sup>∥</sup> = 0.45 ± 0.09 and ν<sup>⊥</sup> = 0.44 ± 0.11, are both equal to 0.5 within the relative uncertainty of 20 - 25 %, indicating nearly to fully incompressible behavior (ν = 0.5), typical for rubbers. In light of the slightly anisotropic mesh size reported above (greater in the parallel direction than in the perpendicular direction by 9 %), this either means that small directional differences in crosslink density do not lead to an anisotropic compressibility, or that their effect on compressibility is too weak to resolve here. By all means, the nanoscopically derived, isotropic Poisson ratios agree well with the macroscopic, also isotropic stress response reported in Section [III.2.1.](#page-11-1) The scattering patterns of the silicone films loaded with CB at a concentration of 9 vol% (Fig. [9\)](#page-24-0)d) show a steep slope in the entire measured q range, indicating that structures (primary CB particles and their aggregates with a very broad size distribution) are present at all length scales covered in the measurement. The scattering is dominated by CB and the contrast of the silicone network is too weak to resolve. No difference in the scattering patterns evaluated parallel and perpendicular to the coating direction is observed. Therefore, coating does not introduce structural anisotropy on length scales between ∼1 and ∼150 nm, i.e., to CB primary particles (about 40 nm [\[25\]](#page-34-1)) and small aggregates. Also stretching the 9 vol%-composite, both parallel (Fig. [9\)](#page-24-0)e) and perpendicular (Fig. [9\)](#page-24-0)f) to the coating direction, did not lead to any structural anisotropy on length scales between ∼1 and ∼150 nm. In both cases, the scattering curves evaluated parallel and perpendicular to the direction of stretching did not deviate from each other. ### <span id="page-23-0"></span>III.4. Simulations In addition to the direct characterization methods introduced above, we performed simulations to assist in identifying the key model features that reproduce the experimental phenomenology. Our simulations are not meant to mimic the experiment as accurately as possible, but to provide evidence regarding the primary mechanism with a model as simple as possible. Two questions are guiding our analysis: To tackle the first question, we adopt the same baseline model as introduced in Ref. [\[25\]](#page-34-1), i.e., we consider stiff CB aggregates as the fundamental building blocks of our simulation. These aggregates consist of primary spheres of diameter σ that are rigidly fused together in the course of a diffusion limited aggregation (DLA) process. As a consequence, aggregates are Brownian trees exhibiting a power-law decaying density profile which corresponds to a fractal dimension of d<sup>f</sup> ≈ 2.5, similar to experimentally observed fractal dimensions of certain CB varieties (cf. [\[66\]](#page-36-8)). The tenuous agglomerate structures (secondary aggregates) are expected to get dispersed in the coating process and may or may not reform in a cured composite. However, as the samples are cured right away, we expect the resulting configuration of CB to resemble a steady state configuration <span id="page-24-0"></span>![Figure 9: 1D SAXS patterns of unloaded silicone films (a,b,c) and silicone films loaded with 9 vol% CB (d,e,f) (hgap = 350 µm, vblade = 20 mm/s) in their unstretched state (a,d), stretched at 40 % uniaxial strain parallel to the coating direction (b,e) and perpendicular to the coating direction (c,f). The patterns are evaluated in the directions as indicated in the graphs, detailed in the experimental section. The black lines are fits according to eq. [\(5\)](#page-22-0).](path) of aggregates in shear flow. For simplicity, we assume that the primary function of the polymer matrix is mediating the momentum transfer between individual carbon black aggregates. Thus, we neglect the microscopic structure of the polymer in favor of including hydrodynamic interactions between carbon black aggregates. with r (i) <sup>m</sup> denoting the mth component of the position of the i th primary particle and the center of mass position r CM, can be diagonalized yielding, in analogy to the tensor of inertia, the principal axes of the aggregate. The corresponding principal moments λ 2 i (eigenvalues of Snm), λ<sup>1</sup> ≥ λ<sup>2</sup> ≥ λ3, quantify the length of the principal axis of an ellipsoid with the same properties. This can be used The aspect ratio provides a simple quantity to estimate how strongly the distribution of primary particles deviates from spherical symmetry. As more primary particles are added to an aggregate the influence of fluctuations in the growth process diminishes in comparison to the aggregate size and the average aspect ratio decreases with N. For our proof-of-principle simulations, we choose a uniform N = 20 for which aggregates have an average aspect ratio of roughly 3. In reality the size of CB aggregates will be broadly distributed with a larger average aggregate size. However, smaller aggregates are easier to simulate and we expect any effects linked to the microscopic anisotropy of aggregates to be relatively more pronounced for smaller aggregates. Each aggregate has three distinguished axes that we label "long", "middle", and "short" axis, respectively. Likewise, the shear flow admits three distinguished directions, i.e., flow, gradient, and vorticity direction, respectively, just like in the experimental setup illustrated in Fig. [1.](#page-3-1) In order to quantify whether the aggregate axes adopt a preferential orientation under shear, we can define the nematic order parameter with θ denoting the angle between an axis of an aggregate and a direction of the shear flow. As a more nuanced indicator of anisotropy, we also depict the relative frequencies of the orientation of each axis as a histogram using a Mollweide projection. # III.4.1. Aggregate alignment We simulate ensembles of aggregates in shear flow with Lees-Edwards boundary conditions using molecular dynamics (MD) with Multi-particle collision (MPC) dynamics [\[67–](#page-36-9)[69\]](#page-36-10). MPC dynamics is a particle-based Navier Stokes solver, providing hydrodynamic interactions and thermal fluctuations using a mesoscale solvent that can be coupled to standard MD simulations. Details on this technique and the parameters we use for our simulations can be found in Appendix [E.](#page-50-0) As MPC simulations require substantial computational effort, these simulations are performed with only 125 CB aggregates, each consisting of 20 primary particles. To quantify the influence of the shear flow, we introduce the P´eclet number as Pe = ˙γR<sup>2</sup> <sup>g</sup>/D, with the shear rate ˙γ, the radius of gyration R<sup>g</sup> and the center of mass diffusion coefficient D of the aggregates. D can be measured via the aggregate mean squared displacement in equilibrium simulations. To assess whether CB aggregates align under shear, we conduct MPC-MD simulations for a broad range of P´eclet numbers The alignment is measured by calculating the nematic order parameters with respect to the flow, velocity gradient, and vorticity axes, respectively, for increasing P´eclet numbers. The results are illustrated in [Figure 10.](#page-26-0) ratio For Pe = 2.3, where shear and diffusion have similar influence, and at lower shear rates, there is no discernible signature <span id="page-26-0"></span>![Figure 10: Nematic order parameters of the long and short axes of aggregates with respect to the flow, gradient and vorticity axes for increasing P´eclet number. Error bars encode the standard deviation.](path) of alignment. However, starting from Pe = 5.8, the particles tend to align their long axis in the direction of flow. The negative order parameter in the gradient direction signifies that the long axes of the aggregates are preferentially orthogonal to the gradient. In combination with the vanishing nematic order parameter in vorticity direction, the overall picture indicates that aggregates tend to orient themselves in the vorticity-flow plane. This is supported by the nematic order parameters of the short axis, which frequently aligns parallel to the gradient and perpendicular to the flow direction. To improve the visualization of this alignment, we use Mollweide projections of the aggregate axes in spherical coordinates. The polar angle θ corresponds to the gradient direction and is chosen such that the poles are represented by θ = ±90◦ . The azimuthal angle ϕ is measured with respect to the flow-gradient plane and thus encodes the vorticity direction at (ϕ, θ) = (±90◦ , 0 ◦ ). The size of bins depends on the polar angle, especially noticeable near the poles of the Mollweide projection. To correct for this effect, we reweight the bin occupancies with 1/ cos(θ), the radius of circles of latitude. Figure [11](#page-27-0) displays the distributions of the aggregate axes. It corroborates that the long axes of aggregates preferentially point in the flow direction and lie in the flow-vorticity plane. The short axis is parallel to the gradient direction, i.e., at the poles of the projection. The behavior of the middle axis is less clear, yet a maximum can be identified that is related to a slight alignment with the vorticity axis. The features of the distributions become more pronounced with increasing shear rate. There is also evidence of a symmetry-breaking effect affecting the polar angle. For example, the maxima of the distribution of the long axis are shifted away from the θ = 0◦ circle of latitude. This indicates long axes at a slight angle with respect to the flow-vorticity plane towards the gradient direction, as known experimentally [\[70\]](#page-36-11) and theoretically [\[71\]](#page-36-12) for rods, an effect that is reduced for stronger flows. Thus, we can conclude that aggregates show alignment in shear flow, exhibiting behavior similar to the phenomenology previously observed for aggregates fracturing in shear flow [\[72\]](#page-36-13) as well as for rod-like particles [\[73](#page-36-14)[–76\]](#page-37-0). Yet, the alignment effect, even in strong flow, is rather weak with <span id="page-27-0"></span>![Figure 11: Mollweide projections of the distributions of the orientation of aggregate axes for weak (Pe = 5.8, top) and strong flow (Pe = 93.4, bottom), respectively.](path) ## III.4.2. Anisotropic conductivity A macroscopically conductive CB composite relies on a percolating network of CB aggregates. We can generate these networks from our simulations by assigning a threshold distance d. If the surface separation between any two primary particles belonging to different aggregates undershoots d, we consider the respective aggregates connected. As we want to calculate the conductivity of the aggregate network, the length d effectively represents a cutoff distance beyond which tunneling transport may be neglected. With this, we can interpret a system configuration as a random resistor network with aggregates as nodes and tunneling junctions as edges. Introducing two electrodes connected to two opposite sides of the simulation box, we compute the resistivity of the composite by solving the Kirchhoff's equations. The Kirchhoff network conductivity provides a crude indicator for the conductivity of the system as we assume that the intrinsic resistance of any aggregate is small compared to the resistance of tunneling contacts and any influence of surface chemistry or the polymer matrix is neglected. Nevertheless, the Kirchhoff conductivity provides an indication of how the geometric arrangement of carbon black within the polymer matrix impacts the properties of the composite. Choosing the location of the electrodes, we can measure the conductivity of the network along different directions and compute the ratio Rx/R<sup>y</sup> of eq. [\(3\)](#page-4-1). Thus, we can test whether the structural changes due to shear during coating induce anisotropy to the conductivity of the network. fortably simulate are small. As a consequence, the conductivity measurements in the resulting networks are subject to strong fluctuations, obstructing a confident assessment of the electrical anisotropy. Therefore, we perform computationally cheaper Monte Carlo simulations without hydrodynamic interactions but instead with an external alignment potential with the orientation of the long axis of an aggregate denoted as ω and the field strength A. These simplified simulations do not accurately capture the CB distribution in shear flow, in particular, because we only adapt the orientation of the long axis. However, they still grant insight into the relationship between microscopic alignment of aggregates and conductivity. As we have access to much larger system sizes and a higher number of independent snapshots to work with, our observations bear higher statistical significance. We consider an ensemble of 1000 DLA aggregates comprising 20 primary particles each. Interaction between different aggregates is hard, i.e., no overlap is permitted between any two aggregates. As there is no simple way to set up aggregates without overlap, we equilibrate by initially allowing overlap at the cost of an energy penalty depending on the depth of interpenetration. We proceed by slowly lowering the temperature while also scaling the alignment potential so that it remains unaltered relative to the thermal energy. Once all overlaps are removed, the annealing potential is removed, followed by a second equilibration at unit temperature. As aggregates are rigid, they are characterized by three positional and three orientational degrees of freedom which are sampled through random translations and orientations of the entire aggregate. The system evolves in periodic boundary conditions but no conductive transport is allowed through the two walls chosen as electrodes. For each alignment field strength A and each density, we generate 100 independent configurations and analyse the corresponding Kirchhoff networks for resistivity parallel and perpendicular to the axis promoted by the alignment field. A more detailed description of the conductivity calculation and the chosen parameters can be found in appendix [E.](#page-50-0) [Figure 12](#page-29-1) illustrates the results. For all field strengths, the average electric anisotropy ratio is smaller than one, indicating that conductivity is robustly enhanced in the direction of preferential alignment. The effect becomes more pronounced as the field strength is increased, inducing stronger alignment as measured by the nematic order parameter S<sup>z</sup> of the long axis relative to the alignment direction. An amplitude of A = 2 kBT roughly reproduces the S<sup>z</sup> order parameter of the MPC-MD simulations for large P´eclet numbers. The absolute resistivities are subject to strong fluctuations and so is the ratio R∥/R⊥, with individual networks exhibiting ratios larger than one. These fluctuations diminish with A as well as the density. As 10 vol-% is just above the percolation threshold without external alignment, the networks, particularly for weak alignment, have little redundancies and thus heavily rely on specific links that break and reform inducing fluctuations. These fluctuations primarily originate from the finite size of the simulation. At larger densities, networks have a lot more connections, heavily reducing the impact of microscopic fluctuations. <span id="page-29-1"></span>![Figure 12: Electrical anisotropy ratio induced by an external alignment field with amplitude A for different filler densities. Points mark the mean ratio averaged over 100 networks and error bars show the corresponding standard variation. The nematic order parameter S<sup>z</sup> is a monotonic function of A (unaltered across the three densities) and is depicted as the non-linear top axis.](path) The trend observed in simulations is again similar to previous studies on rod-like particles. However, it does not agree with our experimental observations that report an electric anisotropy ratio larger than 1 for unstrained films. A more scrupulous glance at individual networks allows us to attribute the drop in the anisotropy ratio to the number and average length of paths linking the electrodes. Especially, at high field strengths the number of independent paths linking the parallel plates in alignment direction is substantially larger than for the perpendicular direction. Likewise, the average length of these independent paths is drastically reduced in alignment direction. A simple Fermi calculation based on the aforementioned metrics can adequately reproduce the Kirchhoff results. Thus, the network nuances like edge weights are largely inconsequential – the network structure is sufficient to predict the trend. Thus, whatever mechanism causes the experimental ratio to behave fundamentally differently, it exceeds the implications of a simple uniaxial aggregate alignment. ## <span id="page-29-0"></span>III.5. Structural model for the observed (piezo-)electric anisotropy In a first step, shear flow during doctor blade coating of the yet liquid CB-silicone mixtures fragments the CB network into (mobile) agglomerates and primary aggregates. As supportive evidence of this basic requirement for structural rearrangements, we know liquid precursors to be strongly shear-thinning in the concerned range of shear rates (see Section [III.1\)](#page-8-0). In addition, we have shown that in cured films, CB is dispersed uniformly within the silicone matrix (Section [III.3.1\)](#page-18-0), i.e., flow of matrix chain segments and of CB particles during coating is linked. a) CB aggregates are increasingly separated along the extensional axis of shear flow and, concomitantly, in the coating direction. This is supported by SAXS data for the silicone matrix which indicates more pronounced flow of chain segments parallel to the coating direction (mesh size in the coating direction is larger than perpendicular to it, Section [III.3.3\)](#page-21-1). Analogous evidence for CB aggregates is lacking so far due to the size limitation of the SAXS measurements, but CB aggregates should experience extensional flow similar to the silicone matrix due to their mechanical coupling. For this mechanism, electrical conductivity is expected to be lower in the coating direction than perpendicular to it, due to larger overall tunneling distances (R∥/R<sup>⊥</sup> > 1). b) Due to their anisometric nature (see Sections [III.3.1](#page-18-0) and [III.3.2,](#page-20-0) CB aggregates are preferentially aligned in the coating direction. This picture is only hinted at for near-surface CB aggregates and agglomerates (segmental analysis of nanomechanical data, Section [III.3.2\)](#page-20-0) but fully supported by simulations of CB aggregate alignment (Section [III.4\)](#page-23-0). The latter show that in the unstrained state, such alignment correlates with increased conductivity in the coating direction (R∥/R<sup>⊥</sup> < 1). In addition, resistance should increase more strongly for strain perpendicular to the coating direction since the conductivity loss is more critical when preferentially aligned aggregates are pulled apart perpendicular to their major axis vs. parallel to it. In the unstrained state, mechanism a) dominates such that electrical conductivity in the coating direction is lower than perpendicular to it, explaining the experimental anisotropy ratio of R∥/R<sup>⊥</sup> > 1. For strain-induced resistance changes, mechanism b) is the governing mechanism, accounting for the observed anisotropy in piezoresistive sensitivity, dR⊥(ϵ⊥)/dϵ<sup>⊥</sup> > dR∥(ϵ∥)/dϵ∥. Concerning the dependence on shear rate, we assume structuration effects from both mechanisms to be close to maximal in our experiments since R∥/R<sup>⊥</sup> > 1 in unstrained films did not notably depend on shear rate. Simulations support this picture of structuration effects becoming maximal at some point, as aggregate alignment and its resulting R∥/R<sup>⊥</sup> saturate for large Peclet numbers. #### IV. SUMMARY, CONCLUSIONS AND OUTLOOK To our knowledge, this paper for the first time reports on flow-induced piezoresistive anisotropy of CB elastomers relevant for sensing applications, as well as underlying structure-propertyrelationships. The results help understand the impact of processing and CB concentration on final properties and thus provide a means to tune CB elastomers accordingly. Doctor blade coating liquid CB-silicone mixtures with CB concentrations above the percolation threshold (7/9/11 vol%) led to significant electrical anisotropy in cured films, with conductivity and piezoresistive sensitivity being higher perpendicular to the coating direction than parallel to it. Anisotropic electrical conductivity in unstrained films was seen for all examined compositions, gap heights and shear rates, stressing the practical relevance of flow-induced anisotropy for conductive CB-filled elastomeric films used in sensing applications. Even if electrical anisotropy appears weak in undeformed samples (R∥/R<sup>⊥</sup> = 1.1 - 1.4 in our experiments, in tune with only weak structuration effects seen in simulations), it can strongly manifest upon stretching as a result of piezoresistive anisotropy (e.g. double as high for stretch parallel to the coating direction than perpendicular to it). Thus, as a practical conclusion, process-induced (piezo-)electric anisotropy must be checked for, and processing and/or CB elastomer composition may need to be adjusted to minimize it. Concerning structure-property-relationships, our results confirm trends for the impact of CB concentration on piezoresistivity and anisotropy found in literature on CB elastomers, and they can be motivated by fundamental considerations based on electrical percolation. In particular, the strain-induced resistance increase and its anisotropy (stretch parallel vs. perpendicular to the coating direction) both increased upon approaching the percolation threshold, consistent with the CB network becoming more sensitive to structural changes (less resilient) when only few conductive pathways are present. In contrast to the electrical response to strain, the mechanical stress response (along with matrix compressibility) was isotropic. We attribute this to the fact that electrical conductivity in CB-silicone films necessitates percolation, while the mechanical reinforcement of CB does not. In conclusion, electrical and mechanical effects of CB anisotropy in CB-filled elastomers can be very different in magnitude, and piezoelectric anisotropy can be minimized in practice by choosing CB concentrations further away (above) the percolation threshold. As a structural explanation of the observed (piezo-)electric anisotropy, we have derived the following hypothesis with the help of structural analysis of the silicone matrix and CB at the aggregate level (SAXS on bulk samples, nanomechanical mapping and segmental analysis for near-surface CB particles) as well as simulations on CB aggregate alignment: Shear flow during coating fragments the CB network and then has a two-fold effect, i.e., it induces a) preferential aggregate alignment, as well as b) increased interparticle distances, parallel to the coating direction. When unstrained, mechanism b) dominates such that R∥/R<sup>⊥</sup> > 1. In contrast mechanism a) dominates upon straining such that the resistance increase is stronger for stretch perpendicular to the coating direction (and the direction of preferential aggregate alignment, respectively). In conclusion, the proposed hypothesis is fully consistent with the observed phenomenology and with trends seen in simulations. Among others, it shows that the common picture on the impact of shear flow on anisometric particles, i.e., preferential alignment in the flow direction, is too simplistic. Since evidence of anisotropy up to the aggregate level is only weak overall, future work should tackle the larger length scales up to the CB network level. For example, our hypothesis could be substantiated by structural analysis of the yet liquid, in situ-sheared material via rheo-USAXS (fresh CB-silicone mixtures in stationary shear flow at gaps of 10<sup>2</sup> µm) as well as cured films via USAXS and FIB-SEM tomography (combined with network reconstruction and statistical analysis). #### V. ACKNOWLEDGMENTS Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2193/1 – 390951807. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 404913146, 457534544, 531007218. The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG (bwForCluster NEMO). We thank Marisol Ripoll and Roland Winkler for helpful discussions regarding MPC. 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Grest, "No-slip boundary conditions and forced flow in multiparticle collision dynamics," Physical Review E, vol. 86, no. 6, p. 066703, 2012. # <span id="page-38-0"></span>Appendix A: Film fabrication (detailed description) CB by ThermoFisher Scientific (Carbon black, acetylene, 100 % compressed, 99.9+%, Thermo Scientific Chemicals) was dispersed in a silicone resin (base component of Sylgard® 184 silicone elastomer, Dow® ) via speedmixing (SpeedMixerTM DAC 600.2 VAC-P by Hauschild, 3 min at 2350 rpm under vacuum). The silicone crosslinker (curing agent of Sylgard® 184) was then added with the concentration recommended by the manufacturer (1/10 of the mass of the base component [\[77\]](#page-37-1)). Final dispersion was achieved by subjecting the mixture to the aforementioned speedmixing protocol. Three compositions above the percolation threshold (∼5 vol% CB [\[25\]](#page-34-1)) were realized: 7/9/11 vol% CB. Volume percentages refer to the volume of the resin and were calculated using the densities reported in [22] (ρCB = 1.7 g/cm<sup>3</sup> , ρresin = 0.965 g/cm<sup>3</sup> ). Doctor blade coating was done in ambient air (22 °C, 39 ± 13 % r.h.) immediately after mixing, using a film applicator (automatic film applicator by TQC) with the doctor blade setup (Microm II Metric Film Applicator by Gardco ®) depicted in Fig. [A.1.](#page-38-1) The reactive mixture was placed in front of the doctor blade with a spatula such that mechanical manipulation is minimal and the sample amount suffices to achieve lateral film dimensions of at least 5 x 7 cm<sup>2</sup> . Two gap heights, hgap = 60 µm and 350 µm, were realized by manually adjusting the micrometer screw of the coating apparatus. In order to assess the impact of dynamic effects during processing on final properties, the blade speed, vblade, was varied between 5 mm/s and 400 mm/s. The resulting global shear rates (γglobal= vblade/hgap) span a large interval of 50 s−<sup>1</sup> to 3000 s−<sup>1</sup> . Polypropylene foil was used as a substrate because of its insulating properties (for electrical four-point probe measurements) and easy detachment of films (relevant for the 'thick' films used for tensile tests). Curing took place in a convection oven for 2 h at 100 °C. Films were prepared using the desired process conditions and removed from the film applicator. Subsequently, the doctor blade was cleaned and the next film was prepared. All films were transferred to the oven as soon as the last film was coated. The resulting time between coating and curing ranged between 10 min (last film) and 45 min (first film). The optical appearance of the films changed neither during this waiting The mean thickness of cured films (df ilm, determined by light microscopy of cross-sectional points) was in the range of 80 – 100 µm for the 60 µm gap and of 260 – 350 µm (see next paragraph for respective ranges for 7/9/11 vol% CB ) for the 350 µm gap. The thickness of the 60 µm-films is thus systematically higher than the gap height, while the thickness of the 350 µm-films tends to be lower. The former can be traced back to inaccuracies in the adjustment of the micrometer screws (offset at the zero-position); the latter is likely due to the films thinning out at the blade exit (as indicated in Fig. [A.1b](#page-38-1), typical of shear-driven flow [\[78\]](#page-37-2)) as well as polymerization shrinkage. In addition to the conductive CB-silicone films, a neat silicone film (0 vol% CB) was fabricated to give specimens for mechanical (uniaxial tensile test) and structural (SAXS) characterization of the silicone matrix. Just as in the case of CB-silicone specimens for electromechanical and structural analysis, the blade speed was set to 20 mm/s. Due to the much lower viscosity compared to CBcontaining mixtures, coating did not take place immediately after mixing resin and hardener, but the mixture was allowed to polymerize till viscosity was similar to the 7 vol% CB-mixture (4 h in ambient air + 5 min at 100°C + cooling down for some minutes). Together with a slightly higher gap height of 400 µm (vs. 350 µm), the resulting film was comparable in thickness (260 - 280 µm) to CB-silicone films coated at 350 µm, 20 mm/s (260 - 290 µm / 280 - 320 µm / 320 - 350 µm for 7/9/11 vol% CB). #### <span id="page-39-0"></span>Appendix B: Processing of electromechanical raw data The electrical resistance response to uniaxial strain at room temperature, just like the mechanical stress response, involves pronounced relaxation (see portions at the strain plateaus in Fig. [B.1\)](#page-40-0). For the first strain plateau (10 %), the hold time (20 min) suffices for tensile force and electrical resistance to fully relax (constant values after loading to 10 % as well as after unloading back to 0 %). For higher strains, force and resistance plateaus are not yet reached. To approximate relaxed resistances, the relaxation phases were fit in Origin 2022 using eq. [\(B1\)](#page-39-1). The desired plateau value equals Rrelax. From the force curves in Fig. [B.1](#page-40-0) it can be discerned that the tensile force generally drops below zero after unloading to 0 % strain, and stays negative after stabilization. This effect is related to plastic deformation and/or slip which reduce the pre-load. For optimal comparison of the stress response, the stress-strain curves presented in the following have been shifted to zero force and zero strain at the beginning of each load phase. According to the unshifted curves, plastic deformation and/or slip becomes apparent only for CB-filled samples (not the neat silicone) and only after loading to 30 % strain (not yet for the 10/20 % strain plateaus). As shown in Fig. [B.2](#page-41-1) for the <span id="page-40-0"></span>![Figure B.1: Examples of raw data of uniaxial tensile tests (load-unload cycles to strain plateaus ϵmax = 0/10/20/30/40 %, 10-2 s -1 strain rate, 20 min hold time at each strain level, 22°C ± 1 K, 30 ± 15 % r.h.) with in-situ electrical two-point measurement at room temperature: electrical resistance, R, and tensile force, F, for stretch perpendicular to the coating direction of samples with 7/9/11 vol% CB. Curves of samples stretched parallel to the coating direction bear the same qualitative features.](path) <span id="page-41-1"></span>![Figure B.2: Unshifted stress-strain curves of uniaxial tensile tests (22 °C ± 1 K, 30 ± 15 % r.h.), cycle 4 (load to ϵ = 40 %, relax, unload), for 11 vol% CB and stretch parallel or perpendicular to the coating direction, respectively. The expressions 'left', 'right', 'start' and 'end' refer to the positions on the film from which the tensile specimens were cut (see Fig. [1](#page-3-1) in Section [II.2\)](#page-3-3). The vertical arrow indicates the maximal plastic strain seen for all samples (from specimen cut from left side of the film, stretched parallel to the coating direction).](path) # <span id="page-41-0"></span>Appendix C: Sensing performance In light of the applicability of CB elastomers as highly deformable resistive strain sensors, we assess our films in terms of linearity, sensitivity and durability/cyclability of the piezoresistive response. Their peculiarity would be their piezoresistive anisotropy, a property which could be exploited to create one-piece sensors with (bi-)directional sensitivity. To evaluate linearity and sensitivity, Fig. [C.1](#page-42-0) shows the first derivative of each trend line for the reversible resistance increase in Fig. [6b](#page-13-0)-e, d(∆R/R0)/dϵ, which is equivalent to the gauge factor for infinitesimally small strains (∆R/R0/ϵ). Since the values are not constant for any CB content and stretching direction in any meaningful interval, the films would perform non-linearly and would need to be calibrated accordingly. Piezoresistive sensitivity increases upon approaching percolation, with gauge factors as high as 20 – 25 (parallel) and 60 – 70 (perpendicular) for our lowest CB content of 7 vol%. According to the review in [\[18\]](#page-33-4), these values compete with the majority of gauge factors reported for silicone-based piezoresistive sensors (mainly between 1.2 and 29, > 100 in only a few exceptional cases involving graphene or carbon nanotubes). Thus, the films would give sensors with good sensitivity and minimal expenses for the filler (low-cost commercial CB, vs. CNTs, silver <span id="page-42-0"></span>![Figure C.1: To assess the CB-silicone films as a potential sensor material, piezoresistive sensitivity was quantified by differentating the trend lines of the reversible resistance increase in Fig. [6b](#page-13-0)-f with respect to strain (equivalent to the gauge factor) parallel and perpendicular to the coating direction, respectively (panel a). To eliminate irreversible effects, one 7 vol% CB-sample for each stretch direction was subjected to cyclic conditioning (at 22 °C ± 1 K, 30 ± 15 % r.h., see main text for details) which results in resistance responses that become stationary toward the end of the procedure (panel b).](path) Concerning the cyclability of the piezoresistive response, the irreversible resistance increase is too high in that it leads to significant differences between the overall resistance increase and the reversible part (except for perpendicular stretch of 7 vol% samples, compare corresponding curves in Fig. [6b](#page-13-0)-e, Section [III.2.2\)](#page-12-0). This calls for cyclic mechanical preconditioning, a common method to eliminate irreversible effects (see [\[8,](#page-32-8) [26,](#page-34-2) [33,](#page-34-13) [56\]](#page-35-14) for examples of CB-silicone elastomers). As a first attempt, we conditioned two specimens of the most sensitive composition (7 vol% CB, one sample for each stretching direction) by subjecting them to 100 load-unload cycles between 1 % and 40 % strain (same strain rate as before, 10−<sup>2</sup> s −1 ). The corresponding resistance responses are shown in Fig. [C.1b](#page-42-0). For both stretching orientations, resistance changes most during the first cycle and approaches a stationary regime toward the end of the procedure. Based on this, we could expect irreversible resistance changes to have ceased. To verify, we again conducted load-unload tensile tests to ϵmax = 10/20/30/40 % with hold times of 20 min. The stress response and the relaxed values of the relative resistance change of the conditioned samples are displayed in Fig. [C.2,](#page-43-0) along with the data for the unconditioned ones. <span id="page-43-0"></span>![Figure C.2: Cycling conditioning hardly modified mechanical properties but clearly reduced both irreversible and reversible strain-induced resistance changes (relative to initial value at zero strain, R0) during subsequent tensile tests, as shown by the comparison of conditioned and unconditioned 7 vol% CB-samples with regard to: a) electrical resistance change and anisotropy ratio during load-unload cycles (strain to ϵmax = 10/20/30/40 %, hold at ϵmax for 20 min), b) stress response to strain of cycle 4 (ϵmax = 40 %), c) overall resistance change to uniaxial strain (relaxed value at ϵmax), d) irreversible resistance change (relaxed value after unloading from ϵmax), and e) reversible resistance change (overall change minus irreversible part). All measurements done at 22 °C ± 1 K, 30 ± 15 % r.h.](path) identical up to ϵ ≈ 20 %, and the stress having dropped only slightly (by ≤ 1/10) for higher strains (Fig. [C.2a](#page-43-0)). In contrast, the modification of piezoresistivity is pronounced (Fig. [C.2c](#page-43-0)-e). This again shows that electrical conductivity of the composite is much more sensitive to microstructural changes than its (visco-)elasticity, probably because the former relies on percolation while the latter does not (see Section [III.2.2\)](#page-12-0). The irreversible resistance increase is, as intended, reduced by cyclic conditioning (Fig. [C.2c](#page-43-0)); however, the reversible part of the strain-induced resistance increase drops even more in comparison (Fig. [C.2d](#page-43-0)). As a result, the irreversible part makes up a bigger portion of the overall resistance increase, and piezoresistive sensitivity (incl. its anisotropy) is greatly reduced overall. In conclusion, our films are not yet suitable as sensor materials, and more work on the stabilization of the piezoelectric response is needed. In addition, tests on the long-term stability of the films apart from mechanical conditioning are necessary to verify whether the flow-induced anisotropy prevails during relevant lifetimes, or whether it is significantly affected by material aging. # <span id="page-44-0"></span>Appendix D: PeakForce QNM maps and value distributions # <span id="page-44-1"></span>D1. General information on PeakForce QNM signals This section is intended to introduce the reader to the origin and informational value of the PeakForce QNM signals, named 'deformation', 'adhesion', 'modulus', and 'dissipation'. Further information is found in publications of the patent holder Bruker [\[43–](#page-35-2)[45\]](#page-35-3). We shall explain the signals using the schematic in Fig. 22. They are derived for each pixel (real-time analysis during scanning) from the load/unload curves that result from indenting the sample with the SFM tip. Contrary to the force-distance curves displayed by the measuring software, the abscissa of the analyzed curves is not the vertical position of the cantilever (z) but the vertical displacement of the tip. Only the latter is fully translated into the indentation depth (δ, corresponding to tip displacement in the contact region, see Fig. [D.1\)](#page-45-0) that is needed for evaluating the deformation and modulus signals. In contrast, the vertical displacement of the cantilever does not only lead to the indentation of the sample but also to the deflection of the cantilever, such that the z-values overestimate δ in the repulsive region and underestimate it in the attractive region. The deformation signal is an approximation of the maximal indentation depth. The latter is equal to the vertical tip displacement between the first tip-sample contact (usually marked by a 'snap-in' of the tip into the sample as a result of attractive interactions) and the lowest tip position (at maximal indentation) of the load curve. The latter usually corresponds to the maximal repulsive force (= PeakForce), unless sample relaxation is so strong that the maximal force is already reached before maximal indentation. Since the contact point is not always readily discerned, the software identifies a point with a low repulsive force instead (e.g. 5 % of the PeakForce), and outputs the vertical tip displacement between this point and the point of maximal indentation. Accordingly, deformation values tend to be smaller than the true maximal indentation depth. The adhesion signal is equal to the pull-off force, i.e., the maximal attractive force during unloading which must be overcome to sever the tip-sample contact. The term 'adhesion' is misleading since the pull-off force is not only the result of adhesive tip-sample interactions in equilibrium, but of dynamic effects, roughness, and additional attractive interactions (e.g. from liquid meniscuses and contaminations). One important methodological consequence is the sensitivity of the adhesion signal to humidity (when measuring in air) and contaminations. The latter usually increase the <span id="page-45-0"></span>![Figure D.1: Visualization of PeakForce QNM signals as derived from force-distance curves (deflection force of the cantilever as a function of vertical tip displacement) by the real-time analysis during the measurement. The reference point of the vertical tip displacement (zero point) is the lowest position, i.e., when indentation is maximal. In the absence of significant sample relaxation during loading, the PeakForce is reached at this point.](path) The modulus signal is a mechanical stiffness value obtained by fitting the unload curve with the so-called DMT model. The latter accounts for adhesive tip-sample interactions in the form of a force offset equal to the pull-off force, and it is valid only for homogeneous, smooth elastic solids. Consequently, the modulus signal generally outputs effective stiffness values rather than the Young's modulus. For polymer composites, the term 'effective' comprises the influence of both heterogeneity and viscoelasticity (dynamic contributions to the mechanical response of the sample). In the case of our CB-silicone composites, the modulus signal appears blurred (reduced lateral resolution) because it averages over heterogeneities with extreme local stiffness differences (i.e., regions where the stiff CB particles and the resilient silicone matrix both significantly contribute to the material response), and because the modulus signal is dominated by the contact region where the deformed sample volume is largest. The dissipation signal is equal to the dissipated mechanical energy per oscillation period. It is obtained by evaluating the hysteresis area between the load and unload curves of a given loadunload cycle, i.e., by integrating their force difference over the traveled distance of the cantilever. It is usually the most robust of all signals because baseline fluctuations cancel out when integrating the difference of the unload curve and the load curve. <span id="page-46-0"></span>![Figure D.2: PeakForce QNM maps mirror the dispersion state of near-surface CB, thanks to the material contrast between CB and the silicone matrix (in addition to topology). As an example, the maps of an unstretched 11 vol% CB-film are shown (scan size 5x5 µm<sup>2</sup> , coating direction from bottom left to top right, explanations on the signals see main text).](path) silicone films, Fig. [D.2](#page-46-0) gives an example of a 5x5 µm<sup>2</sup> scan on a film with 11 vol% CB. We omitted the signal values to obtain a compact representation but assure that they are physically reasonable as well as reproducible (see Appendix [D](#page-44-0) for a quantitative in-depth discussion). Keep in mind that PeakForce QNM yields near-surface information. In our case, the probed regions extend to some 10<sup>1</sup> nm to 10<sup>2</sup> nm into the material (see Section [II.2.3\)](#page-5-0). The height signal (= sample topography) reveals a 'hill-valley' surface morphology. With the help of the other signals, the 'hills' are unambiguously attributed to regions rich in CB and the 'valleys' to regions dominated by the silicone matrix: The 'hills' are less deformable (lower values of the deformation signal) and stiffer (higher modulus values) than the surrounding valleys, as expected for embedded CB in light of its reinforcing effect (see Section [III.2.1\)](#page-11-1). The 'valleys' yield higher dissipation values because the silicone matrix reacts with pronounced viscoelasticity whereas regions dominated by CB do not (see Appendix [D](#page-44-0) for further discussion). In contrast to deformation and modulus, the adhesion signal (= pull-off force) does not correspond to the 'hill-valley' morphology, as a large portion of the CB-rich regions is 'invisible'. The reason for this lies in the pull-off force being governed by adhesive interactions between the tip and the sample surface rather than the mechanical behavior of the indented volume. Since CB particles are always silicone-coated in the initial reactive mixture and since their adhesion to the silicone matrix is good according to the mechanical reinforcement by CB, it stands to reason that most CB particles at the sample surface are covered by at least a thin layer of silicone. As a result, a big portion of CB-rich regions yields similar adhesion values as the silicone-rich regions. In conclusion, PeakForce QNM allows a clear discrimination between the silicone matrix and embedded CB particles thanks to their different mechanical behavior as reflected by the deformation, dissipation and modulus signals. Concerning lateral resolution, the modulus signal is more blurred and thus less suited for structural analysis than the deformation and dissipation signals (see Fig. [D.2,](#page-46-0) plus Appendix [D](#page-44-0) for explanation). We shall therefore focus on deformation and dissipation for discussing CB morphology in both unstretched and stretched states. ### D2. Quantitative discussion of PeakForce QNM data In this section, we provide a quantitative discussion of PeakForce QNM signals along with further nanomechanical information on the CB-silicone films. For this we analyze respective value distributions (frequency densities) of repeatability measurements (scans of 10x10 µm<sup>2</sup> ) for various CB contents (7/9/11 vol% CB) and material states (unstretched/stretched). To roughly assign values to the silicone matrix and CB particles, respectively, Fig. [D.3](#page-48-0) gives an example of signal maps with typical value ranges. Compared to the silicone matrix, regions dominated by CB particles yield low deformation values of about 0 – 20 nm, low dissipation values of about 0 – 20 keV, low adhesion values of about ≤ 20 nN, and high modulus values of some 10 – 100 MPa. (The color scale for the modulus is logarithmic in order to clearly reveal the filler-matrix morphology. In contrast, a linear color scale results in very dark images with a few bright spots, due to the huge stiffness differences between matrix and filler.) Values for the silicone matrix are more difficult to define since the transition between 'filler-dominated' and 'matrix-dominated' is very broad, due to the various degrees of spatial overlap within the deformed volume. Frequency densities for unstretched films (7/9/11 vol% CB) and stretched 7 vol% CB-samples, along with the rough assignment of CB- and matrix-dominated regions, are shown in Fig. [D.4.](#page-49-0) Due to the misleading character of the term 'adhesion', we designated the values of the adhesion signal by the neutral term 'pull-off force' (see Section [D1\)](#page-44-1). Concerning reproducibility, we first point out that after measuring the unstretched 7 vol% CBfilm and one spot on the 7 vol% CB sample stretched parallel, the SFM probe had to be replaced (corrupted force-distance curves, probably due to tip wear). For scans with the same SFM probe, different measuring spots of a given composition and material state have very similar (7/9 vol% CB) or at least similar (11 vol% CB) frequency densities. In conclusion, measurements are repeatable within a measuring series of the same SFM probe, laser alignment and PeakForce QNM calibrations. <span id="page-48-0"></span>![Figure D.3: Example of scaled maps of the PeakForce QNM signals 'deformation', 'modulus' (logarithmic scaling), 'dissipation', and 'adhesion' (11 vol% CB, unstretched, 5x5 µm<sup>2</sup> scan size)](path) For measurements with different SFM probes, variations are bigger, as seen by the curves for the 7 vol% CB-sample stretched parallel to the coating direction (dashed curve from probe #1, other curves from probe #2). This is not surprising as Bruker and other authors [\[79\]](#page-37-3) state that truly quantitative measurements are difficult to achieve. For example, Bruker reports that the variation in mean modulus values introduced by using different probes and re-calibrating equals up to 25 % [\[45\]](#page-35-3). Moreover, tip contamination during scanning can vastly alter achieved values, usually increasing the pull-off force and the measured effective modulus [\[80\]](#page-37-4). For the data displayed in Fig. [D.4,](#page-49-0) the variation from using different probes and re-calibrating is similar to possible differences between the three compositions and between unstretched vs. stretched films. As a result, we cannot discern trends caused by varying CB concentration or by stretching. Concerning absolute values, we comment on some physically interesting differences between the PeakForce QNM signals for embedded CB particles and the silicone matrix. As already hinted at, the embedded CB particles are much stiffer than the silicone matrix, with effective modulus values amounting to 50 – 100 MPa and 3 – 10 MPa, respectively. These big differences in stiffness are a direct (nanomechanical) explanation of the macroscopic reinforcing effect of CB evidenced by mechanical characterization. To underline the plausibility of the absolute values, the modulus range for our PDMS-based silicone matches that for other PDMS samples characterized by PeakForce QNM (mean values of some 100 MPa, up to 10 MPa [\[45\]](#page-35-3), compare to values of peaks in Fig. [D.4b](#page-49-0): <span id="page-49-0"></span>![Figure D.4: Frequency densities of PeakForce QNM signals for unstretched films (7/9/11 vol% CB) as well as 7 vol% CB samples stretched 40 % parallel and perpendicular to the coating direction, respectively (2 – 4 scans of 10x10 µm<sup>2</sup> per material state). Scans of the unstretched 7 vol% CB-film and one scan on the 7 vol% CB-film stretched parallel (green, dashed curves) were acquired with the same SFM probe, the rest of the scans with another probe. The frequency density of the modulus signal is unitless because it is derived from log-scaled values, log(modulus[Pa]), that account for the big differences in stiffness of CB- and matrix-dominated regions.](path) Concerning pull-off force, the adhesion maps (see Fig. [D.3](#page-48-0) and discussion in Section [III.3.1\)](#page-18-0) show that the silicon tip adheres more strongly on silicone than it does on carbon. Since CB particles in the adhesion maps appear to be much smaller and more scarce than in the other signals, we can conclude that the pull-off force is dominated by adhesive interactions between the tip and sample surface, and that a large fraction of near-surface CB is coated by a thin layer of silicone. Concerning dissipation, the much higher values for the silicone matrix can be attributed to its viscoelastic relaxation and higher deformability compared to CB. Both result in a more pronounced pull-off event (more dissipation via dynamic effects and a larger maximal contact area). In addition, the viscoelastic relaxation leads to a larger hysteresis in the repulsive region. Differences in adhesion (silicon tip to silicone vs. to CB) appear to be less relevant since the dissipation values of uncoated portions of CB particles (black spots in the adhesion maps) and coated portions are very similar (compare regions corresponding to the black spots in the adhesion maps with other dark regions in the dissipation maps). ## <span id="page-50-0"></span>Appendix E: Simulation details #### E1. MPC simulations The MPC algorithm alternates between streaming and collision steps. It acts on a group of fictitious point particles representing the solvent that are first propagated ballistically in parallel to the standard MD steps. In certain time intervals, corresponding to an MPC particle free path of h = 0.1 a, MD and MPC particles are sorted into the compartments of a grid. Momentum is exchanged between solvent and solute in these so-called collision cells during the collision step. This is achieved by applying the scheme of Stochastic Rotation Dynamics (SRD), in which all velocities in a cell relative to the center of mass velocity are rotated by an angle of ±90◦ around one of the Cartesian axes. Axis and direction of rotation are drawn randomly from a uniform distribution. Shifting the grid randomly restores Galilean invariance. We work in MPC units, i.e., choose kBT = 1, MPC particle mass m = 1, MPC collision cell size a = 1, and derive the time unit τ = a p m/(kBT). The CB primary particle mass M = ρm with MPC particle number density ρ = 10 accounts for neutral buoyancy. This is a standard choice of parameters, yielding a viscosity of η = 4.6m/(aτ ) and a Schmidt number of Sc = 5.3. For our MPC-MD simulations, we employ a modified [\[81\]](#page-37-5) version of LAMMPS (2 Aug 2023) [\[82](#page-37-6)[–85\]](#page-37-7) and its simple local rescaling thermostat [\[86\]](#page-37-8) that accounts for the shear flow by acting on the velocities relative to the expected flow profile. For the MD simulation of CB aggregates, we restrain point particles representing the primary particles by harmonic potentials of the form U(r) = kb(r−r0) 2 for bonds (k<sup>b</sup> = 450 kBT /σ<sup>2</sup> , r<sup>0</sup> = 1), analogously for angles and dihedrals (k<sup>a</sup> = k<sup>d</sup> = 100 kBT /rad<sup>2</sup> ). The CB particles are propagated with a velocity Verlet integrator, for which we use a small time step ∆t = 10−<sup>4</sup> τ to account for the stiff potentials. Aggregates interact particle-wise with each other, but not with themselves, with a Weeks-Chandler-Anderson potential with ϵ = 10 kBT, σ = 1 a. For the nematic order parameters and the orientation distributions, we measure over 1 · 10<sup>5</sup> τ for Pe > 2 and 2 · 10<sup>5</sup> τ otherwise. <span id="page-51-0"></span>![Figure E.1: Illustration of the Kirchhoff network generated from an aggregate system. Conductivity is measured between artificial electrodes (rectangular nodes on top and below the system). Figure taken from [\[87\]](#page-37-9).](path) #### E2. Conductivity computation The nodes in the Kirchhoff network represent the aggregates, while the edges correspond to connections between aggregates. Fig. [E.1](#page-51-0) illustrates the construction. Each connection is weighted by its individual conductance which decays exponentially with the gap between the two primary particles that form the tunneling contact. Multiple contacts between the same aggregates are considered in parallel so that they can be absorbed into a single edge weighted by the sum of the individual conductances. Tangential aggregates are set to have unit conductance, which is also the maximum conductance between any two aggregates. A contact at separation d, i.e., the threshold distance for contact identification, accounts for 1% of unit conductance to be consistent with discarding longer-range contacts which fixes the conductance model. The network is decomposed into bi-connected components relative to the electrodes in order to eliminate dead ends and Wheatstone bridges from the network. We choose d as <sup>σ</sup> 2 because it induces a percolation threshold just below 10 vol-%, which is comparable to the values observed experimentally. However, a realistic tunneling distance will be much smaller just like the aggregates are realistically significantly bigger, amounting to a similar critical volume fraction. However, our observations for the anisotropic conductivity ratio are robust against larger variations of d with only the absolute values of conductivities changing by orders of magnitude.