Magnetic Manipulation of Spatially Confined Multiferroic Heuslers by Martensitic Microstructure Engineering

Magnetic Manipulation of Spatially Confined Multiferroic Heuslers by Martensitic Microstructure Engineering Milad Takhsha,* Vipin Kumar Singh, Julian Ledieu, Simone Fabbrici,* Francesca Casoli,* Francesco Mezzadri, Michal Horký, Vincent Fournée, Vojtˇech Uhlíˇr, and Franca Albertini 1. Introduction Magnetic shape-memory (MSM) Heuslers are multiferroic materials showing ferroelastic and ferromagnetic properties. [1] They exhibit strong coupling between magnetic and structural characteristics, evidencing a correlation between magnetic, thermal, and mechanical properties through a magnetostructural phase transformation. [2–4] In particular, MSM Heusler thin films are of special interest for the integration into smart micro/nanodevices such as sensors, energy harvesters, and actuators with various promising applications. [5–8] Recently, the successful epitaxial growth of MSM Heusler thin films on silicon substrates using SrTiO 3 buffer layers has facilitated the integration of these materials into micro/nanoelectronics and micro/nanomachining technologies based on silicon. [9] Martensitic microstructure dictates the magnetic characteristics of these films; therefore, controlling the twinning configurations at different length scales is the key to the optimization of the material’s functional magnetic properties. M. Takhsha, S. Fabbrici, F. Casoli, F. Mezzadri, F. Albertini Institute of Materials for Electronics and Magnetism (IMEM) National Research Council of Italy (CNR) Parco Area delle Scienze 37/A, Parma 43124, Italy E-mail: [email protected]; [email protected]; [email protected] V. K. Singh, J. Ledieu, V. Fournée Institut Jean Lamour UMR7198 CNRS-Nancy-Université de Lorraine Campus ARTEM –2 allée André Guinier, BP 50840, 54011 Nancy, France The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/sstr.202500284. © 2025 The Author(s). Small Structures published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. DOI: 10.1002/sstr.202500284 F. Mezzadri Department of Chemistry, Life Sciences and Environmental Sustainability University of Parma Parco Area delle Scienze 17/A, Parma 43124, Italy M. Horký, V. Uhlíˇr CEITEC BUT Brno University of Technology Purkyˇnova 123, 61200 Brno, Czech Republic V. Uhlíˇr Institute of Physical Engineering Brno University of Technology Technická 2, 61669 Brno, Czech Republic Magnetic shape-memory (MSM) Heuslers show a strong coupling between magnetic and structural characteristics, evidencing a correlation between magnetic, thermal, and mechanical properties through a magnetostructural martensitic transformation. This functional aspect makes MSM Heuslers promising for integration into smart micro/nanodevices, including sensors, energy harvesters, and actuators. Controlling the martensitic microstructure, which determines the magnetic characteristics, is among the key points for optimization of the magnetic functional properties of these materials at different length scales. Herein, a strategy is reported for manipulating the magneticpropertiesofspatiallyconfined epitaxial Ni–Mn–Ga films grown on Cr(001)//MgO(001) by twinning configuration engineering in the low temperature ferromagnetic (martensitic) phase. It is demonstrated that the twinning configurations in the continuous film and the micropatterned structures canbeswitchedfromY-type(showingnegligiblemagneticstrayfield) into X-type (presenting significant magnetic stray field) by a postannealing process. Advanced characterization techniques enable the analysis of the atomic structure, the surface quality of the annealed samples and the “twin-switching” phenomenon. The martensitic microstructure engineering reported in this study introduces a simple method for promoting the magnetic stray-field contribution at the surface of Ni–Mn–Ga epitaxial thin films and micropatterns initially showing a negligible magnetic stray field. RESEARCH ARTICLE www.small-structures.com Small Struct. 2025,6, 2500284 2500284 (1 of 12) © 2025 The Author(s). Small Structures published by Wiley-VCH GmbH Upon temperature variation, MSM Heuslers undergo a phase transformation between a high-temperature high-symmetry phase (austenitic) and a low-temperature low-symmetry phase (martensitic). The martensitic phase consists of self-assembled structural elastic domains (twin variants), which accommodate differently oriented cells in complex hierarchical patterns. [10–12] Epitaxial thin films of MSM Heuslers such as Ni–Mn–X (Ga, In, Sn, Al) and their Co-doped compounds, [13–16] having film(001)// underlayer(001)//substrate(001) crystallographic relationship, consist of either both or one of the differently oriented hierarchical twinning configurations in the martensitic phase called X-type and Y-type. [14,17–23] The mechanism leading to the selection of each of the two configurations is not yet well understood. However, it is assumed to be based on stress conditions [24–28] across the phase transformation where the b-axis of the martensitic cells (in the austenitic setting) lies in the plane of the film to construct the X-type or out of the plane of the film to construct the Y-type (Figure 1). If the film is ferromagnetic in the martensitic phase, the coupling between magnetic and structural degrees of freedom gives rise to two different magnetization patterns depending on the twinning configuration. For X-type twinning configurations, the easy-magnetization axis (c-axis) of the cells alternates between out-of-plane and in-plane, while for Y-type twinning configurations, the easy-magnetization axis of the cells alternates in the plane of the film along the MgO diagonal. [14,17] Therefore, different arrangements of twinning configurations give rise to thin films with various magnetic stray-field contributions at the surface (Figure 1). In previous works, we have reported a few strategies for engineering X-type and Y-type in epitaxial Ni–Mn–Ga films. [17,18,27,29] Controlling these twinning configurations in a film is a subtle process depending on many parameters such as composition, thickness, underlayer/substrate, growth conditions, and post-growth treatments, which limit the success of the existing strategies for real applications. For instance, the strategies based on applying mechanical stress [17,27] increase the risk of breaking the sample; whereas playing with composition, [30] thickness, [31] and growth conditions [27] may result in deterioration of the surface quality of the sample and/or undesirable influence on the martensitic phase transformation. In this study, we report a simple strategy for switching the entire martensitic configuration of Ni–Mn–Ga films, epitaxially grown on Cr(001)//MgO(001) with dominant Y-type twins showing negligible magnetic stray-field contribution at the surface. We show that the magnetic stray-field contribution at the surface can be significantly promoted by switching the twinning configurations of the pristine films into X-type as a result of a postannealing treatment while maintaining the high quality of the surface. With the help of advanced characterization techniques, we evaluate the evolution of the microstructure of the sample before and after the annealing process, and we assess the surface characteristics down to the atomic scale. We provide evidence showing that this simple twinning configuration engineering approach works perfectly both for continuous films and spatially confined micropatterns. Therefore, without the need for complicated effort, the twinning configurations can be switched into Xtype while maintaining the high surface quality of the samples, giving rise to considerable magnetic stray-field contribution. Finally, we discuss the observed “switching of twins”by introducing a phenomenological hypothesis, based on which the directions of internal stress in the epitaxial films contribute to the selection of the twinning configurations. Figure 1. a) 3D scheme of the three principal martensitic cell orientations with respect to the substrate plane. b) Simplified representation of the martensitic cell arrangement in Y-type and X-type twins; c) 3D slice of the film showing simplified scheme of Y-type and X-type twins (gray lines), martensitic interfaces (green lines), 180° magnetic domain walls (purple lines), and the easy magnetizationdirectioninthemartensitic cells (black arrows) across the twin boundaries. The resultant magnetic contribution from X-type and Y-type on the surface is shown by the MFM micrographs. www.advancedsciencenews.com www.small-structures.com Small Struct. 2025,6, 2500284 2500284 (2 of 12) © 2025 The Author(s). Small Structures published by Wiley-VCH GmbH 26884062, 2025, 11, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/sstr.202500284 by Brno University Of Technology, Wiley Online Library on [03/12/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 2. Results 2.1. Structural, Magnetic, and Microstructural Characterizations The epitaxial Ni–Mn–Ga films were sputtered on a Cr underlayer/MgO substrate at 623 K. The samples were left in a high vacuum to cool down to room temperature. The annealing process was performed ex situ by heating the pristine films up to 623 K, keeping the samples at that temperature for 1 h, and cooling the films back to room temperature in a high vacuum. The details can be found in the Experimental section (Section 5). The martensitic transformation of the thin film sample before and after the annealing process was traced by electrical resistance measurement as a function of temperature [29] (Figure 2a). Different steps of the phase transformation of thesamplearehighlightedinthefigure, where the cubic austenitic phase, low-symmetry martensitic phase, and the temperature region related to the phase coexistence are labeled. The sample shows a negligible variation in the phase transformation temperature after the annealing process. Through this temperature-dependent characterization, we have confirmed that the sample is in the martensitic phase at room temperature, so the subsequent measurements were performed at room temperature. The backscattered electron (BSE) micrographs give insight into the large-scale distribution of the X-type and Y-type twinning configurations in the sample before and after the annealing process. The large-scale micrographs in Figure 2b show the typical channeling contrast for Y-type (bright) and X-type (dark) twins [27] suggesting the dominance of Y-type twins; in addition, the highresolution micrograph in the panel verifies the presence of these twins, characterized and labeled by their typical relative orientations. [17] Figure 2c shows the micrographs of the same sample after the annealing treatment; no bright contrast is detectable anymore, suggesting the absence of Y-type twins; instead, X-type twins cover the entire scanned area. The angular X-ray maps measured for the thin film before and after the annealing treatment are shown and compared in Figure 3.The2θ=Δωmaps, in this specific case, give us insight into the martensitic cells that are the building blocks for the martensitic twinning configurations (X-type and Y-type). As shown in Figure 1a, if we simplify the orientations of the martensitic monoclinic cells with respect to the substrate, considering them as pseudo-orthorhombic in the austenitic setting, we will have three principal orientations with a-axis, b-axis, or c-axis of the cells out of the plane of the film. To access these cell parameters, the 2θ=Δωangular dataset in the out-of-plane diffraction configuration was acquired to match the diffraction condition of those Ni–Mn–Ga crystallographic planes that are approximately parallel to the film plane. Therefore, the polar angle (χ) was set to zero degrees so that the diffraction from the Ni–Mn–Ga lattice planes was coherent with the substrate. The presence of different diffraction peaks with various intensities in the 2θ=Δωangular maps confirms the coexistence of different martensitic cells. For the sake of simplicity, we have highlighted the obtained peaks based on the aforementioned three principal martensitic cells having the a-axis, b-axis, or c-axis out of the plane of the film (labeled in Figure 3a,b). In particular, the obtained out-of-plane maps reveal reflections at 2θ¼60.2°, 2θ¼64.0°, 2θ¼68.3°, which can be assigned to the (400), (040), and (004) reflections arising from a,b, and c lattice planes of the martensitic cells. [29] In addition, there is a high-intensity reflection at 2θ¼64.4°, which is assigned to the (002) reflection arising from the Cr underlayer lattice planes. [17] Comparing the relative intensity of the obtained reflection peaks for the sample before and after annealing in Figure 3a,b gives us insight into the relative abundance of each of the three principal martensitic cells having a-axis, b-axis, or c-axis out of the plane of the film. The pristine sample in Figure 3a shows a relatively high-intensity reflection for the b lattice planes of the martensitic cells, suggesting the abundance of the building blocks for Y-type twins in this sample. The sample after the annealing process shows a completely different intensity pattern (Figure 3b): the reflection for the blattice planes has disappeared, suggesting the absence of the building blocks for Y-type twins. Based on the obtained data, one can conclude that the annealing process has eliminated the population of the cells constituting Y-type twins in the sample. Figure 2. a) Relative resistance curves as a function of temperature for the sample before (black circles) and after (pink triangles) the annealing process. BSE micrographs of the sample b) before and c) after the annealing process. www.advancedsciencenews.com www.small-structures.com Small Struct. 2025,6, 2500284 2500284 (3 of 12) © 2025 The Author(s). Small Structures published by Wiley-VCH GmbH 26884062, 2025, 11, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/sstr.202500284 by Brno University Of Technology, Wiley Online Library on [03/12/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Details on the annealing-induced magnetic evolution ofthesamplewereobtainedbymagneticforcemicroscopy (MFM) shown in Figure 3c,d. The micrograph after the annealing process clearly shows the dominance of the out-of-plane magnetic domains in X-type twins arising from the perpendicular anisotropy contribution of the martensitic cells having the c-axis out of the plane of the film. The dominant zerosignal areas in the magnetic micrograph obtained for the sample before annealing can be ascribed to the Y-type twins with the c-axis of the martensitic cells in the plane of the film, therefore having negligible perpendicular anisotropy contribution. [14,17,21,22,27] The unique arrangement of the martensitic cells before and after the annealing process gives rise to completely different magnetic characteristics of the sample. Figure 4 shows the magnetization loops of the sample before and after the annealing process as a function of an external magnetic field up to 1.5 T that was applied along three in-plane directions of the sample at room temperature. Among these three applied magnetic field directions, only MgO[110] is parallel to one of the three easymagnetization directions occurring in X-type twins and one of the two easy-magnetization directions occurring in Y-type twins (Figure 1). The magnetization loops in Figure 4a show magnetization jumps in both the first and the third quadrants at around 50 mT when we apply the magnetic field along MgO[100] and MgO[010], which are among the magnetically hard directions of the system (Figure 1). The loops measured for these two directions also show slightly lower remanence and coercivity compared to the measured loop when we applied the magnetic field along MgO[110]. The same magnetization measurements obtained for the sample after the annealing process (Figure 4b) show no trace of the magnetization jump and no considerable variation in the remanence and coercivity when we applied the magnetic field along MgO[100], MgO[010], or MgO[110]. The other noticeable point is the significant reduction of the remanence and coercivity of the loops (76% and 56%, respectively) obtained for the sample after the annealing process. In one of our recent works, [21] we have described the magnetization process of Ni–Mn–Ga epitaxial films in the martensitic phase, where the spatial arrangement of magnetocrystalline anisotropy related to the martensitic twinning configurations plays an important role. The micromagnetic simulations also describe the magnetization jumps in the magnetization loops of the samples as having Y-type twinning configurations. These typical jumps occur when the magnetization reversal proceeds with the formation and expansion of magnetic domains, passing around the zero Figure 3. 2θ=Δωangular maps of the sample, a) before and b) after the annealing process in the martensitic phase at room temperature. MFM micrographs of the sample, c) before and d) after the annealing process in the martensitic phase at room temperature. www.advancedsciencenews.com www.small-structures.com Small Struct. 2025,6, 2500284 2500284 (4 of 12) © 2025 The Author(s). Small Structures published by Wiley-VCH GmbH 26884062, 2025, 11, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/sstr.202500284 by Brno University Of Technology, Wiley Online Library on [03/12/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License field in a closed-flux domain configuration. [21] This microstructuredependent alternation of the magnetization loops in our Ni–Mn– Ga films and its potential connection to the functional properties of the material, such as magnetic-field-induced strain is a very interesting topic for future investigations. 2.2. Surface Study To deeply analyze the atomic structure, superstructure, and surface characteristics of our films after the annealing process, a similarly grown sample was inserted into an ultra-high vacuum (UHV) chamber equipped with advanced surface characterization techniques, including scanning tunneling microscopy (STM) and temperature-dependent low-energy electron diffraction LEED. The LEED pattern of the surface was monitored upon cooling after the annealing process. Figure 5a shows a typical LEED pattern of the sample at 50 eV primary electron beam energy at 348 K as measured by the thermocouple attached close to the sample holder. Sharp diffraction spots with a low background Figure 5. LEED pattern recorded at 50 eV primary electron beam energy at a) 348 K and c) 318 K. b) Scheme of the austenitic unit cell and its orientation with respect to the MgO substrate. d) STM topographic image (300 300 nm 2 ;V b =1.8 V, I t =0.1 nA) of the surface recorded at room temperature after the annealing process. e) Derivative and g) 3D view of an STM image of the same surface (150 150 nm 2 ,V b =1.8 V, I t =0.1 nA). f) The height profile measured perpendicular to the twin lamellae along the blue line in (e). Figure 4. Magnetization reversal curves of the sample at room temperature, a) before and b) after the annealing process, applying the magnetic field along the different in-plane orientations of the sample shown by color code and symbols. www.advancedsciencenews.com www.small-structures.com Small Struct. 2025,6, 2500284 2500284 (5 of 12) © 2025 The Author(s). Small Structures published by Wiley-VCH GmbH 26884062, 2025, 11, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/sstr.202500284 by Brno University Of Technology, Wiley Online Library on [03/12/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License intensity indicate a clean surface with long-range crystallographic order. The pattern is quadratic with p(1 1) symmetry consistent with a cubic austenitic structure with (001) surface orientation. A scheme of the austenitic unit cell and its orientation with respect to the MgO substrate is shown in Figure 5b, together with the surface unit cell corresponding to the experimentally observed unit cell in the LEED pattern. As reported by Leicht et al. [32] for indistinguishable atoms with similar atomic radii, the LEED pattern is expected to show the orientation and periodicity of the nearest-neighbor cell (black dashed square in Figure 5a,b). Instead, the observed surface unit cell is p2 larger (blue dashed square in Figure 5a,b), providing support for a surface termination mainly at (Mn, Ga) planes rather than at Ni planes, with a possible buckling of the top surface plane induced by a vertical relaxation of one sort of atoms. [32,33] Upon cooling in front of the LEED across the transition from the austenitic to the martensitic phase, the sharp spots of the quadratic pattern split into a set of fainter reflections in Figure 5c. This complex LEED pattern has been interpreted to be a consequence of the surface topography and the different twin variants appearing upon the martensitic transformation. [32] The austenitic phase has a cubic L2 1 structure with a L21 =5.82 Å. [17] The martensitic state has a more complicated 7M modulated monoclinic structure, [34] which can be approximately described based on a distorted orthorhombic unit cell having a 7M =6.14 Å, b 7M =5.82 Å, c 7M =5.52 Å, and γ 7M =92.98° (in the austenitic setting) obtained from the Xray diffraction analysis shown in Figure 3. At the surface, twin variants expose either (a,b)or(b,c) surface unit cells in the Xtype twins, each with four simultaneously occurring orientations with respect to the substrate, and tilted at some specific angles with respect to the surface normal. In the case of the Y-type twins, twin variants would expose (a,c) surface unit cells, and no surface tilting is expected in that case, as the twinning planes are perpendicular to the surface normal (Figure 1). Therefore, the observed LEED patterns with split reflections provide evidence that the microstructure of the sample has been switched to pure X-type after the annealing process. Further evidence was obtained by the “large-scale”topographic STM image (300 300 nm 2 ) in Figure 5d, which shows a roof-like surface morphology extending parallel to MgO[110] and MgO[110] directions. A smaller-scale STM image (150 150 nm 2 ) of the roof-like surface morphology isshowninFigure5einaderivativemodetoenhancethecontrast of the atomic steps and defects. The height profile (Figure 5f) measured perpendicularly to the corrugation lines shows a triangular shape, with a peak-to-peak height (in the Zaxis) equal to 1.3 nm and an average periodicity (in the X-axis) of 45 6 nm, from which a misorientation angle ε3.3 0.1° can be deduced. According to previous reports, [32,35] each corrugation line is a twin lamella (Figure 5g), each side of which corresponds to a different twin variant. In the case of so-called a–ctwinning in the 7M martensite, one side of the twin lamella exposes (a,b) type surface unit cells while the other side exposes (b,c) type surface unit cells, and the misorientation angle is ε7M ac¼45° tan1c a ¼3.04° (using the lattice parameters reported above), which is in agreement with the value deduced by STM. The misorientation angles are smaller for a–band b–ctypes of twinning (ε7M ab1.5°and ε7M bc1.5°) and can therefore be disregarded. The a–ctwinning is preferred due to the close match between b 7M and the austenitic lattice parameter, thus limiting the stress. One type of variant in Figure 5e shows additional corrugations (a set of parallel lines perpendicular to the twin lamella). The average spacing between lines (width) is 19.6 0.5 Å, but actual distances are not exactly periodic. Peak-to-peak corrugation between these lines (height) is between 0.4 to 0.8 Å. This extra corrugation is related to the modulated martensite superstructure, [32] which implies a displacement of the (110) basal planes in the [110] directions. The period of this modulation (width) is seven basal planes for the 7M phase (according to the aforementioned cell parameters, this value is 20.4 Å), i.e., close to the value obtained by STM. 2.3. Microfabrication On the last step, a piece of the as-grown thin film sample (5 5mm 2 ) was patterned by UV lithography, fabricating arrays of micropatterns including rings, stripes, square bands, L-stripes, squares, and disks. The martensitic configuration of the fabricated arrays was evaluated before and after the annealing process by a Kerr microscope at room temperature, where different orientations of X-type and Y-type twins interact differently with the polarized light, showing different tonality of the gray color [29] (Figure 6). The fabricated arrays before the annealing process show the dominance of Y-type twins, covering a range of 52–88% of the area of the micropatterns (Figure 6a). After the annealing process, the fabricated arrays were recaptured by the Kerr microscope; they show no evidence of the typical gray contrast arising from the Y-type twins (Figure 6b). This observation is in agreement with all the results obtained for the continuous thin film samples, suggesting the switch of martensitic configuration to X-type as a result of the annealing process. 3. Discussion We have reported a simple strategy for promoting the magnetic stray-field contribution at the surface of epitaxially grown Ni–Mn–Ga thin films and spatially confined structures having negligible magnetic stray-field contribution: an annealing process was sufficient for switching the hierarchical self-assembly of the martensitic magnetic cells from Y-type into X-type twinning configurations with considerable magnetic stray-field contribution at the surface. After the annealing process, the samples underwent several heating (up to 400 K) and cooling (down to 300 K), crossing the phase transformation, and the microstructure remained unchanged (always X-type twins). Therefore, we have concluded that the switching is permanent and not influenced by the temperature variation across the phase transformation. Here, we introduce a phenomenological hypothesis, which describes the observed selection of the twins based on the orientations of internal stress (and strain) in our films. When epitaxial Ni–Mn–Ga film (lattice parameter 5.8 Å) is grown on a Cr layer (lattice parameter 2.9 Å) that is www.advancedsciencenews.com www.small-structures.com Small Struct. 2025,6, 2500284 2500284 (6 of 12) © 2025 The Author(s). Small Structures published by Wiley-VCH GmbH 26884062, 2025, 11, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/sstr.202500284 by Brno University Of Technology, Wiley Online Library on [03/12/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License itself epitaxially grown on an MgO(001) substrate (lattice parameter 4.2 Å), we assume that the primary stress condition mainly originates from the lattice-mismatch at the growth temperature and subsequently the film encounters dynamic thermal stress as a function of temperature because Ni–Mn–Ga, Cr, and MgO have different thermal expansion coefficients. [36] The MgO substrate provides the initial crystallographic orientation but does not impose direct stress on the Ni–Mn–Ga layer due to the buffer effect of the Cr layer. The Cr layer essentially isolates Ni–Mn–Ga from any direct stresses imposed by the MgO substrate. MgO orients the Cr layer such that Cr(001)[110] aligns with MgO(001)[100]. Consequently, the Ni–Mn–Ga film is oriented as Ni–Mn–Ga(001)[110]||Cr(001)[110] (Figure 7a). The Ni–Mn–Ga film’s in-plane stress is primarily influenced by the Cr layer, which generates biaxial stress along Ni–Mn– Ga[100] and [010] directions (Figure 7b) and triaxial strains in the Ni–Mn–Ga cells. Martensitic transformation proceeds with a shear-like mechanism and thus interacts with stress. [37] Moreover, all the possible habit planes and twinning planes in Ni–Mn–Ga cubic cells are close to {110} planes [24] (Figure 7c), which can interact with the stress giving rise to the selection of twins. [24–28] The resolved shear stress (RSS) on the {110} planes of cubic austenite can be calculated from the following general formula τ¼σcosðφÞcosðλÞ(1) where σis the magnitude of the stress, φis the angle between the direction of the stress and the normal to the twinning plane, and λis the angle between the direction of the stress and the twinning direction. The biaxial stress is imposed to cubic Ni–Mn–Ga by epitaxial crystal relation along Ni–Mn–Ga[100] and [010], therefore, the stress along Ni–Mn–Ga[100] (σ 1 ) equals to the stress along Ni–Mn–Ga[010] (σ 2 ). The normal direction and the main involving twinning direction for each of the {110} planes (Figure 7c) are: (110) with normal along [110] and twinning direction along [110], (110) with normal along [110] and twinning direction along [110], (101) with normal along [101] and twinning direction along [101], (101) with normal along [101] and twinning direction along [101], (011) with normal along [011] and twinning direction along [011] and (011) with normal along [011] and twinning direction along [011]. Therefore, the RSS on each of the (110) and Figure 6. Kerr micrographs of the microfabricated Ni–Mn–Ga epitaxial structures a) before and b) after the annealing process. www.advancedsciencenews.com www.small-structures.com Small Struct. 2025,6, 2500284 2500284 (7 of 12) © 2025 The Author(s). Small Structures published by Wiley-VCH GmbH 26884062, 2025, 11, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/sstr.202500284 by Brno University Of Technology, Wiley Online Library on [03/12/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License (110) planes is τ=σ 1 /2 þσ 2 /2. However, the RSS on each of the (101) and (101) planes is τ=σ 1 /2 and on each of the (011) and (011) planes is τ=σ 2 /2 (Figure 7c). Therefore, when Ni–Mn–Ga is epitaxially grown on Cr/MgO, each of the four inclined planes (generating X-type twins) can interact either with σ 1 or σ 2 (Figure 7c), giving rise to either τ=σ 1 /2 or τ=σ 2 /2; whereas each of the two normal planes (generating Y-type twins) can interact with both directions of the biaxial stress, giving rise to τ=σ 1 /2 þσ 2 /2. This makes the normal twinning planes more favorable under biaxial stress along Ni–Mn–Ga[100], [010], and could explain our experimental observation where we reported the dominance of Y-type twins at room temperature in our samples before annealing (Figure 2, 3, and 6). The annealing process involves heating the film to a high temperature and holding it at that temperature. This process typically allows epitaxial stress (and strain) relaxation by the formation of defects (misfit dislocations). This process may cancel out or reduce the primary epitaxial biaxial stress imposed by Cr to Ni–Mn–Ga. To further investigate this effect, we have performed in situ annealing for the as-grown sample in the diffractometer, tracing the out-of-plane lattice parameters of the Cr layer and Ni–Mn–Ga film in the process of heating, annealing, and cooling. The results are shown in Figure 8. At room temperature, the sample is in the martensitic phase, showing the b(040) peak, which corresponds to the Y-type twins (Figure 8a). Upon heating, the sample transforms to the austenite phase, the lattice parameters of Cr(002) and Ni–Mn–Ga(004) Aus expand by around 0.003 and 0.018 Å, respectively, due to the thermal expansion (Figure 8c,e). However, during the annealing process, where the temperature is constant for 60 min, the lattice parameter of Ni–Mn–Ga drops by 0.013 Å while the lattice parameter of Cr is constant, evidencing a strain variation in the out-of-plane direction of the film (Figure 8f), most probably caused by Poisson’s ratio due to in-plane lattice expansion. By cooling, the drop of the out-of-plane lattice parameter of Ni–Mn–Ga accelerates in such a way that the Aus(004) peak almost overlaps the Cr(002) peak at T=380 K (Figure 8d). Finally, by cooling the sample across the phase transition, the martensitic lattice parameter b(040) is not visible anymore due to the “switch of the twins”into X-type, while the Cr(002) peak is detected in its original position at 298 K (Figure 8b). The calculated out-of-plane lattice parameters of Cr and Ni–Mn–Ga are shown in Figure 8g,h graphs. The X-ray diffraction results show the variation of the out-ofplane austenitic cell parameter of Ni–Mn–Ga, most probably due to the relaxation of in-plane epitaxial stresses and strains in Ni– Mn–Ga following thermal annealing, and they confirm the formation of X-type variants. Simple elastic energy calculations based on lattice mismatch indicate that the elastic energy associated with X-type variants is approximately half that of Y-type variants (Figure S1, Supporting Information). In addition, different geometries of the X-type and Y-type twins with respect to the substrate [19] may play a fundamental role in this selection. As a final note, we emphasize that further simulations and theoretical studies are necessary to develop a comprehensive model that Figure 7. a) schematic 3D-view of the epitaxial relation between MgO(001), Cr(001), and Ni–Mn–Ga(001). b) Plan-view of the scheme shown in (a), highlighting the orientations of the biaxial stress imposed by Cr on Ni–Mn–Ga. c) 3D-view of the Ni–Mn–Ga cubic austenite, showing the relative orientations between the six possible {110} habit planes (or twinning planes) and the biaxial stress. The stress directions that do not interact with the twinning planes are shown in red. www.advancedsciencenews.com www.small-structures.com Small Struct. 2025,6, 2500284 2500284 (8 of 12) © 2025 The Author(s). Small Structures published by Wiley-VCH GmbH 26884062, 2025, 11, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/sstr.202500284 by Brno University Of Technology, Wiley Online Library on [03/12/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Figure 8. Out-of-plane 2θ=ωangular maps of the sample a,c,e) before and b,d,f) after the annealing process as a function of temperature. g) Calculated lattice parameter of Cr(002) as a function of temperature. h) Calculated lattice parameter of Aus(004) as a function of temperature. The black circles refer to the lattice parameters before annealing, and the pink circles refer to the lattice parameters after annealing (Uncertainty: 10 3 Å). The black circle at 298 K refers to b(040) in the martensitic phase. www.advancedsciencenews.com www.small-structures.com Small Struct. 2025,6, 2500284 2500284 (9 of 12) © 2025 The Author(s). Small Structures published by Wiley-VCH GmbH 26884062, 2025, 11, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/sstr.202500284 by Brno University Of Technology, Wiley Online Library on [03/12/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License