Insight into the magnetisation process of martensitic Ni–Mn–Ga films: a micromagnetic and vector magnetometry study

This study investigates the magnetisation process of a martensitic Ni–Mn–Ga thin film with microstructure optimized to obtain a unidirectional and reversible magnetisation jump. The study has been realised by a thorough vector magnetometry characterisation and supported by micromagnetic modelling, considering different orientations of the applied field with respect to the symmetry directions of the sample. The model has been built on the film microstructure and experimental characteristics. The main features of the magnetisation curves measured along the film symmetry directions can be well reproduced by the micromagnetic model, that is, neglecting structural and magnetostructural contributions to the free energy. The model also well reproduces the field-dependent behaviour of the transverse magnetisation components. The agreement demonstrates that the spatial organisation of magnetocrystalline anisotropy axes due to martensitic twinning has a dominant effect on the magnetisation process, giving rise to magnetisation jumps when the magnetic field is applied along the alignment direction of the twin boundaries. When a reverse field is applied along this direction, simulations show that magnetisation reversal proceeds through the formation and three-dimensional expansion of magnetic domains, passing around the zero field through a closed-flux domain configuration, with the domain walls showing perpendicular orientation of the magnetic moments.


Introduction
Magnetic shape memory compounds are among the most promising classes of materials for magnetic actuation and multiple-stimuli actuation, offering new capabilities to exploit in MEMS devices for information technology and automotive, or in soft robots for biomedicine, tissue engineering and aerospace [1][2][3][4]. Compared to a conventional shape-memory material, in which a huge elastic deformation is induced by temperature, in the magnetic counterpart the additional occurrence of magnetically ordered states and of a strong spin-lattice coupling also allows magnetically driven deformations [5][6][7].
At the origin of the magnetic shape memory effect is a magnetostructural transition from a high symmetry phase, stable at high temperature, to a lower symmetry phase, stable at lower temperature. In Ni 2 MnGa, a model system within this class of compounds and the object of this work, the phase transition occurs from a cubic L2 1 phase (austenite) to a lower symmetry phase (martensite), which can show different crystal structures depending on compositional variations around the 2-1-1 stoichiometry [7][8][9][10].
The occurrence of the magnetic shape memory effect in Ni-Mn-Ga has been ascribed to two main phenomena: the magnetic field-induced structural transformation and the magnetically induced reorientation of twin variants (MIR) [7]. In the first phenomenon, the shape memory effect takes place when the martensite to austenite transformation is induced by a strong magnetic field [11,12]. Differently, in the MIR effect the reversible deformation of the material occurs due the motion of martensitic twin boundaries to favour the twin variants with easy magnetisation direction oriented parallel to the applied magnetic field [5,13,14]. The latter effect gives rise in single crystals to deformations up to 12%, much larger than the piezoelectric and magnetostrictive effects exploited in competing technologies [15].
The importance of magnetic shape memory thin films gains relevance for micro-and nano-scale devices, in which the direct integration of the thin film material offers clear advantages compared to the sizing and bonding of bulk single-crystals [1,4,[16][17][18][19]. The capability to grow thin films with high structural quality and clear magnetostructural transition has been demonstrated by different authors [20][21][22][23][24]. The transition temperature values mostly depend on composition and film thickness, while the sharpness of the transition and thermal hysteresis are also related to structure quality and microstructural characteristics [22,[24][25][26][27][28][29]. After transforming to the martensitic phase, the thin film shows a complex pattern of twin variants and, depending on substrate/underlayer and film thickness, different types of twinning and different twin patterns have been observed [24,26,27,30,31]. The support of detailed XRD analyses and electron microscopy investigations (e.g. BSE, HRTEM) has been fundamental to disentangle the details of such complex patterns.
In previous works, the magnetisation curve measured in the martensitic phase as a function of an external magnetic field has been used to deduce the occurrence of the MIR effect in thin films, in analogy to what occurs in bulk materials, where the appearance of magnetisation jumps at definite field values marks the field-induced twin boundary motion that can be directly visualized by microscopy techniques [5,32,33]. Instead, visualizing twin boundary motion in thin films imposes major challenges on the experimental side, also due to the much smaller typical twin size. For this reason, the occurrence of the MIR effect in Ni-Mn-Ga films has been often deduced indirectly, mostly from the appearance of sudden slope variations in the first quadrant of the magnetisation curve [7,23,24,26,28,34]. It has to be noted that this kind of slope variation is often associated to a specific type of twin microstructure: clear magnetisation jumps occur in thin films with full or partial Y-type twinning. Here, 'Y-type' refers to a specific orientation of the crystal cells within the twin variants and of the twinning planes with respect to the substrate symmetry directions, as originally defined in the paper by Diestel et al [35]. For this specific type of microstructure, Laptev and co-authors have demonstrated that micromagnetic simulations can give account of magnetisation jumps similar to those measured experimentally, raising doubts about the unconditional attribution of magnetisation jumps to the occurrence of the MIR effect [36]. Another kind of twinned microstructure occurring in thin films is X-type microstructure; micromagnetic modelling has been usefully employed to interpret the experimental results, here too, by simulating the MFM contrast in a Ni-Mn-Ga thin film [37]. Finally, by resorting to micromagnetic modelling, C Phatak et al could associate the appearance of nanoscale skyrmions in a thinned Ni-Mn-Ga crystal to the spatial modulation of magnetic anisotropy originating from martensitic twinning [38].
In our previous works, we have been able to grow epitaxial films with different twinned microstructures, thanks to the careful control of growth conditions and the application of external stimuli [26,29,39]. In this work, we focus on the magnetisation process of a martensitic thin film with microstructure optimized to obtain a unidirectional and reversible magnetisation jump. The study has been realised by an in-depth vector magnetometry characterisation, in which the longitudinal and transverse magnetisation components are measured, while applying the magnetic field along the symmetry directions of the film. The study is extensively supported by micromagnetic simulations. A micromagnetic model similar to those employed in [36][37][38] has been developed, but tailored to the specific thin film sample studied by magnetometry, in order to obtain the best possible correspondence between model and real sample. The model microstructure in fact reproduces the one observed in the thin film, i.e. a twinned microstructure with oriented Y-type twin boundaries. In addition, the experimentally measured characteristics have been introduced in the micromagnetic model. The comparison between measurements and simulations has been drawn for different experimental configurations, i.e. for the longitudinal and transverse magnetisation components obtained with magnetic field applied along the symmetry directions of the sample and at tilt angles to these directions, also analysing the three-dimensional evolution of the micromagnetic configurations as a function of field. Our paper presents and compares an extensive range of experimental and simulation results, allowing us to confirm and expand the conclusion of [36], also clarifying some of the doubts emerging due to the limits of the approaches previously adopted.

Samples preparation and characterisation
Ni-Mn-Ga(200 nm)/Cr(50 nm) films were grown by RF sputtering on a single-crystal MgO substrate with (100) orientation. The films were grown in a system with a base pressure of 2 × 10 −6 Pa, setting the Ar pressure to 1.5 Pa and the growth temperature to 623 K. First, a Cr underlayer with 50 nm thickness was  [26].
Samples microstructure was studied by means of scanning electron microscopy (SEM, FEI Inspect-F) in both conventional and backscattered modes. The surface magnetic configuration was investigated by means of a Dimension 3100 atomic force microscope (AFM) equipped with Nanoscope IVa controller (Bruker Instrument). The surface topography was obtained in tapping mode, whereas magnetic domain images were acquired in tapping-lift mode with a lift scan height of 35-50 nm measured above topography.
A vector VSM magnetometer (vVSM, MicroSense Model 10) equipped with a rotating electromagnet (µ 0 H max = 2 T) and two sets of fixed pick-up coils perpendicular to each other was used to measure at room temperature (RT) the in-plane components of the magnetisation vector along the directions parallel, M x , and perpendicular (transverse signal), M y , to the external magnetic field H. The two directions x and y constitute a reference system that rotates with respect to the samples together with the field (figure 1). Comparing the parallel, M x , and perpendicular (transverse signal), M y, signals allows the evolution of the magnetisation vector to be investigated while sweeping the external magnetic field in the film plane, thus providing information about the magnetisation reversal mechanism.

Micromagnetic simulations
Micromagnetic simulations were performed by using the object oriented micromagnetic computational framework (OOMMF) code, where a time evolver that tracks the Landau-Lifshitz-Gilbert dynamics was implemented [40]. The material parameters used in the simulations were taken from our experimental data on Ni-Mn-Ga film and bulk samples [26,41], except for the exchange stiffness constant A, which was deduced from a paper reporting on inelastic neutron scattering measurements performed on a bulk single crystal with composition Ni 50 Mn 28 Ga 22 [42]. The following values were used in the simulations: a saturation magnetisation M S of 490 kA m −1 , as measured by VSM on the thin film sample; a uniaxial second order anisotropy constant K 1 of 1 × 10 5 J m −3 , as evaluated by Singular Point Detection measurements on bulk polycrystals with composition similar to the film studied in the present paper (higher order anisotropy coefficients were neglected) [41]; an exchange stiffness constant A of 5.07 × 10 −12 J m −1 [42]. The geometrical distribution of magnetic easy axes in the sample was deduced from the microstructural and magnetic analyses reported in [26]. Micromagnetic simulations obtained on two different types of microstructural patterns will be presented in this paper, as explained in the 'results and discussion' and 'supplementary information' sections. In both cases, the model sample has a size of 1.  time, for each applied field direction only the first half of the hysteresis loop was simulated (from +2T to −2T), and the remaining part was obtained by symmetry.

Results and discussion
At RT the Ni-Mn-Ga films show a twinned martensitic microstructure, which forms on cooling the film grown at high temperature through the martensitic transformation temperature [26,29]. At the growth temperature, the compound grows epitaxially in the L2 1 austenitic phase with epitaxial relationship (001)[100] Ni-Mn-Ga //(001)[100] Cr //(001)[110] MgO . On cooling to RT, a structural transformation occurs from the austenitic to the martensitic structure, with a transformation temperature of approximately 335 K. The martensitic structure is twinned, as typically shown by ferromagnetic shape memory compounds, which accommodate the transformation stress with the formation of twin variants [24,26,27,30,31]. In general, these twin variants are organized into characteristic patterns, but the type of twinning and details of the twin variants pattern depend on the specific martensitic structure and on a complex balance between elastic, magnetic and magneto-elastic energy terms.
If we compare a thin film with a bulk crystal, we have to consider that the substrate constraint and surface energies influence the formation of twin variants [24]. It is also evident that magnetostatic energy, which in thin films is comparable in intensity to magnetocrystalline anisotropy energy, plays an important role in determining the twin variants pattern, due to the strong magneto-elastic coupling of the compound. Indeed, it has been demonstrated that in free-standing thin films the magnetostatic energy can drive a stray-field-induced shape memory effect [43].
Depending on the composition of the compound, the martensitic structure can be tetragonal, monoclinic (7M, 5M, non-modulated) or orthorhombic [7,8]. In all these cases, the symmetry reduction determines an increase in magnetocrystalline anisotropy compared to the austenitic cubic phase. In our case, i.e. Ni-Mn-Ga thin films with Ni-rich composition grown on Cr/MgO(100), the structure is 7M-modulated monoclinic with lattice parameters a' = 4.23 Å, b' = 5.52 Å, c' = 4.32 Å and angle β = 93 • . The cell parameters in the austenitic setting are the following: a = 6.20 Å, b = 5.88 Å, c = 5.52 Å with γ = 91 • , with the c axis corresponding to b' of the monoclinic setting. Two types of twinning can occur: X-type and Y-type [26,35]. These are sketched in figure 2, both in the monoclinic and austenitic settings; for sake of simplicity, the latter setting will be preferred in the following description.
In X-type twins, the twinning planes are 101 planes and the short c-axis exchanges its orientation with a-axis, flipping from the in-plane to the out-of-plane directions, while the b-axis remains in the film plane. The c-axis is the easy magnetisation axis (e.a.) of the compound, as also observed in bulk materials with similar composition and the same crystal structure [26,41].
In Y-type twins, which are also a-c twins, the twinning planes are 110 planes and the c-axis switches between two symmetric in-plane directions, corresponding approximately to the diagonals of MgO basal plane, while the b-axis always remains perpendicular to the film plane ( figure 2).
Interestingly, a-c twins give rise to a typical microstructure with twin lamellae oriented at characteristic angles with respect to the film plane [30,44]. This allows an easy identification of the specific type of twins by SEM and AFM [26].  X-type twins give rise to a surface corrugation and to a height contrast measurable by AFM, while both types of twin lamellae can be visualized in SEM images recorded by backscattered electron detector (BSE) [26,30].
The two types of twin microstructure described above are energetically competitive for Ni-Mn-Ga films, and, depending on the substrate/underlayer, thickness and growth parameters, a full X-type, Y-type, or mixed X/Y-type film can be grown [26,29,31,35,37,39]. Also a stress applied to the film during or after the growth can influence the microstructure, giving rise to specific orientations of the twin lamellae or inducing Y-type twins in a full X-type film [26,39]. Figure 3(a) shows the X/Y microstructure of a film with Y-type twin boundaries oriented along one direction, that is, the [100] direction of the MgO substrate. We have been able to obtain such an oriented microstructure by applying a stress to the film during the growth [26]. Light stripes in the figure correspond to sample areas with Y-type twins, while dark areas correspond to X-type twins. The average width of the Y-type twin lamellae is 25 nm and a fraction of Y-type areas to X-type areas of 80% was estimated from large-scale SEM images, as reported in [26].
For the same thin film sample, the hysteresis loops measured along different directions of the MgO substrate are reported in figure 3(b). The easy-magnetisation direction of the system is parallel to the MgO [010] direction, while the [100] direction, i.e. the Y-type boundaries alignment direction, is the hard direction of the system. Along this latter direction, a sharp magnetisation jump was measured between 60 and 30 mT (values measured on the branch from +2T to −2T). The [010] and [100] MgO directions therefore correspond to the maximum and minimum remanence values, respectively. Intermediate remanence values were measured along the [110] direction, which is the other symmetry direction of the system and corresponds to one of the two easy-magnetisation directions occurring in Y-type twins ( figure 1 and [26]).
The occurrence of a magnetisation jump in the first quadrant of the magnetisation curves has been previously attributed, in analogy to what occurs in bulk single crystals, to the magnetic field induced reorientation of twin boundaries [5,24,[26][27][28]34]. In this phenomenon, moving the boundaries between neighbouring twins is more advantageous for the magnetic shape memory crystal than rotating the magnetisation of the system along the field direction, thanks to the high magnetocrystalline anisotropy and strong magneto-elastic coupling [5,7,13,14]. Along with the twin boundary motion, an abrupt change of slope in the magnetisation curve occurs. On the other hand, twin boundary motion has been directly visualized in bulk single crystals [32,33], while in thin films on substrate, also due to the much smaller typical twin sizes, it has never been directly observed to the best of our knowledge.
In order to understand the magnetisation process of the Ni-Mn-Ga film, we have realized micromagnetic simulations for a sample with distribution of the easy magnetisation axes as sketched in figure 3(c). Such a geometrical organization of the easy-magnetisation axes is the one imposed by Y-type twin variants, as described in figure 2(b). Our choice is justified by the experimental observation that Y-zones constitute the largest fraction of martensitic twin variants in the studied sample. The model sample is thus made of alternate stripes with easy-magnetisation axes along the [−110] (green) and [110] (light blue) directions of the MgO crystal, respectively. Stripe width corresponds to twin size. As described in the experimental section, compared to the approaches previously adopted, and in particular to [36], the magnetic and geometrical parameters used in the micromagnetic model were taken from our experimental data on the same thin film (M S , thickness and twin size) or on bulk materials with similar composition (symmetry and value of the magnetocrystalline anisotropy). The exchange stiffness constant A was deduced from a thorough study on inelastic neutron scattering measurements taken on a bulk sample with composition similar to our thin film [42]. Differently from [36], the organization of Y-type twins is in our case unidirectional and we have performed OOMMF simulations applying the magnetic field along all the symmetry directions of the sample.  figure 4, on further decreasing the applied field value, magnetisation decreases due to the fast evolution of domain configuration and abruptly drops between 30 and 20 mT, when a closed flux configuration is reached, which is stable also at zero external field and for small negative field values. In this configuration, the areas with resultant magnetisation along the positive and negative [010] directions are separated by domain walls with moments oriented perpendicular to the film surface.
An animated gif image collecting the micromagnetic states of several points along the hysteresis loop is reported in the supporting information (figure S1 (available online at stacks.iop.org/JPMATER/3/045003/ mmedia)).
The micromagnetic configuration at zero applied field gives an interpretation of the experimental results reported so far about the local magnetic configuration in Ni-Mn-Ga films with Y-type microstructure, as deduced by electron holography and MFM [26,35]. In particular, magnetic microscopy measurements realized in the demagnetised state have been previously interpreted considering magnetic domains identical to the red and light-blue areas in the micromagnetic configurations in figure 4(a), separated by zigzag shaped 180 • domain walls. Moreover, the domain walls appearing in the simulation explain the out-of-plane  [26,35]. Flux-closure domains instead (corresponding to purple and green areas in the micromagnetic configurations) have not been evidenced, to the best of our knowledge, in any experimental work so far. In the simulation these domains remain at zero field in order to minimize the magnetostatic energy in a sample with all Y-type microstructure, while in the experimental works, differently from the simulation, Y-type areas always border X-type areas, which might work as flux-closure zones.
Domain nucleation and expansion occur in the simulation in the field range 50-30 mT, which well corresponds to the experimental field range for the magnetisation jump. The good correspondence between the simulation and experimental measurements indicates that a purely magnetic domain nucleation and expansion process, and not only the field-induced twin boundary reorientation, can give account of the magnetisation jump.
No hysteresis is present in the central portion of the simulated curve, although irreversible processes clearly occur in the real system in the corresponding portion of the loop. In general, for the three symmetry directions of the system, the agreement between experiment and simulation with regard to the coercive field values is not perfect. The irreversibility of the magnetisation process is primarily influenced in the real system by microstructural defects, which are not taken into account in the micromagnetic model. = 0.75, 0.9 for experimental and simulated curves, respectively) can be attributed to the simplicity of the model sample, characterized only by Y-type stripes of constant size that are perfectly aligned. Both the presence of misoriented easy-axes in the Y-type twin areas of the real sample and the contribution of X-type twin areas to the magnetisation process could determine the difference in simulated and measured M r /M S values. X-type twin microstructure was not considered in the simulation for the following reasons. First of all, there is a large predominance of Y-type twin areas, corresponding to an areal fraction of 80% of the sample, as reported before. Typically, X-type films are spontaneously magnetised in the film plane, despite showing perpendicular stripe domains in MFM, which are weak stripe domains since for Ni-Mn-Ga films the quality factor Q = 2K 1 /µ 0 M S 2 is less than 1 [29,45]. All the in-plane directions are equivalently easy in an X-type film, coercive force and M r /M S are around 10 mT and 0.2-0.3, respectively. Given these aspects, we can consider that in the sample under investigation the contribution of X-type areas, which superimposes to the dominant Y-type contribution, is isotropic with respect to the direction of the in-plane field. It is thus reasonable to conclude that, in the real sample, the contribution of X-type twins and of some degree of misorientation of the easy-axes also in Y-type twins, determine a small reduction in remanent magnetisation compared to an all Y-type sample with Y-type twins oriented along one direction. As a further step towards understanding the details of the reversal process, the evolution of both the parallel and transverse components of the magnetisation vector were measured by using vector VSM magnetometry and the results were compared to the predictions of the micromagnetic model (see figure 1 for the measurement geometry). When the external field is applied along the hard [100] MgO direction ( figure  5(a)), the transverse M y /M s signal is close to zero in the whole H-range due to the symmetrical arrangement of the in-plane easy-axes with respect to the [100] MgO direction (insets in figure 5(a)). To break the symmetry, the same measurement was performed by applying the external field at an angle θ of ±5 • to the [100] MgO direction ( figure 5(b)). A structured transverse component emerges, the two curves measured at −5 • and +5 • being symmetric to each other as expected on the basis of the easy-axes configuration of the sample. The slight tilt of the external magnetic field does not affect the M x /M s signal and then the overall magnetisation reversal mechanism, which was investigated by simultaneously studying the evolution of the M x and M y signals. To support the interpretation of the experimental data, micromagnetic simulations were performed under the same conditions, i.e. with the external field applied at angles θ = ±5 • to the [100] direction ( figure 5(c)). The simulated M y /M s signal is symmetric with respect to the [100] MgO direction and both the parallel and transverse components of the magnetisation present characteristic features that are in good agreement with the experimental data. The quality of the simulation is also confirmed by the agreement between experimental and simulated M x and M y curves collected with the external field applied along different directions as described in the supporting information (figures S2 and S3). The main difference between the experimental and simulated curves concerns the trend of the M x and M y signals in the central portion of the loop, where irreversible processes occur in the real system, while a fully reversible behaviour is observed in the simulated curves. With a tilt of +5 • , starting from the positive saturated state, the rapid increase of M y in correspondence to the abrupt drop of the M x signal is followed by a sudden decrease of the M y signal to a negligible value in the simulated curve, while it remains almost constant in the experimental loop before dropping in correspondence of the steep increase of the M x signal in the opposite direction. This discrepancy, as better discussed later in the text, is likely a consequence of the presence of inhomogeneities in the real system that cause irreversible processes in the central portion of the loop.
To analyse in detail the reversal mechanism, we focused on the simulated and experimental curves collected at θ = +5 • . Figure 6 shows the micromagnetic configuration states of representative points in the simulated field-dependent magnetisation loops. Similarly to the case where the field is applied along the   As a further step to improve the comprehension of magnetisation processes in magnetic shape memory thin films, we have extended the comparison between micromagnetic model and experimental magnetisation curves to a film with a different type of microstructure, i.e. a film with Y-type twin lamellae oriented along two orthogonal directions corresponding to the [100] and [010] directions of the MgO substrate. The results are reported in the supplementary material ( figure S4) and confirm the good agreement between magnetometry and micromagnetic results. This specific microstructure resembles the one assumed in [36], but with two areas for each orientation of the Y-type variants instead of one only.
For Y-type twin microstructure, and for the two kinds of twin variant orientations that we have investigated, a pure micromagnetic model is able to give a clear and consistent interpretation of the experimental magnetisation curves and of magnetic microscopy results. The magnetisation jumps occurring when the magnetic field is applied along the alignment directions of Y-type twin boundaries also appear in the simulated curves, showing that the spatial organisation of magnetocrystalline anisotropy axes originated by twinning can give rise to steep magnetisation variations similar to those associated to the MIR effect in bulk single crystals. The magnetic domain configuration shown by micromagnetic simulations explains some of the details of transverse magnetisation measurements and also of previously reported magnetic microscopy results. Further progress towards understanding shape memory effects in Ni-Mn-Ga films could be made by modelling the specific twin microstructures occurring in epitaxial thin films with including also the structural and magnetostructural energy terms in the free energy, as done in [46,47] with the aim of modelling the general magneto-mechanical behaviour of magnetic shape-memory alloys.

Conclusions
We have investigated, in-depth, the magnetisation process of a 200 nm martensitic Ni-Mn-Ga thin film, showing at RT a Y-type twin microstructure with unidirectional alignment. The study has been carried out by vector magnetometry and micromagnetic modelling, introducing the experimental characteristics in the model, and considering different orientations of the applied field with respect to the symmetry directions of the sample. The comparison has been extended to a thin film showing Y-type twin boundaries with two orthogonal orientations.
The main features of the magnetisation curves measured along the film symmetry directions are well reproduced by the model, which is purely micromagnetic, i.e. neglecting structural and magnetostructural contributions to the free energy. The model shows a three-dimensional evolution of the micromagnetic structure during magnetisation reversal, which is consistent with the detailed vector magnetometry characterisation and with magnetic microscopy. The good agreement between experimental results and simulations demonstrates that the spatial organisation of magnetocrystalline anisotropy axes due to martensitic twinning has a dominant effect on the magnetisation process, giving rise to magnetisation jumps when the magnetic field is applied along the alignment direction of Y-type twin boundaries. Along this direction, simulations show that, in order to minimize magnetocrystalline anisotropy and magnetostatic energy, magnetisation reversal proceeds through the formation and three-dimensional expansion of magnetic domains, passing around zero field through a closed-flux domain configuration, in which the domains with resultant opposite magnetisation directions in the film plane are separated by domain walls with perpendicular orientation of the magnetic moments.