Materials Science and Engineering Axxx (2004) xxx xxx. Magnetically driven shape memory alloys

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1 Materials Science and Engineering Axxx (24) xxx xxx Magnetically driven shape memory alloys J. Enkovaara a, A. Ayuela a,b,, A.T. Zayak c, P. Entel c, L. Nordström d, M. Dube e, J. Jalkanen a, J. Impola a, R.M. Nieminen a a Laboratory of Physics, Helsinki University of Technology, Helsinki, Finland b Donostia International Physics Center (DIPC), Spain c Institute of Physics, Gerhard-Mercator University, Duisburg, Germany d Condensed Matter Theory, Uppsala University, Uppsala, Sweden e Department of Physics, McGill University, McGill, Canada Received 15 June 23; received in revised form 17 October Abstract Significant progress has been made both in experimentation and theoretical modelling of the magnetic shape memory (MSM) effect, where magnetic field can induce strains of 1%. The theoretical models used to analyze and interpret the different experiments provide reliable information and insight into the physical changes involved in the magnetically driven shape memory alloys. The aim of this review is to discuss the presents status of the computational modelling we have done. First, the basic MSM requirements and a brief summary of the experimental results for the prototype material Ni Mn Ga are given. Then, in the context of atomic-scale calculations, we focus primarily on the understanding of the structural variants, magnetic anisotropy, and Curie temperatures. Finally, we discuss modelling related to mesoscopic scales where we develop a phase field model for the description of twins. 24 Published by Elsevier B.V. Keywords: Ferromagnetism; Shape memory; Martensitic transformation; Ternary alloys 1. Magnetic shape memory effect The phenomenon of magnetostriction where an external magnetic field can change the dimensions of the sample was observed already in 1842 by Joule. In normal ferromagnets such as Fe or Ni the strains associated with the magnetostriction are of the order of 1 4 % while materials with exceptionally large magnetostriction, for example Tb Dy Fe alloys (Terfenol-D), show strains of the order of.1% [1]. In contrast, MSM materials can show magnetic field induced strains of 1% [2]. Not only are the strains in the MSM effect two orders of magnitude larger, but also the mechanism is different from ordinary magnetostriction. While ordinary magnetostriction is observed in structurally homogeneous samples, the MSM effect requires a special microstructure. This microstructure is provided by a martensitic transformation. The martensitic transformation [3,4] is a displacive, diffusion free structural transformation from a higher sym- Corresponding author. Tel.: ; fax: address: aay@hugo.hut.fi (A. Ayuela) /$ see front matter 24 Published by Elsevier B.V. 2 doi:1.116/j.msea metry structure (austenite) to a lower symmetry structure 4 (martensite) upon cooling. For example, in Ni 2 MnGa the 41 high symmetry phase is cubic while the lower symmetry 42 phase can be tetragonal or orthorhombic. In order to mini- 43 mize the total shape change (and the macroscopic strain en- 44 ergy) over the whole sample, some microstructure develops 45 in the martensitic phase. A common way to create this kind 46 of microstructure is twinning: because there are usually sev- 47 eral crystallographically equivalent ways to deform the high 48 symmetry structure, the deformation may take different di- 49 rections in different regions of the sample. These structural 5 domains have well defined boundaries and they are called 51 twin variants. A schematic example of the twinning is seen 52 in Fig Twin boundaries are often mobile which is exploited in 54 the temperature driven shape memory effect [5]. Due to the 55 easy movement of the twin boundaries the sample can be 56 deformed easily in the martensitic phase. When the material 57 is heated back to the austenitic phase the sample will recover 58 to its original shape, i.e. it will remember the shape it 59 had before cooling. Even though strains in the temperature- 6 driven shape memory effect can be several percent, the heat- 61

2 2 J. Enkovaara et al. / Materials Science and Engineering Axxx (24) xxx xxx Martensitic transformation Variant II Variant I Twinned microstructure Fig. 1. Schematic illustration of the martensitic transformation and twinning in two dimensions. ing and especially the cooling are relatively slow processes. Therefore a way to drive the shape change with a faster response would be desirable for many applications. This can be achieved by taking the magnetic degrees of freedom into play. Magnetic materials such as ferromagnets, antiferromagnets and ferrimagnets are characterized by local magnetic moments. Also, in the absence of an external magnetic field their magnetisation has a certain preferable direction with respect to the crystal lattice, the so-called easy direction. If the easy direction is parallel to the twin direction, then in a twinned microstructure the lattice orientations of the twin variants are different and therefore the magnetisation directions also differ, as shown in Fig. 2a. When an external magnetic field is applied, the magnetic moments try to align in the field direction. If the energy required to rotate the magnetisation out of the easy direction, the magnetic anisotropy energy MAE, is higher than the energy required to move a twin then, it is energetically more favourable to move the twin boundaries instead of rotating the magnetization. The fraction of twins where the easy axis is in the direction of the field will grow at the expense of the other twin variants. This process results in large shape changes as shown schematically in Fig. 2b. Based on the above discussion the basic requirements for the appearance of the MSM effect can be summarised: The material should be (ferro)magnetic and exhibit a martensitic transformation. The magnetic anisotropy energy must be higher than the energy required to move a twin boundary. Fig. 2. Magnetic moments without the external field. Redistribution of the variants in an applied field. From the point of view of practical applications the material 92 should be in the martensitic phase at room temperature. The 93 strength of the required external field depends on the local 94 magnetic moment of the material, so the magnetic moment 95 should be high. 96 At the moment, the MSM effect has been observed in 97 Fe Pd [6], Co Ni Ga [7], La Sr CuO 4 [8] and Ni Mn Ga 98 [2,9,1] alloys. La Sr CuO 4 is interesting as it is not a ferro- 99 magnet but an antiferromagnet, confirming that the magnetic 1 anisotropy is more important than the macroscopic magnetic 11 moment. For practical applications the most promising ma- 12 terial is Ni Mn Ga. As most of the work presented here con- 13 cerns Ni Mn Ga, some of its experimentally known prop- 14 erties are reviewed next Experimental studies of Ni Mn Ga 16 Numerous studies of Ni Mn Ga alloys have appeared in 17 recent years. Here, a brief summary of the experimental 18 results is given, concentrating on the properties which will 19 be discussed from the theoretical point of view later on. 11 At the martensitic phase transformation, energy is re- 111 leased or absorbed, and this can be measured by differential 112 scanning calorimetry (DSC). Also, the response to the ex- 113 ternal magnetic field changes at the transition, allowing the 114 phase transformations to be observed in magnetic suscepti- 115 bility measurements. These methods allow detection of the 116 occurrence of the transformation and determination of the 117 corresponding temperatures. The advantage of the magnetic 118 susceptibility is that the Curie temperature is also observed 119 clearly as the susceptibility changes from the ferromagnetic 12 to paramagnetic order. On the other hand, DSC measure- 121 ments allow the energetics of the transition to be studied. 122 Some examples of these measurement are shown in Fig Fig. 3. Examples of magnetic susceptibility and differential calorimetry, showing the martensitic and austenitic transformation temperatures M s and A s and the Curie temperature T C. Courtesy of Heczko [11].

3 J. Enkovaara et al. / Materials Science and Engineering Axxx (24) xxx xxx 3 5M NM T 7M NM M s 7M 5M Fig. 5. The stability of the martensitic phases. Relation between the first martensite structure and the transformation temperature M s. Ni Mn Ga Fig. 4. Cubic L2 1 structure with atomic positions of the stoichiometric composition. The martensitic transformation temperatures span over a wide range, from 16 to 62 K [12 16] and they vary strongly with composition as a 1% change in the composition can alter the transformation temperature by 5.In addition to the austenite martensite transformation, up to two intermartensitic transformations can be observed. The Curie temperature is less sensitive to the composition and it is between 32 and 38 K [12,17,18]. The crystal structures of the different phases can be studied with X-ray and neutron diffraction. The high temperature austenitic phase has the cubic L2 1 structure which is shown in Fig. 4, with a lattice constant a L21 = 11.1 a.u. [18,19]. This structure has the fcc Bravais lattice with a four atom basis. As the constituent atoms have similar atomic numbers it can be difficult to distinguish them from the X-ray data. In this case the structure can be interpreted as a bcc structure with a lattice constant half from that of the fcc lattice. Three different martensite structures are observed. Two of them have a basic tetragonal symmetry and one has an orthorhombic symmetry. In the first tetragonal structure the ratio of the c and the a lattice constants is c/a.94 [18]. In addition, there is a shuffling of the atomic planes. The (1 1 ) planes show a modulation in the [1 1 ] direction with a period of five atomic planes and the structure can be designated as 5M [15,2]. The other tetragonal structure has the deformation c/a 1.2 and it can be denoted as non-modulated (NM) because there is no modulation of the atomic planes [15,21,22]. The orthorhombic structure has the lattice constant ratios b/a =.94 and c/a =.89 and it is designated as 7M since it has a seven layer modulation similar to the other tetragonal structure [2,11,15,23]. The volume remains approximately constant during all the transformations. The first martensite which appears on cooling depends on the composition, but the stability of the structures i.e. the order in which they appear on cooling seems to be always the same. This is shown schematically in Fig. 5a. The NM structure is the most stable before the 7M structure [15,21,24]. If the 5M structure is to be observed it is trans- formed directly from the austenite. There is also an em- 164 pirical correlation between the austenite martensite trans- 165 formation temperatures and the first martensite structure as 166 shown in Fig. 5b [16,25]. The alloys transforming directly 167 to the NM structure typically have transformation tempera- 168 tures which can be higher than the Curie point [26,27] and 169 the 7M phase appears first only in a narrow temperature 17 range [2,25,28]. 171 The lattice constant ratios give the maximum strain which 172 is available from the twin rearrangement. This limit is 6% 173 in the 5M structure, 1% in the 7M structure, and more than 174 2% in the NM structure. Up to now, the maximum strain 175 has been realised as magnetic-field-induced both in the 5M 176 [9,29] and in the 7M structures [2]. In the NM structure the 177 MSM effect has not been observed. 178 As neutrons carry a magnetic moment, neutron diffrac- 179 tion can provide information about the value of the local 18 magnetic moments. The total magnetic moment in the sto- 181 ichiometric composition is found to be 4.1µ B per formula 182 unit and it originates mainly from Mn [18,19,3]. Other 183 experimental tool for the determination of the saturation 184 magnetisation is the vibrating sample magnetometer (VSM) 185 also used to obtain the magnetic anisotropy energy. With 186 VSM one measures the magnetisation as a function of 187 the external field, as shown in Fig. 6. By applying this 188 field in different directions with respect to the crystal axis 189 the magnetic anisotropy energy can be determined as the 19 area between the two magnetisation curves. The magnetic 191 anisotropy energy and the easy axis are different for the 192 particular martensites. In 5M, the [ 1] direction (the short 193 c-axis) is the easy axis and the magnetic anisotropy energy 194 is around J/m 3 at room temperature [29,33,34]. 195 In the non-modulated tetragonal structure [ 1] is the hard 196 direction and there is an easy plane with an anisotropy 197 energy of J/m 3 [21,32]. In the orthorhombic 7M 198 structure there are three inequivalent directions. The shortest 199 axis has the lowest energy, the longest axis the highest en- 2 ergy and the energy of the intermediate axis is in-between. 21 The largest energy difference is about J/m 3 22 [2,32]. 23 In the following section the above discussed properties are 24 studied from a theoretical point of view, in some cases also 25 for other materials than Ni Mn Ga. It will be seen that the 26 calculations can predict and explain with simple arguments 27 much of the experimentally observed behaviour. 28

4 4 J. Enkovaara et al. / Materials Science and Engineering Axxx (24) xxx xxx Fig. 6. Magnetisation as a function of the external magnetic field with different field directions in the NM (T in the figure) structure and the 7M structure. Courtesy of Straka [31,32] Theoretical results Theoretical work has been done both on microscopic and mesoscopic scales with a double faceted task. First, we have a clear mission in trying to understand and interpret the experimental results in the light of the basic theoretical principles. The second task is to motivate new experiments so that we can increase the understanding necessary for further progress. Next we give some results about the points that arose during the last years of studies Atomistic calculations All the conclusions presented in this subsection have been obtained by first principles atomistic calculations within the density functional theory (DFT). We will refer to the corresponding publications for further details. As said before, a required property of these materials is that they show local magnetic moment, which gives the possibility to obtain the magnetic shape memory effect in them. In a brief course to atomic magnetism, there are two main quantities that describe most of the magnetic properties. The first parameter is the exchange which is connected with the magnetic moment, and implicitly considered always in this section unless stated otherwise. The second quantity is the magnetic anisotropy which is studied later in Section Structural properties As many properties of these alloys are based on the structure, a first question to be answered is what are the reasons which drive them to these structures. By calculating the total energy for various structures the possible martensitic phases can be identified as energy minima. In refs. [35,36] several Heusler alloys have been studied from this aspect concentrating in tetragonal and orthorhombic structures. None of the studied materials show energy minima for orthorhombic structures, but there are some minima for tetragonal deformations. Ni Mn Ga alloys show rubber like behaviour because a tetragonal transformation is allowed in contrast to other Heusler alloys. In Fig. 7 we show the total energy of 244 the L2 1 structure of Fig. 4 versus the tetragonal distortion as 245 measured by the ratio of c and a lattice constants. There are 246 minima at c/a =.94, 1. and 1.25 which correspond to ex- 247 perimentally observed structures. The appearance of energy 248 minima is ascribed to the electronic structure which is de- 249 scribed in further details in refs. [35,36] as band Jahn Teller 25 effect. However, the relative energetic depth of the minima is 251 different from the stability of the martensitic phases shown 252 in Fig Theoretically, a tetragonal crystal structure is stabilised 254 around c/a =.94 with respect to the L2 1 structure when, 255 in addition, modulation shuffles with a period of five atomic 256 planes is taken into account [37]. This is in agreement with 257 the observed structures in experimental works. The modu- 258 lation appears to be critically important for stability of the 259 tetragonal structure with c/a < 1. Here it is found that 26 the optimum amplitudes of the modulation vary in different 261 atomic planes. 262 The transition from the tetragonal variant with c/a >1 to 263 the cubic L2 1 phase is driven by vibrational entropy when 264 finite temperature effects are included [38]. A more detailed 265 study about the vibrational properties of the L2 1 structure 266 shows a complete softening of the transverse acoustic mode 267 E (mry) c/a Fig. 7. Total energy of Ni 2 MnGa as a function of the tetragonal distortion.

5 J. Enkovaara et al. / Materials Science and Engineering Axxx (24) xxx xxx has been found around the wave vector q =.33(2π/a) along the [1 1 ] direction [39]. The softening of this TA 2 phonon mode leads to the premartensitic modulated super-structure which is observed also experimentally. Further phonon anomalies, related to other structural transformations in Ni 2 MnGa, have also been found and examined. However, these anomalies appear to be due to the coupling of particular acoustic and optic phonon modes. Some microstructures can also be studied because they are accessible with the actual computing power. We have simulated here a [1 1 ] twin in the atomistic scale for the variant with c/a.94. Preliminary results from this simulation give a twin formation energy of 31 mj/m 2, which is between the ones in Ni and Cu Magnetic order We are interested here in the coupling between the magnetism and the lattice which is described by the magnetic anisotropy energy. Some dependences for the magnetic anisotropy energy have been drawn in Fig. 8 described in ref. [38,4] in more detail. Around c/a = 1, the magnetic anisotropy in Fig. 8 follows mostly a linear trend and changes its sign. Also the angular dependence shows an almost perfect uniaxial behaviour as shown in Fig. 8. However, there can be several twins in the martensitic phase which may complicate the interpretation of the experiments. Simple estimate for this effect is obtained by averaging over two twins whose easy axes are perpendicular to each other [38]. This enables us to explain the puzzling experiments in ref. [41] which reports values for the anisotropy constants K 1 = 1.3µeV and K 2 = 1.1µeV. According to our interpretation this points clearly to an average over different twins with K 1 = 179µeV and K 2 = 1µeV which is in agreement with other experiments. A ferromagnetic material loses its long range magnetic order and becomes paramagnetic at the Curie temperature. This temperature is therefore one of the key parameters determining the operation range of the MSM effect. From a more fundamental point of view the Curie temperature is a MAE (µev) c/a E (µev) θ Fig. 8. Magnetic anisotropy energy as function of the tetragonal distortion c/a and of the magnetisation direction θ. E (mry).5 1 [ q] [q q q].5 1 [q q ] Fig. 9. Total energy as a function of the spiral vector q in the units of 2π/a. ( ) Ni 2 MnAl, ( ) Ni 2 MnGa. basic property for the theoretical understanding of a ferro- 36 magnetic material. The decrease of macroscopic magnetisa- 37 tion with increasing temperature is caused by longitudinal 38 and transverse fluctuations of magnetic moments. It is as- 39 sumed that transverse excitations of magnetisation are the 31 dominant origin for the Curie temperature in transition met- 311 als because large local magnetic moments are shown to 312 exist above the Curie point [42,43]. These transverse excita- 313 tions are quantised as magnons. In ref. [44] the magnon re- 314 lated properties of Ni 2 MnGa and Ni 2 MnAl are studied with 315 total-energy calculations of spin spirals. The spin spiral is 316 a magnetic configuration where the magnetisation direction 317 varies with a well-defined period. The period is determined 318 by the wave vector q. The total energy as a function of the 319 spiral vector q is shown in Fig. 9 for the high symmetry 32 directions [ 1], [1 1 1] and [1 1 ]. 321 It is seen that the energies are very similar both in 322 Ni 2 MnGa and in Ni 2 MnAl. For small values of q the en- 323 ergy grows quadratically, but the dispersion flattens for

6 6 J. Enkovaara et al. / Materials Science and Engineering Axxx (24) xxx xxx c/a MAE (µev) n v n v Fig. 1. Tetragonality and the MAE for c/a =.94 as a function of the number of valence electrons n v. The stoichiometric composition corresponds to n v = 3. Experimental values from ( ) ref. [34], ( ) ref. [29], ( ) ref. [46] larger q especially in the [1 1 ] direction. The theoretical spin wave stiffness D obtained from the total energies, ω q = Dq 2, shown in Fig. 9 is D = 77 mry a.u. 2 and it is in good agreement with the experimental value of 79 mry a.u. 2 measured in Ni Mn Ga films [45]. Also the Curie temperature can be estimated from the calculated energy dispersions in Fig. 9 [44]. We obtain the Curie temperature 48 K in good agreement with the experimental one 38 K Role of composition and ageing process It was clear on the first steps of this research that the key properties of these alloys showed a marked composition dependence. At a first step the composition dependence can be studied within the rigid band approach which considers only the change in the number of valence electrons. As seen in Fig. 1, the c/a ratio changes in an appreciable way for the variant with c/a > 1. The experimental trend of decreasing MAE with increasing electron concentration is reproduced also by the rigid band approximation + Mn Ga Ni Extra Mn µ total (µ Β ) as seen in Fig. 1. The inclusion of electronic structure is 343 needed in order to explain these trends, for details see Refs. 344 [38,4]. 345 In ref. [47] we have gone beyond the rigid band ap- 346 proximation and studied the alloying effects with supercell 347 calculations of Ni 2 Mn 1.25 Ga.75. Experimentally, composi- 348 tions close to Ni 2 Mn 1.25 Ga.75 have good MSM properties 349 as large strains are obtained around room temperature. This 35 composition is obtained by replacing one Ga atom by an Mn 351 atom in the 16-atom supercell, as shown in Fig. 11. The 352 main difference in the supercell approach to the actual ex- 353 perimental composition is that in the supercell the extra Mn 354 atoms are perfectly ordered while in the real material they 355 can be distributed randomly in the Ga sites. The most impor- 356 tant result obtained from the calculations is that the magnetic 357 moments of the extra Mn atoms favour anti-ferromagnetic 358 alignment with respect to the neighbouring Mn atoms. The 359 magnetic moment of a Mn atom is approximately a con- 36 stant 3.5µ B regardless of the direction of the moment or 361 electron concentration. Hence, the antiferromagnetic order e/a Fig. 11. L2 1 supercell of Ni 2 Mn 1.25 Ga.75. Saturation magnetisation vs. the average number of valence electrons per atom (e/a). The dashed line is a linear fit to the experimental data and the solid line is the theoretical prediction assuming a 5% Ni content

7 J. Enkovaara et al. / Materials Science and Engineering Axxx (24) xxx xxx ing of the extra Mn means that every additional Mn reduces the total moment by 3.5µ B. The magnetic moment of Ni 2 Mn 1+x Ga 1 x can be described with a simple model as µ total = 2µ Ni + (1 x )3.5µ B, where the Ni moment µ Ni varies around the stoichiometric value of.µ B with the electron concentration according to the rigid band results. As seen in Fig. 11 the model describes well the experimental variation of the magnetisation. From this agreement it can be concluded that the Mn atoms substituted at Ga sites are antiferromagnetically coupled to the Mn atoms at Mn sites. The extra Mn has also important consequences for the appearance of the martensitic phases. With the antiferromagnetic alignment, the structure with c/a =.94 is stabilised and it has 5 mev lower energy per formula unit than the cubic structure. In addition, an energy minimum appears for the orthorhombic structure with lattice constant ratios c/a.93 and b/a.97. The energy of the orthorhombic structure is between the energies of the two tetragonal minima, so that the theoretical order of phases agrees with the experimental findings discussed in Fig. 5b of Section 2. On the other hand ageing has to due with the ordering of atoms between the different sub-lattices present in the compounds [48]. A basic ordering process is the atomic exchanges involving only two atoms. For Ni 2 MnGa stoichiometry the Mn Ga, Mn Ni and Ga Ni atomic exchanges constitute the whole spectra to be taken into account. The energetically most favourable exchange pair corresponds to Mn Ga sublattices with an energetic cost of about 1 K per-formula-unit. This energy difference is in agreement both with the annealing temperature for the thermal treatment in the experimental samples as well as with the onset of disorder phases [49]. These results encourage further experiments. An example is the measurement of saturation magnetic moment before and after thermal treatment, because the Mn atom in the Ga sites align antiferromagnetically when large Mn Ga disorder is present. Also it seems interesting to look at the change, induced by local order, of the X-ray peaks during ageing. h twin 3.2. Mesoscopic modelling 42 Based on our previous atomistic results we proposed a 43 Ginzburg Landau model to describe the behaviour of a twin 44 under a magnetic field for the case of Fig. 12. Our ex- 45 pression for the free energy is given by: 46 f = f [e] + f [θ] + f [e, θ] + c( e) 2 + d( θ) 2 (1) 47 and in more detail when c = d = as for the homogeneous 48 phase, 49 f = a 2 e2 e4 4 + e6 6 hµ cos (θ) + k ap(e) sin (θ) (2) 41 where as in refs. [5,51], the relevant order parameter in 411 one dimension is the strain e, and we add also the angle θ 412 of the magnetic moment with the applied field. This model 413 includes: (i) in the three first terms f [e] an elastic contribu- 414 tion together with the magnetic contribution due to strain, 415 which can be fitted to curve of Fig. 7 in the c/a < 1 region. 416 (ii) in the magnetic part the interaction with the field (Zee- 417 man term), f [θ], and (iii) in the so called magnetoelastic 418 part the rotating cost of the magnetic moment f [e, θ]. In the 419 following the magnitude of µ is considered together with 42 the field, and is dropped to µ = 1. In (iii) the turning cost is 421 given by the anisotropy energy that depends on the uniaxial 422 anisotropy constant k a and the strain in a linear way around 423 c/a = 1. With these order parameters we construct several 424 models for the term p(e) involving the magnetic anisotropy 425 and the strain, which can be fitted to linear or higher order 426 polynomials around c/a.94 variant from the data given 427 in Fig Anyhow, upon application of a magnetic field, the vari- 429 ants around a twin boundary become non-equivalent, be- 43 cause one of them will rotate its magnetic moment, as seen 431 in the horizontal regions of Fig. 12. This asymmetry will 432 have also consequences around the twin, for instance in 433 the twin shape after certain time of applying the field. For 434 large enough values of the uniaxial magnetic anisotropy, 435 and the magnetic field the variants can become so asym- 436 metric that the so-called Zeeman energy becomes equal to h/k a another minima= austenite? elastic+magnetoelastic=magnetic Fig. 12. Twin scheme. Different regions in the order parameter diagram as function of field h and anisotropy constant K a (see text). K a

8 8 J. Enkovaara et al. / Materials Science and Engineering Axxx (24) xxx xxx the other energetic terms, long dashed line in Fig. 12. In such case a more structured twin can appear, which can give rise to the nucleation of other phases in more complex models. Our actual code is ready to deal with microstructures in two dimensions where following ref. [5] the elastic compatibility conditions have been included. Here the analysis of twin patterns will allow us to address a new kind of problems. 4. Conclusions In this review, simulations for magnetic shape memory alloys are presented. On the atomistic level, structural properties and phonon dispersions as well as magnetic properties such as magnetic anisotropy energy and Curie temperature are described accurately. We are also able to identify the basic mechanisms behind these properties. Some conclusions about the role of constituent atoms in stoichiometric Ni 2 MnGa can be given. The magnetic moment originates mainly from Mn, but regarding the other properties investigated here, Ni is more important. This is related to the fact that most of the electronic states near the Fermi-level are due to Ni. The Mn-states are at lower energies making Mn less important concerning both structural and transition properties as it can be seen in refs. [35,36]. The fact of having several elements in order to form ternary alloys rises the question about the role of the sublattices. This issue brings into attention two new aspects to be taken into account. First, when we move out of the ideal L2 1 stoichiometry, it means that we are doping some of the sublattices which bring new characteristics for example in the form of different magnetic orderings. Second, the atoms can jump between the sites corresponding to different substructures, although the composition remains constant as in ageing and annealing processes. In addition, superseded to these atomistic effects, the twins can assemble and form more complicated patters. At this point, we require to address this question within longer length scales beyond the atomistic approach. A phase field model including both magnetic field as well as elastic and magnetic parameters has been introduced for the research in the mesoscopic scale. Last but not least, we wish to remark that these alloys are promising for industrial applications up to now. Acknowledgements This work has been supported by the Academy of Finland (Centers of Excellence Program 2 25), by the National Technology Agency of Finland (TEKES) and the consortium of Finnish companies (ABB Corporate Research Oy, AdaptaMat Oy, Metso Oyj, Outokumpu Research Oy), and by the Graduate School Structure and Dynamics of Hetero- geneous Systems of the German Science Council (DFG). 487 Computer facilities of the Centers for Scientific Computing 488 (CSC) Finland and Research Centre Juelich are greatly ac- 489 knowledged. 49 References 491 [1] A.E. Clark, in: E.P. Wohlfarth (Ed.), Ferromagnetic Materials, vol , North Holland, Amsterdam, 198, p [2] A. Sozinov, A.A. Likhachev, N. Lanska, K. Ullakko, Appl. Phys. 494 Lett. 8 (22) [3] Z. Nishiyama, Martensitic Transformation, Academic Press, New 496 York, [4] D.A. Porter, K.E. Easterling, Phase Transformations in Metals and 498 Alloys, Van Nostrand Reinhold, New York, [5] H. 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