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1. Introduction

Martensitically transforming ferromagnetic Heusler alloys form an important class of materials since they exhibit a number of application-relevant functional properties such as the magnetic shape memory effect (MSME), or the magne-

[*]M. Siewert, Dr. M. E. Gruner, Dr. A. Hucht, Dr. H. C. Herper, Dr. A. Dannenberg, Prof. P. Entel

Faculty of Physics

University of Duisburg-Essen and CENIDE 47048 Duisburg, Germany

E-mail: mario@thp.uni-due.de Dr. A. Chakrabarti

Raja Ramanna Centre for Advanced Technology Indore 452013, Madhya Pradesh, India

Dr. N. Singh, Dr. R. Arro´yave Department of Mechanical Engineering Texas A&M University, College Station Texas 77843, USA

tocaloric effect (MCE).[1] Among the materials that have been investigated in the past, Ni2MnGa is one of the most promising compounds,[2] showing strains of about 10% in the martensite phase in magnetic fields of less than 1 T.[3] Since the MSME depends on the presence of a modulated martensitic phase which displays predominantly ferromag-

[**]M. S., M. E. G., and P. E. would like to thank S. Fa¨hler, T. Hickel, J. Neugebauer, S. Ener, J. Neuhaus, W. Petry, S. M. Shapiro, S. R. Barman, M. Wuttig, and M. Acet for helpful discussions. Work at Duisburg-Essen is supported by the Deutsche Forschungsgemeinschaft within the Priority Programme SPP 1239. R. A. and N. S. would like to acknowledge the National Science Foundation (Grant No. DMR-0844082 through IIMEC as well as Grant No. DMR-0805293) for the financial support provided. R. A. and N. S. acknowledge Texas A&M Supercomputing Facility for computational resources provided. M. S., M. E. G., and P. E. acknowledge the use of the supercomputing hardware kindly maintained by Center for Computational Sciences and Simulation (CCSS) at University of Duisburg-Essen and the John von Neumann Institute for Computing (NIC) and Ju¨lich Supercomputing Centre (JSC) at Forschungszentrum Ju¨lich.

This paper was amended in issue 8 of Advanced Engineering Materials because there was a mistake in the Early View publication.

netic order, it is limited by the martensitic 10M and 14M and magnetic transition temperatures. While stoichiometric

Ni2MnGa has a Curie temperature that is higher than room temperature (TC 380 K),[4] the martensitic transition temperature is much lower (TM 202 K).[5] The latter can be improved by switching to Mn-rich off-stoichiometric compositions.[6] For application in many sectors still higher operating temperatures and a higher ductility will be needed. Therefore, an intensive search for new materials with transformation temperatures higher than room temperature and better elastic and mechanical properties has been undertaken during the last decade. In this context, a variety of different compounds have been targeted, ranging from the Heusler Ni-based magnetic shape-memory alloys (MSMAs) like Ni Mn Z (with Z ¼ Ga, In, Sn, Sb),[6] Fe-based alloys like Fe Pt or Fe Pd,[7,8] Co-based alloys like Co Ni Al[9] and Co Ni Ga,[10–15] to Cu-based alloys like Cu Mn Ga,[16] and also quaternary alloys such as Ni Mn In Co[17] and Pt Ni Mn (Ga,Sn).[18] The necessity of expanding the repertoire of shape memory alloys (SMAs) has recently been highlighted by Ma and Karaman.[19] The requirement of better workability and higher operation temperature addresses all MSMA based on Heusler structures as well.

In the search for improved functional applicability, we need to determine the characteristics of the phase-transformations over a broad range of compositions as was recently done for the quaternary Ti Ni Cu Pd thin films in the search for SMA with near-zero thermal hysteresis and functional stability.[20] Similar theoretical work with application to the MSMA in mind requires the prescription of how to deal with the complexity of structural and magnetic phase transitions in a simplified way. We have recently shown how first-principles calculations can be used to design new SMAs by calculating phase diagrams based on composition dependent energy differences.[18] This is a convenient and rather quick method which can be used to avoid high experimental costs associated with the fabrication of samples. In this contribution we pursue the idea by describing how the phase transformation characteristics and related changes of magnetic properties can be derived from ab initio calculations. We focus the discussion on the MSMA series Ni Mn Z with Z ¼ Ga, In, Sn, and Sb. Further results refer to the Co Ni Ga alloy system and quaternary Ni-based Heusler alloys containing Pt. The investigations culminate in a general discussion of the electronic features, which stabilize the austenite and martensite phases relevant for the magnetic shape-memory behavior.

2. Experimental Characterization of

(Ni, Co) Mn (Ga, In, Sn, Sb)

Among the alloy series Ni Mn (Ga, In, Sn, Sb), Ni48.8Mn29.7Ga21.5 shows the largest magnetic field induced strain (MFIS), which is of the order of 10% at room temperature.[3] The martensitic transformation temperature TM quickly decreases when approaching the stoichiometric composition while simultaneously a premartensitic 3 M

phase appears below 340 K. The premartensitic phase can be manipulated under compressive stress and even disappears in some off-stoichiometric samples. Chernenko et al. have shown that in single crystals with composition Ni52.6Mn23.6Ga23.8 and in crystals of similar composition a sequence of martensitic transformations exists between austenite (L21) and martensite (L10) having modulated monoclinic structures (with monoclinic angle b) 10 M ð32Þ2 or 14 M ð52Þ2,[21] which may be compared with the nonmodulated tetragonal structures and can be linked to the adaptive features of the martensitic twin-structure. This appears on the nanoscale due to the complex interaction between elastic strain energies

and interfacial energies between austenite and martensite variants.[22–25] For a detailed discussion see the review by Niemann et al. in this issue.[26]

The appearance of the martensitic phase in Ni2MnGa is accompanied by a softening of the acoustic TA2 branch along the [110] direction in the phonon spectrum. In particular, ab initio calculations carried out at 0 K exhibit imaginary frequencies along this direction.[27] Supporting experimental evidence comes from neutron scattering measurements at finite temperature which reveal that the phonon dispersion becomes increasingly softer at a specific wave vector when approaching the (pre)martensitic transformation temperature.[28] The temperature dependence of the softening is however not only determined by the structural transformation but depends also on the magnetic transition temperature.[29] The softening is often interpreted as a precursor phenomenon of the (pre)martensitic transformation in the system and the investigation of dynamical properties in terms of phonon anomalies gives a first hint of the transformation path into the martensitic phase in Ni2MnGa and other SMAs.[30] The origin of the softening has been related to Fermi-surface nesting, since the phonon wave vector associated with the softening coincides with the nesting vector of the Fermi surface of the minority spin-channel of Ni2MnGa.[31]

The substitution of atoms largely affects the martensitic transition temperature TM as well as TC.[32,33] A common

approach, which is used to increase the martensitic transition temperature of Ni2MnZ, is the partial substitution of the Z element by Mn. However, due to the fact that Mn atoms necessarily become nearest neighbors in this case, antiferro-

magnetic tendencies are introduced, leading to a decrease of the magnetic moment.[34,35] In a recent effort, the experimental

phase diagrams for stoichiometric and Mn-rich compositions of Ni Mn Z (Z ¼ Ga, In, Sn, Sb) have been compiled from many sources.[6] In this work, we show that the linear dependence of the transformation temperature on the valence electron concentration e/a which was observed experimentally can be related to the increasing number of antiferromagnetically aligned magnetic moments of Mn with increasing e/a. The effect of Mn excess in Ni2MnZ (Z ¼ Ga, In, Sn, Sb) leads to competing antiferromagnetic interactions which we studied in some detail by ab initio calculations. This study has been performed using supercells with 16 atoms in order to model the off-stoichiometric compositions. The supercells

allow discrete changes of the composition in steps of 6.25 at.% by substituting a single Mn atom at the Z sites. The central idea is to relate the structural energy differences for these compositions which appear due to tetragonal distortions to the martensitic transformation temperatures while neglecting vibrational and magnon contributions (to the entropy) in a first approximation. This simple procedure allows us to model the experimental phase diagram of Ni Mn Z (Z ¼ Ga, In, Sn, Sb) and gives information about the influence of the valence electron concentration e/a on the martensitic transformation temperature TM. It also allows us to point out the important role of antiferromagnetic interactions on TM. We have carried out similar studies for the Co Ni Ga alloy system which has also been considered as a ferromagnetic SMA in the past.[10,11] Due to the absence of manganese in this compound, antiferromagnetic interactions are not prevalent in Co Ni Ga. The same simple way of considering the structural changes as a function of e/a has also been used to

predict the phase diagrams for new types of materials such as (Ni,Pt) Mn Z (Z ¼ Ga, Sn).[18] The addition of

platinum enforces the antiferromagnetic tendencies which help to stabilize the martensitic phase and is responsible for increasing the martensitic transformation temperature.

Our discussion will proceed in the following way. Section 3 contains information about the parameters which have been used in the calculations and the codes employed in the study of the materials. In Section 4 we present results of the variation of the total energy as a function of tetragonality, the magnetic order and the transition temperatures for Ni Mn Z, Co Ni Ga, and Ni Pt Mn Z. The results are compared to the experimental trends which are available from literature. In Section 5 we critically discuss the influence of the valence electron concentration on the martensitic transformations as well as the role of magnetic interactions and electronic structure on the appearance of tetragonal transformations.

3. Computational Details

Our total energy calculations were carried out using density functional theory as implemented in the VASP code,[36,37] using PAW pseudopotentials[38] and the generalized gradient approximation (PBE) as exchange-correlation functional.[39] For all the compounds an energy cutoff of 459.9 eV was used (except Pt2MnGa where we used 353.4 eV).

The calculations for the stoichiometric compositions were performed by using 15 15 15 k-points[40] and the tetrahedron method with Blo¨chl corrections.[41] The supercell calculations of off-stoichiometric compositions were performed by using a 10 10 10 k-point mesh. The Fermi surfaces were calculated for the stoichiometric compositions

In the case of supercell calculations the E(c/a)-curves were calculated by keeping the volume constant corresponding to the ground state volume of the cubic structure. After the determination of the E(c/a)-curve for constant volume the cell

volume was relaxed again for each c/a-ratio which corresponds to a minimum in the E(c/a)-curve. The magnetic

exchange interactions presented here have been calculated using the Munich SPR-KKR code.[42] For the phonon dispersions the PHON code from Dario Alfe` and a supercell size of 4 4 4 (3 3 3 in case of the free energy calculations) was used.[43]

4. Computational Results

4.1. Ni Mn Z

The martensitic transformation in Ni2MnGa was first discussed by Webster et al. in 1984.[44] Density functional theory calculations of the ground state reveal that Ni2MnGa possesses a nonmodulated martensitic structure with a tetragonal distortion at c/a ¼ 1.25 at very low temperatures.[45,46] Although the MSME that appears in Ni2MnGa is related to the more complex 5Mor 7M-phases, it has recently been suggested that these modulated structures can be understood as rearrangements of the nonmodulated structure by allowing for twin boundaries on the nanometer scale[24] in accordance with the concept of adaptive martensite.[25] The intrinsic instability of the cubic structure with respect to a tetragonal distortion has been motivated in terms of the band Jahn–Teller effect.[47] Regarding how far phonon softening influences the details and especially dynamical aspects of the transformation still requires further microscopic investigation, for instance, by using the time dependent density functional theory concept. The accompanying reconstruction of the electronic density of states (DOS) curves has been experimentally verified by photoemission experiments.[48,49]

Density functional theory calculations of the ground state energy for different tetragonal distortions along the Bain path from the bcc-type to fcc-type structure provide information about the ground state of the corresponding material. If the global energy minimum of the total energy curve, E(c/a) (where c/a is the ratio between the two different axis parameters of the crystal cell) appears at c/a ¼6 1, the ground state corresponds to a tetragonally distorted martensitic structure. The energy difference between the tetragonal and the cubic structure with c/a ¼ 1 can be regarded at first glance as the energy which is needed to transform the tetragonal structure to the cubic one, and is therefore tentatively related to the transformation temperature.

In order to give an impression of the importance of tetragonal distortions on the total energy of Heusler alloys, we show in Figure 1 the energy as a function of tetragonality for the stoichiometric compositions of Ni2MnZ (Z ¼ Ga, In, Sn, Sb). Since the stoichiometric compounds do not require supercells, we can evaluate the energy surface for each system as a function of tetragonality and volume with feasible computational effort. The phase diagrams compiled by Planes et al. show that Ni2MnGa is the only compound of the alloy series Ni2MnZ (Z ¼ Ga, In, Sn, Sb) which undergoes a martensitic transformation at stoichiometric composition.[6]

Fig. 1. Total energy of the stoichiometric compounds Ni2MnGa, Ni2MnIn, Ni2MnSn, Ni2MnSb as a function of the c/a ratio (tetragonal distortion). Each of the curves represents an intersection of the corresponding energy surface with the curve of vanishing pressure. Thus, the volume in this representation is not fixed.

The total energy curves which are plotted in Figure 1, validate this result on a theoretical basis since only Ni2MnGa (e/a ¼ 7.5) shows a pronounced energy minimum at c/a > 1.

In order to model the phase diagram for the whole e/a region of interest, calculations of off-stoichiometric compositions are needed. In this work, we applied the supercell approach to model the disordered compositions. To keep the computational effort tractable, supercells consisting of only 16 atoms were used. The Ni atoms are known to occupy one of the two simple cubic sublattices of the L21 Heusler structure already at high temperatures after cooling down from the melt and transforming to the B2-structure.[50] Therefore, the occupancy of the Ni sublattice was not changed when modeling off-stoichiometric compositions by substituting atoms.

The valence electron concentration e/a can be increased

either by adding Mn or Ni to the stoichiometric composition in the experimental phase diagram.[6,46] As the former yields larger martensitic transformation temperatures, we mainly stick to this case, here. It should be noted that both types of

substitutions affect the number of 3d valence electrons in the system while the increase of e/a by changing the Z element changes the number of p-type valence electrons. We considered three off-stoichiometric Mn-rich compositions for each alloy system, namely Ni2Mn1.25Z0.75 (Ni8Mn5Z3 supercell), Ni2Mn1.5Z0.5 (Ni8Mn6Z2), and Ni2Mn1.75Z0.25 (Ni8Mn7Z1). In addition, we calculated the binary alloy NiMn (e/a ¼ 8.5) where all of the Z atoms have been replaced by Mn.

Since the Ni sites were fixed, the number of possible configurations is small in our limited supercell. However, as the Mn atoms have a non-vanishing magnetic moment, the alignment of the atomic spins introduces an additional degree of freedom to the configuration space. Therefore, for each composition, the ground state volume was calculated for different types of magnetic orderings considering a ferromagnetic type and different types of antiferromagnetic orderings which are commensurate with the respective lattice.

It turns out that for all compounds with excess Mn except the binary alloy NiMn, the energetically favored magnetic ordering is of antiferromagnetic nature. In all cases, two types of Mn atoms have to be considered. In particular, the additional Mn atoms that occupy the Z sites order antiferromagnetically with respect to the Mn atoms on the original Mn sites of the stoichiometric alloys. The antiferromagnetic exchange can be explained by the smaller distance between these two types of Mn atoms. Compared to the Mn Mn distance in the stoichiometric composition the distance between the Mn atoms on the original Mn sublattice and those on the Z sublattice is reduced by 29%. The appearance of antiferromagnetic interactions naturally goes hand in hand with a decrease in magnetization. Figure 2(a) shows the behavior of the magnetization as a function of e/a. When taking into account a composition with excess Ga, in particular, Ni2Mn0.75Ga1.25 (Ni8Mn3Ga5 supercell), the crossover from ferroto antiferromagnetism can be well observed in the Slater–Pauling type of curve in accordance with previous results for this composition.[51] Since the magnetic moments of the Mn atoms in Heusler compounds like Ni2MnGa can be considered as localized,[52,53] the

Fig. 3. Total energy as a function of c/a for different compositions of Ni Mn Sb. The volume was fixed to the groundstate volume of the respective composition during the calculations. The two different curves for Ni2Mn1.5Sb0.5 (denoted as ‘‘para’’ and ‘‘ortho’’) originate from the fact that due to the limited size of the supercell two crystallographic inequivalent directions appear in the supercell. In particular, the c-axis can be aligned parallel or orthogonal to the plane which is defined by the remaining two Sb atoms in the supercell at this concentration.

decrease of magnetization varies linearly with the number of additional Mn atoms in the supercell and therefore also linearly with e/a. This relationship is plotted in Figure 2(b).

After the determination of the energetically favored magnetic state, the behavior of the total energy under a variation of c/a was calculated for the specific magnetic ordering. For all alloy series it was observed that tetragonally distorted structures become favored with increasing e/a ratio. This is also in agreement with results for Ni Mn Al from Bu¨sgen et al.[54] Figure 3 shows the influence of excess Mn on the total energy curves in case of Ni Mn Sb. The two different curves that appear in case of Ni8Mn6Sb2 can be traced back to the fact that due to the finite size of the supercell, which limits the number of possible configurations, two

crystallographic inequivalent directions appear in the supercell, while for all other compositions all three crystallographic axes are symmetrically equivalent. In particular, the tetragonal distortion can be applied in-plane or perpendicular to the plane which is defined by the two remaining Z atoms in the case of Ni8Mn6Z2. In all of these cases, both types of distortions were taken into account and the martensite structure with the lower energy was considered when determining the structural energy difference.

From the behavior of the total energy under a tetragonal distortion, the structural energy differences between austenite and martensite, i.e., the energy difference between c/a ¼ 1 and the minima of the energy curves at c/a > 1, can be calculated and associated with a temperature scale in order to get an estimate of TM. The binding energy surfaces reveal that volume changes in the relevant c/a interval are small (less than 1%) for the stoichiometric cases of the Ni Mn Z compounds. For the off-stoichiometric compositions the optimal volume was determined for all c/a values corresponding to a minimum of the E(c/a)-curve as well as for c/a ¼ 1. In this case, the ground state energies corresponding to the so obtained ground state volumes of the different c/a-ratios were then used to calculate the structural energy differences.

The structural energy differences for all considered compositions allow us to obtain an approximate theoretical phase diagram which is shown in Figure 4. In agreement with the experimental findings (see ref.[6]) the transformation temperature increases nearly linearly with e/a. The increasing slopes of the lines with increasing mass of the Z element which is found in the experiment can also be reproduced using our rough approximation. The binary alloy NiMn also fits into the scheme as extrapolated point of all other curves. However, it is to be noted that in the case of binary NiMn the considered antiferromagnetic order, where the Mn atoms on the Z site

align antiferromagnetically with respect to the Mn atoms on the original Mn sublattice, is not the energetically favored magnetic order. In particular, another type of antiferromagnetic order, where the magnetic moments of the Mn atoms are aligned layerwise in an fct lattice has been considered as the ground state of the binary compound.[58]

The remaining deviation in the absolute values of transformation temperatures between theoretical and experimental data in Figure 4 can be traced back to different reasons. One of this reasons is that the small supercells used here are not suitable to model statistical disorder. The substituted atoms are periodically repeated and do not experience on average the environment of real disordered samples. Another point is that defects and impurities that might appear in the alloys, are also not considered in the calculations. Note that the composition with Ga excess (Ni2Mn0.75Ga1.25) can be discussed along the same lines. In particular, the minimum which appears at c/a ¼6 1 is shifted to energies lying above the minimum which appears at c/a ¼ 1 for this composition. Thus, at e/a ¼ 7.25 no martensitic transformation is predicted theoretically for the Ni Mn Ga system. The addition of Mn results in a reduction of the lattice parameter. Thus, TM increases with decreasing volume which is in agreement with experimental results.[59] Atomic disorder influences in particular the magnetic configuration. For the alloy series Ni Mn (Ga, In, Sn, Sb) the antiferromangetic

tendencies become stronger when passing from Ga to In, Sn and Sb. Ab initio results for the case of Ga and In are shown in Figure 5.

The replacement of Ga by Mn has similar effects as the replacement of Mn by Ni in Ni Mn Ga. Both types of substitutions increase the valence electron concentration while at the same time TC is decreased and TM is increased. The phase diagram of Ni2 þ xMn1 xGa is shown in Figure 6.

4.2. Co Ni Ga

Another interesting class of magnetic Heusler materials are the Co Ni Ga alloys where manganese has been replaced by nickel which suppresses the antiferromagnetic interactions found in Ni Mn (Ga, In, Sn, Sb) alloys. Furthermore the presence of Co insures sufficiently large magnetic moments so that these Heusler alloys may be further candidates for materials with multifunctional properties. Recent calculations predict a competition of two different types of crystal ordering, which can be both represented in a four-atom primitive cell. This can be tentatively interpreted as a preference for a partially disordered type of structure (B2-Structure) in agreement with experimental findings.[13,14,61] These crystal orderings are the conventional L21 Heusler structure and the so called inverse Heusler structure. The latter structure can be obtained from

Fig. 6. Phase diagram of Ni2 þ xMn1 xGa where TP denotes the transition temperature of the premartensitic phase. Curie-temperatures that have been measured experimentally are marked as brown triangles, while theoretical values obtained by Monte Carlo simulations are denoted as black open circles. Theoretical data adapted from ref.[60] Experimental phase diagram taken from ref.[46]

the conventional Heusler one by exchanging one of the atoms occupying the X site with either the Y or the Z atoms. This results in the fact that the X atoms become nearest neighbors. The inverse structure is denoted as XY(XZ) in distinction to X2YZ for the conventional one. The inverse structure was already discussed in 1978 for the case of Fe2CoGa.[62] Iron and cobalt based Heusler alloys have also been investigated by our group using density functional theory calculations in the past.[63,64] A computational combinatorial study of inverse Heusler structures was recently published by Gillessen and Dronskowski.[65] The authors observe that the inverse structure is energetically favored over the conventional one for many Heusler systems.

For Co Ni Ga alloys, the interesting question is whether in the same way as for the Ni Mn Ga alloys, an increase of the structural energy differences between austenite and martensite structures with increasing e/a is found although no predominant antiferromagnetic interactions appear in this material. In order to probe this, we used a similar approach as for Ni Mn Z, i.e., we modeled off-stoichiometric

compositions by substituting single atoms in a supercell with 16 atoms. In case of Co Ni Ga the gallium atoms were substituted by nickel atoms in order to stepwise increase the valence electron concentration from e/a ¼ 7.31 to 8.19. Due to the competition of conventional and inverse Heusler structures we considered both, conventional and inverse structures when calculating energy differences, see Table 1. In case of Co Ni Ga the substitution of atoms is connected with some issues that do not appear in the case of Ni Mn Z. Especially for the inverse structures the substituted atoms are also favoring the Co sites although Co is not directly affected by the substitution. Therefore, the number of possible configurations is larger. A second issue is related to the fact that atomic relaxations are more important in this material than in the Ni Mn Z system.

In Table 1 calculated structural and magnetic properties for various compositions are gathered. The fact that no dominating antiferromagnetic interactions appear for increasing e/a can be extracted from the magnetic moments of both the conventional and the inverse Heusler structure, which increase with e/a. For all compositions which have been investigated so far, the magnetic moments are larger for the inverse structure compared to L21. This means that a larger magnetic moment is observed, when the Co atoms become nearest neighbors. Table 1 also contains the energy differences between the two different types of crystal orderings for the cubic case, c/a ¼ 1. For all considered compositions the inverse structure is favored over the conventional Heusler structure in this case.

The result of total energy differences between austenite and martensite, i.e., between cubic and tetragonally distorted lattices is shown in Figure 7. Our results clearly show that also in case of Co Ni Ga, tetragonally distorted structures become favored with increasing e/a. The inverse Heuslerstructure exhibits larger energy differences between cubic and tetragonal structures for practically all investigated e/a ratios compared to the conventional arrangement. However, a clearly linear dependence of the energy differences on e/a as found for the Ni Mn Z system is only reproduced for the conventional structure. Our results are in agreement with results from Li et al. for Ni Fe Ga where an increase of TM was observed when Ni was added to the system in exchange for Fe.[66]

Although increasing energy differences between austenite and martensite structures with increasing valence electron concentration are observed for both, Ni Mn Z and

4.3. Ni Pt Mn Z

In the search to increase TM without loosing essential features of the MSME we have investigated various other alloy systems. An interesting choice is the partial replacement of Ni by Pt. This isoelectronic substitution keeps the e/a ratio of 7.5

for L21 Ni2MnGa to 6.05 A for NiPtMnGa and 6.23 A in the

case of Pt2MnGa.

We would like to remind that the magnetic order of Pt2MnGa was already discussed in 1981 by Uhl.[67] In this context it was found to be antiferromagnetic with TN ¼ 75 K. In contrast, our ab initio calculations using the gradient corrected PBE exchange-correlation functional as implemented in the VASP code[36,37] rather indicate that the ferromagnetic state is energetically preferred over an antiferromagnetic configuration consisting of alternating ferromagnetic layers. These results suggest that the compound is either ferromagnetic or has a more complex antiferromagnetic structure, which cannot be represented within our rather small supercell. Compared to Ni2MnGa the ferromagnetic state of Pt2MnGa has a slightly increased magnetic moment of M ¼ 4.14 mB/f.u. which originates mainly comes from the Mn moments which increase to 3.75 mB per atom while the Pt moments are significantly smaller than the Ni moments (0.12 mB).

Figure 10 shows the E(c/a)-curves of Ni2 xPtxMnGa for different values of x. The figure shows that the minimum at c/a > 1, which is related to the nonmodulated tetragonal martensite structure, becomes more and more pronounced as Pt is added to the system. At the same time, a second

minimum for c/a < 1 emerges as the amount of platinum is increased. Even for the compositions containing only 6.25 and 12.5 at.% Pt the energy difference between the L10 and L21 structures significantly increases. The magnetic moment of quaternary NiPtMnGa amounts to M ¼ 4.16 mB/f.u. which is larger than that of Ni2MnGa and is also larger than the magnetic moment of Pt2MnGa. This behavior can be explained by the two opposing trends which appear when Ni is replaced by Pt. On the one hand, the platinum atoms have a smaller induced magnetic moment compared with Ni which leads to a decrease of the total magnetization. On the other hand, the volume is increased due to the Pt which goes hand in hand with an increase of the localized Mn moments. The largest magnetization was obtained for Ni1.5Pt0.5MnGa (Ni6Pt2Mn4Ga4 supercell) (M ¼ 4.17 mB/f.u.).

The phase diagram in Figure 4 also contains the estimated martensitic transformation temperatures of NiPtMn1 þ xZ1 x (Z ¼ Ga, Sn) for 0 x 1. As in the case of Ni Mn Ga the transformation temperature increases linearly with e/a while simultaneously antiferromagnetic tendencies appear which decrease the magnetization. In fact, the transformation temperatures are higher than for the systems without Pt which may be explained by the softness of the magnetic lattice which allows for higher TM. At the same time, also the magnetization increases when substituting Ni by Pt. This suggests, that the Ni Pt Mn Z alloys are suitable candidates for future magnetic shape memory devices.

This statement is underlined by the observation that the

same type of nanotwin boundaries form in the monoclinic ð52Þ2 unit cell corresponding to c/a < 1 as in Ni Mn Ga.[26]

Fig. 11. The 14M structure of Pt2MnGa which can be constructed from the tetragonally distorted structure with c/a ¼ 0.86 when allowing for a monoclinic distortion with an angle of 83.58 and two nanotwin boundaries. The two-dimensional rectangle inserted into the figure is nearly identical with the unit cell of the nonmodulated tetragonal martensite at c/a ¼ 1.31. The appearance of the nanotwins and the similarity of the 14M

structure with the tetragonal structure has been discussed in the frame of adaptive martensite.[8,18,24,25] Figure adapted from ref.[57] and created using VESTA.[68]

Also for the Pt enriched alloys the ð52Þ2 structure is close in total energy to the nonmodulated tetragonal martensite. This implies that the formation energy of twin boundaries is rather low and the generation of an adaptive phase is likely. The 14M-structure of Pt2MnGa, which is plotted in Figure 11, is very similar to the corresponding structure found for NiPtMnGa.

5. Origin of the Martensitic Transformation

In agreement with previous calculations and experimental observations, our results obtained for the Ni Mn Z alloys show that tetragonal transformations become energetically favored (corresponding to increased transformation temperatures) when e/a is increased by adding Mn to the systems. In this context it should be noted that TM does not depend on e/a alone. In particular, Ni2MnGa is the only compound that undergoes a martensitic transformation at stoichiometric composition, while neither the isoelectronic Ni2MnIn nor Ni2MnSn and Ni2MnSb do undergo a martensitic phase transition at stoichiometric composition although the e/a ratios of these compounds are even larger for the latter two. Instead, the linear increase of the energy differences between austenite and martensite structures might rather be related to the linear decrease of the magnetization due to antiferromagnetic interactions introduced by increasing e/a when adding Mn to the system. This behavior is expressed in the resemblance between Figure 2(b) and the behavior of the structural transformation temperatures with e/a in Figure 4. The observation that TM is determined by the magnetic interactions in the material is in agreement with a recent thermodynamic investigation from first principles that suggests indeed that the martensite structure in Ni2MnGa is stabilized by magnetic interactions while the high

temperature austenite phase is stabilized by lattice vibrations i.e. phonons.[18,55] Figure 12 disentangles the different

contributions to the free energy as a function of tetragonality c/a. The total energies (left diagram) obtained directly from our DFT calculations yield a tetragonal ground state (A). This is also true, if we model a reduction of the magnetization which is encountered at finite temperatures by the so-called fixed spin moment (FSM) approach,[69] although the difference in energy between austenite and tetragonal martensite is considerably reduced (B). Here, we model an average finite temperature reduction of the magnetization by 10%. In a