530_ftp

previous study we have shown that this value yields a nearly stable TA2 phonon branch.[70] The FSM approach constrains the total moment of the simulation cell, which results in an overall reduction of both, Ni and Mn moments. While with respect to the localized Mn moments this is certainly an oversimplification, it reproduces well the Stoner-like character of the Ni-moments, as these are induced by the interaction with the surrounding Mn.[71] Since the Ni-eg states close to the Fermi-level are deemed to play a dominant role in the phase transformation behavior (for a critical discussion, we refer to the section that concentrates on the band Jahn–Teller effect within this work), one may thus still assume to grasp the essential physics correctly by this simplified treatment. The central diagram of Figure 12 contains the vibrational contributions for three different temperatures. For T ¼ 0 (C) these comprise only the zero point energy, which already gives a noticeable contribution favoring the austenite, although still considerably smaller than the energy difference for the equilibrium moment in the left diagram. At finite temperatures, notably here T ¼ 200 K (D) and T ¼ 300 K (E), the vibrational part also comprises entropic contributions. These also favor austenite and become increasingly important with increasing temperatures. However, adding the vibrational free energy to the ground state magnetization curve (right diagram: A þ D, A þ E) does not yield a phase transformation in the experimentally observed temperature range. Instead, the inversion of the energetic order of the minima is only achieved if both finite temperature effects, reduction of the magnetization and vibrational excitations are taken together (B þ D). Please be aware that the right panel of Figure 12 does not depict the complete free energies, as magnetic and electronic contributions to the entropy are missing as well as cross-coupling contributions (for a more complete discussion of the free energies of austenite, martensite and premartensite, see ref.[56]). Nevertheless, this schematic picture already proves that a magnetoelastic coupling, which is present in this material, contributes significantly to the free energy in favor of the austenite and effectively reduces the onset temperature of the martensitic phase transformation.

Another aspect of the role of e/a becomes apparent when considering the impact of electron addition and depletion in Mn-rich systems. Figure 13 shows the E(c/a) curve and the corresponding magnetic moments of Ni2Mn1.25Sb0.75 for different values of e/a. While the e/a ratio corresponding to this composition is e/a ¼ 8.125, compositional changes were modelled by changing the number of valence electrons in the density functional theory calculations. In this type of calculation, the addition or removal of extra charge is compensated by a homogenous background charge which keeps the system electrically neutral. Our results show that when changing the valence electron concentration, the originally flat energy landscape can be changed into a landscape which contains two minima or a landscape which consists of only one minimum depending on the e/a ratio. At the same time, the total magnetic moment of the cell is decreased or increased in accordance with the supercell calculations for

Fig. 13. The influence of the number of valence electrons used in the density functional theory calculations on the total energy E and the magnetization M for different c/a ratios of Ni2Mn1.25Sb0.75.

different e/a ratios. This underlines that magnetism is indeed responsible for the appearance of martensitic transformations when e/a is increased. In agreement with these results Ye et al. recently showed that the martensitic phase transformation in Ni Mn Sn is related to a band Jahn–Teller effect which originates from the hybridization of Ni 3d minority-spin states with antiferromagnetically coupled Mn states.[72]

In case of Ni2MnGa the situation is somewhat special. Figure 14 shows results of total energy calculations for different valence electron numbers e/a and different c/a ratios for Ni2MnGa. Similar to the other compounds containing In, Sn, or Sb, tetragonal distorted structures become more favored when increasing e/a (to values larger than e/a 8). However, the energy landscape also contains an energy minimum at a tetragonally distorted structure for which the e/a ratio corresponds to the stoichiometric composition (e/a ¼ 7.5).

Fig. 14. The energy landscape of Ni2MnGa defined by the difference E(c/a,e/a)E(1.0,e/a). The total energies have been derived from density functional theory calculations. The volume of the unit cell is fixed to the ground state volume of the

¼ ¼ ˚

L21 structure (e/a 7.5) with lattice constant a0 5.81 A.

This shallow minimum in the energy landscape corresponds to the minimum that is shown in Figure 1 and explains why Ni Mn Ga already favors a martensitic transformation at e/a ¼ 7.5. Between e/a ¼ 7.6 and 7.9 the landscape appears to become flatter, however. This indicates the limited value of such a simple approach, which distributes the extra charge uniformly over all atoms.

The band Jahn–Teller effect is observed for all compositions of the Ni Mn Z alloys which have been considered here. However, it becomes much less pronounced for the offstoichiometric compositions as compared to Ni2MnGa. Figure 15 shows the variation of the electronic DOS (minority spin-channel) under a tetragonal transformation for the case of stoichiometric Ni2MnGa and the Mn-rich composition Ni2Mn1.75Ga0.25. For the stoichiometric composition the band Jahn–Teller behavior can be monitored if one concentrates on the peak which is situated about 0.2 eV below EF at c/a ¼ 1. This peak is splitted and partially shifted to the energy region above EF when increasing or decreasing the c/a ratio, i.e., when

breaking the cubic symmetry. In case of Ni2Mn1.75Ga0.25 a corresponding peak is situated just below the Fermi level at c/a ¼ 1. Again, parts are also shifted to energies above EF as c/a deviates from unity. However, the valley below this peak does not fill up and also the corresponding amount of states above EF decreases with the distortion. This is untypical for the band Jahn–Teller effect and cannot motivate the corresponding changes in total energy. Thus, the band Jahn–Teller effect does not explain the increase of TM with increasing valence electron density and should consequently not be regarded as the origin, but rather as an accompanying feature of the martensitic transformation in the magnetic shape memory composition range, which supports the formation of modulated or adaptive martensite. The latter is a prerequisite of the MSME since it enforces the formation of highly mobile twin interfaces. For a detailed explanation of this interrelation, see the review of Niemann et al. in this issue.[26]

Finally, we want to add a few remarks considering the change of Fermi surfaces of the Ni Mn Z alloys with e/a.

Figure 16 displays the Fermi surfaces of Ni2MnZ (Z ¼ Ga, In, Sn, Sb) for the minority spin channel for different values of e/a using the simple rigid band model. From the full band structures in k-space, energy surfaces belonging to different e/a-ratios can thereby be extracted. The change of e/a compared to the Fermi surface that belongs to EF can be described by

EF

where the prefactor 1/4 arises from the fact that the DOS D(E) refers to one formula unit which contains four atoms. Figure 16 shows that the plane sheets, which are connected

to the nesting features, partly vanish as the valence electron concentration is increased. Another feature, which can be derived from Figure 16, is the similarity of the Fermi surfaces of different materials with identical e/a ratio. This is eye-catching in particular for e/a ¼ 7.5 where the Fermi surface of Ni2MnGa shows pronounced nesting behavior.[27,74] It should be noted that the four different materials, which reveal rather similar Fermi surfaces in the rigid band picture, only differ in the number of p-type valence electrons defined by the Z atoms. Since the p electrons are much less present at the Fermi level than the d electrons the similarities that turn up in the Fermi surfaces are somewhat expected. Nevertheless it has to be noted that also some more

or less pronounced differences between the different materials can be observed in the rigid band picture. For example, the case e/a ¼ 8.0 shows that the Fermi surface of Ni2MnGa differs significantly from the Fermi surfaces of the other materials for this valence electron concentration. In fact, a more detailed description taking into account the antiferromagnetic interactions in the off-stoichiometric alloys is needed for a better understanding of the different contributions, which are responsible for the martensitic transformation. These are neglected by the rigid band approach where we use a band structure calculated for the stoichiometric composition.

The influence of antiferromagnetic interactions on the Fermi surface is visualized in Figure 17 where we plotted the spin-down Fermi surfaces for different compositions of Ni Mn Ga. As the size of the supercell is four times as large as the size of the primitive unit cell the extension of the Brillouin zone is correspondingly reduced. The electronic bands, which are part of the old zone, but outside the extensions of the new one, are therefore folded back. The same applies naturally for the electronic states of the Fermi-level, which define the Fermi surface. In order to allow for a direct comparison, we thus folded the Fermi surface of the

stoichiometric case Ni2MnGa (Ni8Mn4Ga4 supercell) in the same fashion. After addition of manganese to the stoichiometric system the electronic states meeting at the new zone boundary tend to repel each other, which leads to a splitting of the surfaces belonging to different bands. This makes it difficult to unfold the Fermi surface properly in a straightforward fashion.[75] The splitting of the bands also goes hand in hand with a reduction of the parallel parts of the Fermi surface, which are responsible for the nesting in Ni2MnGa. This gives us, apart from the disappearance of the Band Jahn–Teller effect at large e/a another argument against a causal link between Fermi surface nesting and the temperature dependence of the martensitic transformation with e/a. Instead, the antiferromagnetic tendencies, which arise from nearest neighbor Mn Mn pairs in off-stoichiometric Mn-rich compositions, appear to be of such high importance for the martensite transformation temperature. It should be no surprise that the addition of manganese also influences the spin-up Fermi surface significantly, since the spin-up and spin-down Fermi surfaces necessarily approach each other with increasing Mn content as they become identical for the binary NiMn compound being an antiferromagnet.

The situation is rather different for the Co Ni Ga system. Here, similarly strong antiferromagnetic tendencies are not present due to the absence of Mn. Therefore, the addition of Ni for Ga results in an increase of the magnetic moment and of e/a. It has already be pointed out in the past that Fermi surface

nesting and the band Jahn–Teller effect do not appear to play an important role in case of Co Ni Ga.[14]

The different magnetic interactions appearing in Ni Mn Z and Co Ni Ga are related to the crystal structure. While the L21 structure is clearly favored over the inverse Heusler structure for Ni Mn Z alloys, the two structures have approximately the same energy in case of Co Ni Ga alloys. This does not only favor atomic disorder but also strengthens the ferromagnetic interactions.

6. Conclusions

In this work the phase diagram containing the martensitic transformation temperatures of the alloy series Ni Mn Z (Z ¼ Ga, In, Sn, Sb) has been derived by density functional theory. We found that the energy differences, which are obtained for different compositions, can directly be related to TM. This shows that the composition dependence of the

vibrational and magnetic contributions to the entropy in austenite and martensite cancels to a large extent.[18,56,76] Thus,

our results are in good agreement with previously published experimental results.[6] When we apply the method to another alloy system such as Co Ni Ga, we obtain rather similar results as for Ni Mn Z alloys. In both cases, TM increases with the valence electron concentration e/a. This is remarkable as the type of magnetic interaction as well as the details of the martensitic transformation are different for both systems. While antiferromagnetic interactions due to the Mn atoms as well as the band Jahn-Teller effect appear in Ni Mn Z, the system

Co Ni Ga is ferromagnetic and also no band Jahn-Teller effect has been identified in this material in previous studies. A comparison of the minority spin Fermi surfaces under variation of the number of valence electrons indicates that the increase of TM with increasing e/a should not be related to Fermi surface nesting. Although Fermi surface nesting and the band Jahn–Teller effect are still present for the off-stoichiometric compositions of Ni Mn Z, both effects do not appear as pronounced as in stoichiometric Ni2MnGa. In case of Co Ni Ga the magnitude of the energetic preference of the tetragonal structure, which can be tentatively associated with the transformation temperature, does not only depend on e/a, but also on the arrangement of the atoms in the unit cell, i.e., the type of Heusler structure, conventional or inverse. While the inverse Heusler structure shows rather large energy differences between the cubic and the tetragonal structure for all compositions, the increase of the energy differences is linear for the conventional Heusler structure, just as in Ni Mn Z.

Finally, our results describe a possible route for the computational design of materials with martensitic transformation temperatures higher than room temperature by studying various types of atom substitution. This has been exemplified at the Pt-doped Ni Mn Z (Z ¼ Ga, Sn) systems, where it turns out that the addition of platinum is beneficial. Ab initio calculations and experimental measurements suggest that the transformation temperatures of these alloys will be higher compared to systems without Pt while the mechanisms of twin boundary formation is very similar to the one that is observed for Ni2MnGa.

Received: February 15, 2012

Final Version: April 4, 2012

Published online: May 29, 2012

[1]P. Entel, M. E. Gruner, A. Hucht, A. Dannenberg, M. Siewert, H. C. Herper, T. Kakeshita, T. Fukuda, V. V. Sokolovskiy, V. D. Buchelnikov, in Springer Series in Materials Science, Vol. 148 (Eds: T. Kakeshita, T. Fukuda, A. Saxena, A. Planes,), Springer, Berlin 2012, pp. 19–47.

[2]S. Fujii, S. Ishida, S. Asano, J. Phys. Soc. Jpn. 1989, 58, 3657.

[3]A. Sozinov, A. A. Likhachev, N. Lanska, K. Ullakko,

Appl. Phys. Lett. 2002, 80, 1746.

[4]S. Wirth, A. Leithe-Jasper, A. N. Vasilev, J. M. C. Coey, J. Magn. Magn. Mater. 1997, 167, L7.

[5]T. Fukuda, T. Kakeshita, Adv. Mater. Res. 2008, 52, 199.

[6]A. Planes, L. Manosa, M. Acet, J. Phys.: Condens. Matter 2009, 21, 233201.

[7]T. Kakeshita, T. Fukuda, Mater. Sci. Forum 2002, 531, 394.

[8]J. Buschbeck, I. Opahle, M. Richter, U. K. Ro¨ßler,

P. Klaer, M. Kallmayer, H. J. Elmers, G. Jakob,

L.Schultz, S. Fa¨hler, Phys. Rev. Lett. 2009, 103, 216101.

[9]K. Oikawa, L. Wulff, T. Iijima, F. Gejima, T. Ohmori,

A.Fujita, K. Fukamichi, R. Kainuma, K. Ishida, Appl. Phys. Lett. 2001, 79, 3290.