Journal of Alloys and Compounds 632 (2015) 822–829
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Effect of substitution on elastic stability, electronic structure and magnetic property of Ni–Mn based Heusler alloys: An ab initio comparison Tufan Roy a, Markus E. Gruner b, Peter Entel b, Aparna Chakrabarti a,c,⇑ a b c
HBNI, Raja Ramanna Centre for Advanced Technology, Indore 452013, India University of Duisburg-Essen, D-47048 Duisburg, Germany ISUD, Raja Ramanna Centre for Advanced Technology, Indore 452013, India
a r t i c l e
i n f o
Article history: Received 18 November 2014 Received in revised form 21 January 2015 Accepted 28 January 2015 Available online 4 February 2015 Keywords: DFT Elastic constants Density of states Curie temperature Heusler alloy
a b s t r a c t First-principles density functional theory based calculations have been used to predict the bulk mechanical properties of magnetic shape memory Heusler alloy Ni2MnGa substituted by copper (Cu), platinum (Pt), palladium (Pd) and manganese (Mn) at the Ni site. The elastic constants of Ni2MnGa alloy with and without substitution are calculated. We analyze and compare in detail the bulk mechanical properties for these alloys, in particular, the ratio between the calculated bulk and shear modulii, as well as the Poisson’s ratio and Young’s modulii. This analysis further based on an empirical relation, indicates that Pt2MnGa may inherently be the least brittle material, among the above-mentioned alloys. Interesting difference has been observed between the shear modulii calculated from Voigt’s and Reuss’s method. This 0 has been explained in terms of the values of the tetragonal shear constant C of the materials. Study of Heisenberg exchange coupling parameters and Curie temperature as well as density of states of the materials shows the effect of substitution at the Ni site on the magnetic and electronic properties, respectively. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction One important signature of shape memory alloys (SMAs) is a thermoelastic transition from a high temperature cubic austenite phase to a low temperature martensite phase of lower symmetry. Most of these SMAs also show interesting electronic and magnetic properties [1–5]. Magnetic shape memory alloys (MSMAs) are considered to be technologically important following the discovery of 10% magnetic ﬁeld induced strain (MFIS) in the martensite phase of Ni2MnGa [1–3], and 4% MFIS in the martensite phase of Mn2NiGa . Besides MFIS, magnetocaloric effect (MCE) [7,8] and inverse MCE [9–12] are observed in some of these MSMA including Ni2MnGa and Mn2NiGa, suggesting possible practical applications of these well known Heusler alloys. Partial or complete substitution of atoms by various elements or changing of relative concentrations of manganese (Mn), gallium (Ga) and nickel (Ni) of these well known compounds can help in tuning of their various properties [13–30]. Changes in the properties by partial replacement of Mn by copper (Cu) as well as the effect of the Cu-substitution at
⇑ Corresponding author at: HBNI, Raja Ramanna Centre for Advanced Technology, Indore 452013, India. E-mail address: [email protected]
(A. Chakrabarti). http://dx.doi.org/10.1016/j.jallcom.2015.01.255 0925-8388/Ó 2015 Elsevier B.V. All rights reserved.
the Ga and Ni site in Ni2MnGa with small to substantial degree of substitution have already been studied in the literature both from theoretical and experimental methods [19–29]. Substitution up to 100% at different sites in both Ni2MnGa and Mn2NiGa has been studied from ﬁrst principles calculation, the substituent being a Cu atom . It has been predicted that large substitution of Cu at Ni site, leads to a more stable austenite phase, rendering a martensite transition impossible unlike the cases of substitution at Mn or Ga sites. Recently, a maximum MFIS of 14% has been predicted for Pt doped Ni2MnGa (composition Ni1.75Pt0.25MnGa)  on the basis of ﬁrst-principles calculations. Neutron diffraction study establishes the existence of modulated structure in Ni1.8Pt0.2MnGa, which is a prerequisite for a large MFIS in small magnetic ﬁelds . It is also shown very recently  that Ni1.8Pt0.2MnGa exhibits sizable MCE near room temperature. A large increase in martensite transition temperature has also been reported earlier for Pt-doped Ni–Mn–Ga . Hence, by studying the literature it appears that, Pt-doped Ni2MnGa is expected to have similar or better properties in comparison to Ni2MnGa for technological applications. The disadvantage related to poor mechanical properties of Ni2MnGa in terms of its technological suitability has been already addressed in the literature and studies aiming to improve the
T. Roy et al. / Journal of Alloys and Compounds 632 (2015) 822–829
ductility of SMA alloys have been carried out earlier . The elastic properties are amongst the most important physical properties relating to the structure and mechanical stability of materials. These properties are also associated with some other fundamentally important properties, e.g. hardness, speciﬁc heat and Debye temperature. There are two elastic modulii important for applications, namely, bulk modulus and shear modulus. First-principles calculations can give reliable values of the elastic constants and modulii of materials. In the literature, there are some studies of ﬁrst-principles calculations of elastic and related properties of materials, including Heusler alloys [34,35]. However, in spite of the outstanding, both experimental and theoretical, work on the magnetic Heusler alloys and their functional properties in conjunction with the structural martensitic transformation, relatively less studies have been carried out on the elastic properties of these alloys. In this paper, we carry out in depth study of the bulk mechanical stability in terms of the elastic stability criteria  and elastic constants of Ni2MnGa, substituted at the Ni site by Cu and isoelectronic elements Pt as well as Pd. Though the material parameters at a larger length scale, such as grain boundaries, structural defects like vacancies and dislocations inﬂuence the mechanical properties substantially, the non-directional metallic bonding is thought to contribute signiﬁcantly to the ductility of most of the metallic solids. In crystalline materials in the cubic phase a high value of ratio of shear and bulk modulii gives a fairly good phenomenological indication of the inherent brittleness of the material . This inherent brittleness is probed here using the empirical relationships proposed by Pettifor  and Pugh . Both Cu and Pt are highly ductile materials. Further, because Pt and also Pd are isoelectronic with Ni while Cu has only one electron more than Ni, we expect interesting results if we substitute Ni with Pt, Pd and Cu. Hence, we study and compare here the elastic properties of Ni2MnGa and its substituted alloys. We also study the electronic structures in terms of the density of states (DOS). In addition, we evaluate the magnetic properties of these alloy materials in terms of the Heisenberg exchange coupling parameters from which we derive the Curie temperature (TC) following the literature . In what follows, ﬁrst, we give a brief account of the methodology used in this work and then we present our results on elastic, electronic and magnetic properties of the above-mentioned materials along with the related discussion. Finally, the results of our work are summarized in the last section.
2. Method Ni2MnGa is known to assume an ordered X2YZ type (conventional Heusler alloy structure) of structure while Mn2NiGa has a XYXZ type (inverse Heusler alloy structure) of structure (X, Y typically elements with d-electrons and Z typically elements with s, p electrons). In the austenite phase, Ni2MnGa has cubic L21 structure that consists of four interpenetrating face-centered-cubic (fcc) lattices with origin at fractional positions (0.25, 0.25, 0.25), (0.75, 0.75, 0.75) (0.5, 0.5, 0.5), and (0.0, 0.0, 0.0). In the following, we label these sub-lattices as A, B, C and D, respectively. In L21 Ni2MnGa, the Ni atoms occupy the A and B sub-lattices. Mn and Ga occupy the C and D sub-lattices, respectively. To simulate Cu, Pt and Pd substitution at Ni site in Ni2MnGa, we have replaced the Ni atom at the respective site by the substituent atoms. We also compare some results when Cu substitutes Mn or Ga atoms in Ni2MnGa. Commonly used structures, the conventional Heusler and inverse Heusler alloy structures have been tested for all these materials and conventional structures are found to be lower in energy. On the other hand, for Mn2NiGa, we have considered the inverse Heusler structure as in our earlier work [13,41]. Hence, Mn2NiGa has one Mn atom at C sub-lattice (fractional positions with origin at 0.5, 0.5, 0.5, referred to as MnMn), while the other Mn atom (referred to as MnNi) occupies fully one of the Ni sub-lattices (B) of Ni2MnGa. The rest of the atoms Ni and Ga in Mn2NiGa occupy the same fractional atomic positions as in Ni2MnGa, the A and D sub-lattices, respectively. For comparison, we list here the energy differences between the inverse and conventional structures. The conventional Heusler alloy structure of Ni2MnGa, Ni2CuGa, Cu2MnGa, Ni2MnCu, Pt2MnGa, Pd2MnGa are lower in energy by 459 meV/f.u., 299 meV/f.u., 49 meV/f.u., 285 meV/f.u., 761 meV/f.u., 661 meV/f.u. respectively. For Mn2NiGa inverse Heusler alloy structure is lower in energy by 423 meV/f.u. compared to its conventional counterpart.
For the ﬁrst principles calculations of the geometry optimization of these substituted systems, we employ the Vienna Ab Initio Simulation Package (VASP)  in combination with the projector augmented wave (PAW) method  and the generalized gradient approximation (GGA) of Perdew, Burke and Ernzerhof (PBE) for the exchange–correlation (XC) functional . GGA yields a better agreement with experimental data for most of the Heusler alloy systems compared to the results of local-density approximation (LDA). We have used an energy cut-off of minimum 460 eV for the planewaves and the convergence of the results has been tested. The ﬁnal energies have been calculated with a k-mesh of 9 9 9 for the cubic case. The number of k-points for the self-consistent ﬁeld (SCF) calculations has been tested up to a mesh of 13 13 13. The energy and the force tolerance for our calculations were 10 leV and 10 meV/Å, respectively. The equilibrium lattice constants are obtained by geometry optimization using VASP package  and these agree well with literature wherever data are available [13,6,18]. The electronic stability of the substituted and parent materials has been conﬁrmed by calculating the mixing energy for each system as described in our previous work . The optimized geometries of the systems studied are compared with the results obtained in the literature, wherever results are available, and these agree well with literature. The bulk modulus has been calculated from the second derivative of the total energy, by using the Murnaghan’s equation . The elastic constants of a material describe its response to an applied stress or, the stress required to maintain a given deformation. Both stress (r) and strain () have three tensile and three shear components, giving six components in total. The linear elastic constants form a 6 6 symmetric matrix, which has 27 different components, such that ri ¼ C ij j for small stresses, r, and strains, . Elastic constants of all the materials are calculated from the second derivative of the energy with respect to the strain tensor. For the calculation of elastic constants, we have used the software package CASTEP (Materials Studio 6.1) . It involves the variation of total energy of the system induced by the strain. We use ultrasoft pseudopotential and PW91 GGA over the LDA for the exchange–correlation functional . For ﬁne calculations, required for response properties, the number of k-points and the energy cut-off have been increased from the values used in SCF calculations till the convergence of the mechanical properties of each individual material has been achieved. Maximum mesh of k-points has been taken as 17 17 17 and energy cut-off has been increased up to 1500 eV according to the requirement. After the calculation of elastic constants, the elastic modulii have then been calculated from these constants for each material. The Debye temperatures have been calculated from the averaged sound velocity which is dependent on longitudinal and transverse elastic wave velocities of the material which are in turn derived from the bulk and shear modulii . The optimized geometries have been calculated using CASTEP  as well and have been veriﬁed with the results obtained from VASP  and these two match very well. For magnetic properties, the results from all-electron calculations are typically very reliable for systems containing ﬁrst row transition metal atoms. Hence, we have carried out relativistic spin-polarized all-electron calculations on the optimized geometries of the substituted and unsubstituted systems using full potential linearized augmented planewave (FPLAPW) program  with PBE GGA for the XC functional . We have also used this method for calculating the electronic density of states (DOS). For obtaining the electronic properties, the Brillouin zone (BZ) integration has been carried out using the tetrahedron method with Blöchl corrections . An energy cut-off for the plane wave expansion of about 16 Ry is used (RMT K max ¼ 9:5). The cut-off for charge density is Gmax ¼ 14. The number of k-points for the self-consistent ﬁeld (SCF) cycles in the reducible (irreducible) BZ is about 8000 (256) for the cubic phase. The convergence criterion for SCF calculation of the total energy Etot is about 0.1 mRy per atom. The charge convergence is set to 0.0001. To gain detailed insight into the magnetic properties of the systems, we calculate the Heisenberg exchange coupling parameters. We use the Spin-polarized-relativistic Korringa–Kohn–Rostoker method (SPR-KKR) to calculate these exchange coupling parameters, Jij, within a real-space approach, which is proposed by Liechtenstein et al.  and implemented in SPR-KKR programme package . Fullpotential method has been used for the SCF calculations and the results of the same have been used further for calculating the exchange parameters. The number of kpoints for the SCF cycles has been taken as 500 in the irreducible BZ. The angular momentum expansion up to lmax = 3 has been taken for each atom. TC is derived from the Heisenberg exchange coupling parameters using mean-ﬁeld-approximation (MFA) . Convergence of the TC has been tested with respect to the cluster radius, where the cluster radius (Rij ) is the radius of a cluster of atoms (j) surrounding a central atom, i.
3. Results and discussions 3.1. Elastic stability of substituted Ni2MnGa First we discuss the mechanical stability of the cubic austenite phase of Ni2MnGa and its substituted alloys. For cubic lattices, there are only three independent elastic constants, C 11 ; C 12 and C 44 , where, from symmetry, we have the following conditions:
T. Roy et al. / Journal of Alloys and Compounds 632 (2015) 822–829
C 11 ¼ C 22 ¼ C 33 ; C 12 ¼ C 23 ¼ C 13 and C 44 ¼ C 55 ¼ C 66 . Table 1 lists all the relevant elastic constants along with the lattice parameters calculated for all the materials studied here using the CASTEP programme package . The optimized geometries are found to be very similar to the geometries obtained using the VASP programme package . It is well-known that the elastic constants C 44 , C 0 ¼ 0:5 ðC 11 C 12 Þ and C L ¼ 0:5 ðC 11 þ C 12 þ 2C 44 Þ can be directly measured from experiments and are related to the TA1, TA2 and LA accoustic phonon modes. Here we compare our calculated data with the experimental results wherever available in the literature. For Ni2MnGa, we observe that the values of C 11 ; C 12 and C 44 agree quite well with the available experimental values of Worgull et al.  and Stenger et al. . However, these results are different from the data obtained from an earlier experiment of Vasilev et al.  and the issue of this mismatch has already been addressed in detail by Worgull et al. . For Mn2NiGa also, we observe that the values of C 11 ; C 12 and C 44 agree well with the available experimental values of Jian-Tao et al. . For cubic crystal it is well known that the elastic stability criteria are as follows : C 11 > 0; C 44 > 0; ðC 11 C 12 Þ > 0; ðC 11 þ 2C 12 Þ > 0. From Table 1, it is observed that while most of the criteria are satisﬁed for all the materials, for some of them, the 3rd one is not, which then leads to a negative C 0 value for those (Ni2CuGa, Cu2MnGa, Mn2NiGa and Pt2MnGa). For the remaining materials, the computed value of C 0 , the tetragonal shear elastic constant, is positive but quite close to zero, indicating that, for all the materials studied here, the cubic austenite phase tends to become unstable with respect to a tetragonal distortion. To this end, calculation of elastic constants for the tetragonal martensite phase for the same materials can be interesting to compare. In a further work , we plan to discuss these results and compare the same with available results from the literature . Now we discuss the two isotropic mechanical parameters, which are also important modulii for applications, namely, bulk modulus and shear modulus. The bulk modulus, B, which is a ratio of volume stress and volume strain and represents the resistance to fracture. It is connected to the elastic constants as follows: B = (1/3) ⁄ (C 11 + 2 C 12 ). The calculated values of B have been listed
in Table 1. The isotropic shear modulus, G, which is the ratio of shearing stress and shearing strain, is related to the resistance of the material to the plastic deformation. Typically, G is calculated as an average value of the shear modulii given by formalisms of Voigt (GV )  and Reuss (GR ) , which means G = (GV + GR )/2. In calculating the G, Voigt made an assumption that homogeneous strain is maintained throughout the stressed sample while calculating the modulii. On the contrary, Reuss’s calculation of the modulii was based on the assumption that homogeneous stress is maintained throughout the stressed sample in all directions. However, Hill  later concluded that the true values of the shear modulii would lie in between the two values given by Voigt  and Reuss  and can be considered as an average of the two. The argument given by him is that the samples cannot be in equilibrium under the assumption of constant strain or constant stress. In many cases the difference between the values given by these two methods is within the experimental accuracy. Hence, although there is no particular justiﬁcation given in the literature for using the Hill’s averaging method, it is generally used and, by and large, it seems to work well. Analytical expressions of GV and GR in terms of elastic constants are given as: GV = (1/5) ⁄ (C 11 C 12 + 3 ⁄ C 44 ) and GR = (5 ⁄ C 44 ⁄ (C 11 C 12 ))/(4 ⁄ C 44 + 3 ⁄ (C 11 C 12 )). Table 1 lists all the GV and GR values for all the materials. We observe that the GV values agree well with the experimental values (which are derived from the experimental elastic constants C 11 ; C 12 and C 44 for both Ni2MnGa and Mn2NiGa). The GR values are found to differ considerably from the GV values. In all cases studied here, including the parent compounds, GV is found to have reasonable value, but the value of GR is very different which is contrary to the general trend where the two values are very close: while GV gives the higher limit, GR sets the lower limit. We note that GV and GR being close to each other is a general trend which is observed in the literature including for predominantly half-metallic or semiconducting full and half-Heusler alloys [66–69]. Therefore, we probe now the reason behind the large underestimation of GR values for all the compounds studied here. A well-known observation is: for stable cubic crystals, typically the value of C 11 is 2–2.2 times greater
Table 1 Calculated bulk mechanical properties.a
Lattice parameter (Å)
5.82 5.81b, 5.82c
5.87 5.843b, 5.907e
Bulk modulus (B)(GPa)
158.78 161b, 146d, 170g, 156h, 147i
114.76 114b, 115.51f
Shear modulus (GV) (GPa)
66.26 63.6d, 61.4j, 57.96k, 80.4l
Shear modulus (GR) (GPa)
6.19 10.56d, 14.78j, 18.44k, 77.69l
162.20 152.0d, 156j, 139.4k, 213l, 153m
108.72 103d, 103i, 98j, 91k, 92l
157.07 143d, 143j, 122.6k, 87l, 148m
48.35 40d, 45j, 31.6k, 5l
C0 (GPa) Poisson’s ratio
2.57 4.5d, 14i, 6.5j, 8.4k, 63l, 2.5m
Comparison done with experiments or previous calculations, wherever data are available as given: b Ref. . c Ref. . d Ref. . e Ref. . f Ref. . g Ref. . h Ref. . i Ref. . j Ref. . k Ref. . l Ref. . m Ref. .
T. Roy et al. / Journal of Alloys and Compounds 632 (2015) 822–829
than each of the remaining constants. But from Table 1, it can be noted that the values of C 11 are only slightly different from those of C 12 and this small difference is either positive or negative for all the alloys. Further, as a result of this small difference between C 11 and C 12 , the constant C 0 is seen to possess either a negative value or a value close to zero when positive. From the expressions of the shear modulii, it is seen that the difference between C 11 and C 12 (= 2 ⁄ C 0 ) comes as a multiplicative factor in the numerator in the expression of GR , unlike GV . Hence, a small value of C 0 manifests itself in bringing down the value of GR . On the contrary, 2 ⁄ C 0 comes only as an additive factor in the numerator of the expression of GV , so a large dominance of the C 44 term in the expression yields reasonable values of GV for all the compounds. Since the experimental GV values (derived from experimental elastic constants) agree with our calculated data, we consider only the GV value as the shear modulus for further consideration, though it is typically expected to be the higher limit of the same. We note here that since in our further discussion, we are only interested in the relative values or in the trend of different mechanical properties, the slightly higher value of the shear modulus is not expected to affect the analysis of the following results. We now focus on the value of Cauchy Pressure, C p , which is deﬁned as C p ¼ C 12 C 44 . According to Pettifor , a simple relationship exists between the sign of C p and the metallicity of the materials. A large negative value of C p indicates strong covalent bonding in the material, or in other words, represents a more directional character of the bonds. On the other hand, increasing positivity of this term means increasing non-directional bonding as in metals. It is found that all these materials have positive C p values. Further, along this line, there is another simple and empirical relationship, proposed by Pugh  which says that the plastic property of a material is related to the ratio of the shear and bulk modulus of the material. A high value (greater than 0.571) of ratio of shear and bulk modulus, namely, G/B, is connected with the inherent brittleness of a crystalline material. By analyzing the GV /B values from Table 1, we ﬁnd that Pt2MnGa is expected to be inherently less brittle than Ni2MnGa. The GV /B ratio of the former turns out to be 0.17, which is much lower than all the substituted alloys, including the parent compound Ni2MnGa. It is indeed interesting to note that the GV /B value of Pt2MnGa is comparable or lower than the GV /B value of metal Cu and Pt, and slightly higher than that of Au as is found from the standard literature. The Poisson’s ratio is calculated from the expression, m = 0.5 ⁄ (3 ⁄ B/G 2)/(3 ⁄ B/G + 1). A value smaller than 0.33 is considered to be associated with an inherently more brittle material. As per the GV /B as well as the m values (calculated using B/GV values), Mn2NiGa seems to be inherently more brittle compared to
the other materials. Further, in Fig. 1, we plot all the available data for C p versus GV /B calculated in case of Ni2MnGa and the partially and fully substituted compounds. We ﬁnd that overall, there is a clear trend of inverse (linear) relationship between GV /B and C p . In the literature also, it is observed that, the higher the C p , the lower the ratio G/B; in particular, a nearly-linear inverse correlation between the Cauchy pressure and the G/B ratio is well established for various related compounds . Along with the alloys studied, we have also plotted the C p and GV /B values for Cu bulk calculated by us in Fig. 1, for comparison. 3.1.1. Comparative study of Ni2MnGa and Pt2MnGa Since recent studies indicate the importance of Pt2MnGa from studies related to MFIS and MCE, we probe its elastic properties in detail while comparing the same with the parent compound Ni2MnGa. From Table 1, it is evident that Pt2MnGa has a very high Poisson’s ratio and Cauchy pressure as well as a very low GV /B value when compared to Ni2MnGa. Further, the calculated value of Young’s modulus for Pt2MnGa is found to be 85.15 GPa, while Ni2MnGa has a value of 181.01 GPa. This is consistent with the trend of the comparative values of C p and Poisson’s ratio as well as GV /B values of the two compounds (see Table 1). Furthermore, we compare the Debye temperature (HD ) values calculated from the elastic constants for Ni2MnGa and Pt2MnGa. For the former, the HD value has been obtained to be 411 K (which is close to the value 419 K obtained from the phonon density of states  calculated using CASTEP programme ). This value, however, is somewhat higher from the reported experimental value of 261 K  and earlier theoretical result of 323 K . Our calculated HD value for Pt2MnGa is observed to be lower than the value of Ni2MnGa and it is found to be 200 K obtained from the averaged sound velocity . From our present work, we only predict the trend and note that Pt-doping in Ni2MnGa is likely to give rise to lowering of Debye temperature. 3.1.2. Elastic stability of substituted Pt2MnGa It is observed from Fig. 1, that the inherent brittleness decreases with Mn replacement by Cu in Ni2MnGa. Taking a cue from this ﬁnding, we probe the effect of Cu replacement of Mn in Pt2MnGa and also Pd2MnGa on the brittleness of the materials. We mention here that though there are studies in the literature on substitution of various elements in different sites in Ni–Pt–Mn–Ga compounds [18,72], Cu doping at the Mn site in Pt2MnGa (or Pd2MnGa) was not considered earlier. Here ﬁrst we study the stability of all the Ptderived materials. Fig. 2(a) shows the percentage of Cu-substitution at Mn site versus mixing energy for Pt2MnGa. It is found that this substitution leads to stable compounds with negative mixing
Fig. 1. Cauchy pressure, C p , versus GV /B; a linear ﬁtting of the data is carried out and shown here. An inverse linear-type relation is seen to exist between the two parameters (see text).
T. Roy et al. / Journal of Alloys and Compounds 632 (2015) 822–829
Ni2MnGa Pt2MnGa Pd2MnGa Cu2MnGa
10 5 0 10 5 0 −5 −10
Energy (eV) Fig. 3. The upper panel gives the total DOS of Ni2MnGa, Pt2MnGa, Pd2MnGa and Cu2MnGa at the respective optimized lattice constants. For Ni2MnGa there is a double peak structure around the Fermi level, with one peak at 0.199 eV and the other at 0.482 eVi. There is a dip or pseudo-gap at 0.453 eV. For Pt2MnGa, the peaks are at 0.418 and 0.643 eV, while for Pd2MnGa these peaks are at 0.465 and 0.6 eV. The respective pseudo-gaps are at 0.661 and 0.799 eV for Pt2MnGa and Pd2MnGa, respectively. The lower panel shows the spin-resolved DOS for the same materials. The solid, dotted, dashed and dot-dashed lines represent the Ni2MnGa, Pt2MnGa, Pd2MnGa and Cu2MnGa, respectively, in both the panels. For further details, see text.
energies. Further, we ﬁnd that the material Pt2CuGa possesses a low GV /B ratio of 0.12 and a high Poisson’s ratio of 0.44. Pd2CuGa has a GV /B value of 0.26 and Poisson’s ratio is 0.39. We have also tried to assess the effect of Ga replacement by Cu on properties of Pt2MnGa and Pd2MnGa . By summarizing all the data, we ﬁnd that interestingly Pt2CuGa is expected to possess the least GV /B value and highest Poisson’s ratio among all the compounds studied here. Furthermore, in context of MSMA properties of these materials, we have probed the probability of martensite transition for the compounds with Cu-substitution at Mn site in both Pt2MnGa and Pd2MnGa. In Fig. 2(b) we plot the energy versus the tetragonality (lattice constant ratio c/a) for the compounds with Cu-substitution at the Mn site in Pt2MnGa. It is gratifying to ﬁnd that all the substituted compounds are likely to be prone to martensite transformation as is seen in case of the parent compound . Similar is the case for the Pd-derived compounds . It is to be noted that for all the Pt and Pd-derived compounds, the conventional Heusler alloy structure has been found to have a lower energy compared to the inverse structure and hence we have taken the former structure for all these compounds. 3.2. Electronic stability of substituted Ni2MnGa We discuss now the electronic stability of Ni2MnGa, and the derivatives of this alloy with substitution at Ni site. As in our previous work  we have taken the conventional Heusler structure for Ni2MnGa and its respective Cu, Pt and Pd substituted compounds. On the contrary, we have taken the inverse Heusler structure for Mn2NiGa, where one Ni is substituted by Mn atom. To analyze the electronic stability, we plot the total DOS of Ni2MnGa, Pt2MnGa, Pd2MnGa and Cu2MnGa in Fig. 3 at the respective optimized lattice constants. A double peak structure at as well as a clear pseudogap close to the Fermi level for one of the spin channels are observed in all the materials except in Cu2MnGa. These
features arise due to hybridization between the Ga 4p and the outermost d-electrons of the transition metal atoms (Ni, Pt and Pd) and this is known to play a crucial role in the stability of these types of materials [73,74]. On the other hand, the peak occurring just below the Fermi level (at about 0.2 to 0.4 eV) may give rise to a band Jahn–Teller instability giving a lower symmetry structure in Ni2MnGa, Pt2MnGa and Pd2MnGa. To gain further insight, we plot the partial DOS (PDOS) of the materials. Fig. 4 shows the PDOS of Ga atom and its nearest neighbor Pt atom in Pt2MnGa. For comparison, in Fig. 5, we show the PDOS of Ga atom and its nearest neighbor (NN) Ni atom in Ni2MnGa. From analyzing Figs. 4 and 5, we observe a double peak structure at the Fermi level for down
Fig. 2. (a) Mixing energy in eV/f.u. as a function of Cu substitution at Mn sites in Pt2MnGa. The line is only guide to the eyes. (b) Energy in meV/atom as a function of c/a for Pt2MnGa system, Cu-substituted at Mn site. The energy (E) of the martensite phase is normalized with respect to the austenite phase. c/a = 1 corresponds to the austenite phase.
Ga −0.4 3
Pt −3 −6
Energy (eV) Fig. 4. PDOS of Ga atom and its nearest neighbor Pt atom in Pt2MnGa. The dotted line represent the total DOS and the solid lines represent up and down spins, respectively. While the total and up spin DOS is given in the upper panel, lower panel gives the down spin DOS. The extent of hybridization between the Ga and its nearest neighbor atom is shown (see text). The PDOS of Mn atom is also provided for completeness.
T. Roy et al. / Journal of Alloys and Compounds 632 (2015) 822–829
Table 2 Total magnetic moments per formula unit for the austenite phases of Cu, Pt, Pd and Mn substituted Ni2MnGa as well as TC values are listed here. The average moments are given in the parenthesis. In the ﬁrst four rows, ﬁrst value in the parentheses corresponds to the average moment of Mn atom, second value to the Ni, Pt, Pd and Cu atom, whichever present. For Mn2NiGa, ﬁrst value in the parenthesis corresponds to the MnNi atom, second to the MnMn atom and third to the Ni atom.
Mn DOS (states/eV/fu)
Ga −0.4 5
Ni −5 −6
Energy (eV) Fig. 5. PDOS of Ga atom and its nearest neighbor Ni atom in Ni2MnGa. The dotted line represent the total DOS and the solid lines represent up and down spins, respectively. While the total and up spin DOS is given in the upper panel, lower panel gives the down spin DOS. The extent of hybridization between the Ga and its nearest neighbor atom is shown (see text). The PDOS of Mn atom is also provided for completeness.
spin channel in the PDOS of both Ga and its NN atom, Ni and Pt. This indicates hybridization of Ga 4p and Ni (or Pt) 3d (or 5d) electrons in down spin channel leading to stability of the material as discussed in the literature . Similar features appear when the PDOS of Ga and its nearest neighbors in Mn2NiGa (NN of Ga: Ni and MnNi) as well as Pd2MnGa (NN of Ga: Pd) are analyzed . From the PDOS of Ga as well as its NN atoms, we observe the double peak structure at and a pseudogap close to the Fermi level . This is a signature of hybridization between Ga and its NN atoms. On the contrary, the total DOS as well as PDOS of Cu and its nearest neighbor Ni, in pure Cu2MnGa, shows no such double peak structure at the Fermi level as well as there is no clear pseudo-gap close to the Fermi level for either up or down spin. Further, due to the (nearly) full d-band of Cu, the d-orbitals are centered at different energies compared to Ni. These lead to a lack of strong hybridization between the atoms in the X and the Z positions in the Cubased material and consequently a weaker covalent character. The ﬁlling up of the pseudo-gap and decrease in the intensity of the peak near Fermi level indicates a weaker Jahn–Teller instability. The overall electronic stability of the cubic phase is a resultant of these effects. Notably the mixing energy calculations by Chakrabarti et al. have shown  that the Cu-derived material is much less stable compared to the parent Ni2MnGa compound.
3.3. Magnetic properties of substituted Ni2MnGa In Table 2 we give the magnetic moments calculated using allelectron wien2k package . The total moments increase as Ni is replaced by Pd and Pt at the Ni site in Ni2MnGa. This increase is due to the larger moment at the Mn site. As the Ni atom is replaced by Pd or Pt, the lattice constant increases, leading to larger a Mn–Mn distance. This probably gives rise to a stronger atomic-like moment in Mn. On the contrary, the moment of the X atom (Pt or Pd) is smaller than the one found in case of Ni2MnGa. We have further calculated the Heisenberg exchange coupling constants using SPR-KKR package . It is observed that for all the materials, in comparison to the other magnetic coupling terms, the nearest inter-site exchange interaction term is the most important one. Typically the dominant exchange interactions in most cases are conﬁned to a cluster of radius (Rij ) less than or equal to the lattice
Moments (lB /f.u.)
Ni2MnGa Pt2MnGa Pd2MnGa Cu2MnGa Mn2NiGa
4.096 4.155 4.151 3.626 1.188
414 176 206 800 775
(3.413, 0.361) (3.845, 0.139) (3.928, 0.121) (3.578, 0.04) (2.371, 3.172, 0.343)
constant (a) . In Table 2, we also present the TC calculated from the Heisenberg exchange coupling constants as has been done earlier in the literature . It is to be noted that the experimental TC of Ni2MnGa and Mn2NiGa are 376 K and 588 K, respectively. Keeping in mind that the TC calculated from a mean-ﬁeld-approximation always gives an overestimation of the TC, we ﬁnd that while the agreement between the experimental and our theoretical result is good in the former material, the mismatch in case of the latter is quite evident. We ﬁnd that Pt2MnGa has a much lower TC value compared to that of Ni2MnGa. This is in agreement with the experimental (as well as theoretical) trend of lowering of TC values as a function of Pt-doping in Ni2MnGa . We now probe the reason behind this observation. From Table 2, it is noted that the X atom in Pt2MnGa has a lower moment (0.139 lB ) compared to the moment on Ni atom (0.361 lB ) in Ni2MnGa. Subsequently, the X-Mn exchange interaction is found to be weaker in the former compound . Since the X-Mn exchange interaction is the dominant exchange coupling in these materials, the TC is much smaller in case of Pt2MnGa. Further, in Heusler alloys, it is seen that the TC increases upon hydrostatic pressure, i.e., dTC/dp is greater than 0 . In other words, a positive pressure coefﬁcient of T C is likely to exist in these materials due to an increased overlap. In the present case the larger lattice constant in the case of Pt-material may have led to a reduction of overlap, in turn, leading to a smaller TC. Furthermore, we observe that the TC shows an anomalous dependence on the total magnetic moment  when Ni is replaced by Pt or Pd in the material Ni2MnGa, namely, the TC value decreases when the total magnetic moment increases as a result of Pt or Pd substitution at the Ni site. Though for the isoelectronic elements, Ni, Pd and Pt, where the number of valence electrons are the same, the total moments increase from Ni2MnGa to Pt2MnGa (Table 2), the individual moment on the Ni is more than that on Pt/Pd. This causes a reduction of the strength of direct interaction in the Pt/Pd-based materials and the Jij values, hence the value of TC. This trend is in agreement with the literature . On the other hand, for Mn2NiGa, since Mn atom (MnMn) is the next nearest neighbor of another Mn atom (MnNi) with a ferrimagnetic conﬁguration between the two, this leads to a low value of total magnetic moment, but individual moments on each type of Mn (MnMn and MnNi) are high as seen in Table 2. In this material, hence, the exchange interaction (Jij) between two types of Mn atoms (MnMn, MnNi) is much stronger  which may have given rise to a high Curie temperature as reported in the literature . Hence, to summarize, we observe here that, there seems to exist no consistently linear relationship between the value of the total moment and the TC of a material as is also observed in case of experimental results of Ni2MnGa  versus Mn2NiGa  as well as observed in other Heusler alloys . Furthermore, the inﬂuence of choice of the XC potential on TC has also been probed. The listed TC values have been obtained using LDA XC potential . It has been observed that in most of the cases, the values of TC obtained using GGA are close to but slightly higher compared to the values obtained using LDA.
T. Roy et al. / Journal of Alloys and Compounds 632 (2015) 822–829
4. Summary and conclusion Using ﬁrst-principles density functional theory based calculations, we investigate the effect of Pt, Pd, Cu and Mn substitution at the Ni site in Ni2MnGa alloy system. We study in depth the mechanical and electronic stability of these substituted compounds. The elastic constants agree well with the literature wherever the data are available. It is observed that the tetragonal shear constant, C 0 , has values close to and/or below zero for all the compounds studied here indicating that, for all these materials, the cubic austenite phase might be prone to an elastic instability. In the literature it has been shown that shear modulii obtained from Voigt formalism (GV ) and Reuss formalism (GR ) for many materials, including few Heusler alloys, are very close. Notably most of these half and full Heusler alloys are predominantly halfmetallic or semiconducting. It is observed that for all the Heusler compounds studied here the shear modulus obtained using Reuss formalism is much lower compared to the one given by Voigt formalism. We have explained this difference in terms of the values of 0 the tetragonal shear constant C of the materials. However, it remains a conceptually open question as to why the GR values for Ni2MnGa and the substituted alloys are so much underestimated compared to the GV values. Detailed analysis is warranted for the same, which is beyond the scope of the present work. When the Cauchy pressure (C p ) of different materials is compared it is observed that, a trend of a nearly linear inverse correlation exists between the C p and the GV /B ratio among all the materials studied. This trend is already phenomenologically established in the literature for various other compounds. Based on the relative values of GV /B and C p , we predict that Ni2MnGa is expected to be inherently less brittle than Mn2NiGa, while Pt2MnGa is the least brittle one among the three. Further, a Cu-substitution at the Mn site for Pt2MnGa seems to have led to an increase of the C p value and Poisson’s ratio whereas there is a reduction in the GV /B value. From analyzing our results of total and partial DOS of the materials, existence of a double peak structure at the Fermi level and a pseudo-gap close to the Fermi level is found for the stable materials. This indicates hybridization and interaction of p states of the Z atom and d electrons of the X atom leading to electronic stability of the material as observed in the literature. Further, our result of the magnetic properties is in agreement with the already observed trend of lowering of TC values as a function of Pt-doping in Ni2MnGa. Furthermore, we observe that the TC shows an anomalous dependence on the total magnetic moment when Ni is replaced by Pt (or Pd) in the material Ni2MnGa , namely, the TC value decreases while the total magnetic moment increases as a result of Pt (or Pd) substitution at the Ni site as has been observed in the literature for similar Heusler alloys . Overall, the present study gives a detailed comparison of the bulk mechanical and also magnetic as well as electronic properties of some of the well-known Heusler alloys. The present study along with the studies in the literature indicates that, Pt substitution at the Ni site, along with the substitution of Cu at the Mn site, may lead to reduction of the GV /B and Young modulii values and increase of the C p value and the Poisson’s ratio indicating a lowering of the inherent crystalline brittleness in the substituted materials. Acknowledgments Authors thank S.R. Barman, S.W. D’Souza and S. Singh for useful discussions and Manfred Wuttig for providing results of unpublished work. P.D. Gupta, S.K. Deb and G.S. Lodha are thanked for facilities and encouragement throughout the work. The scientiﬁc computing group and the computer centre of RRCAT, Indore are
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