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Physics Letters A www.elsevier.com/locate/pla

Ab initio study of effect of Co substitution on the magnetic properties of Ni and Pt-based Heusler alloys Tufan Roy a,b,∗ , Aparna Chakrabarti a,b a b

Theory and Simulations Lab, HRDS, Raja Ramanna Centre for Advanced Technology, Indore 452013, India Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India

a r t i c l e

i n f o

Article history: Received 20 December 2016 Received in revised form 24 January 2017 Accepted 16 February 2017 Available online xxxx Communicated by M. Wu Keywords: Heusler alloy Density functional theory RKKY interaction Martensite transition

a b s t r a c t Using density functional theory based calculations, we have carried out in-depth studies of effect of Co substitution on the magnetic properties of Ni and Pt-based shape memory alloys. We show the systematic variation of the total magnetic moment, as a function of Co doping. A detailed analysis of evolution of Heisenberg exchange coupling parameters as a function of Co doping has been presented here. The strength of RKKY type of exchange interaction is found to decay with the increase of Co doping. We calculate and show the trend, how the Curie temperature of the systems vary with the Co doping. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Full Heusler alloys (with typical formula A2 BC) have drawn considerable attention of the researchers over the last decades because of their possible technological applications. Upon cooling, some of the Heusler alloys undergo a structural transition from a high temperature cubic phase, namely austenite phase to a lower symmetry phase, called martensite phase below a certain temperature. This type of structural transition is referred as martensite transition, and the particular temperature at which the transition takes place is called martensite transition temperature. Ni2 MnGa belongs to this category of Heusler alloys [1–3]. The Heusler alloys of this category may ﬁnd their application as various devices, such as actuator, antenna, sensor etc. For the application purpose it is always desired that martensitic transition temperature is above the room temperature. In case of conventional shape memory effect, which is governed by the temperature, the actuation process is much more slow compared to a magnetically controlled actuation. So it is desirable to have a magnetic shape memory alloy with the Curie temperature (TC ) higher than the room temperature. It has been observed that both T M and TC values are very much dependent on the composition of a particular Heusler alloy [4–16]. There is also another category of full Heusler alloys, which are known to be metallic for one kind of spin channel and insula-

*

Corresponding author. E-mail address: [email protected] (T. Roy).

http://dx.doi.org/10.1016/j.physleta.2017.02.020 0375-9601/© 2017 Elsevier B.V. All rights reserved.

tor for the other kind of spin channel because of their very high spin polarization (HSP) at the Fermi level. They are often called as half metallic Heusler alloys [17]. Most of the Co-based Heusler alloys, like Co2 MnSn, Co2 MnGa belong to this category [18,19]. These Heusler alloys may have potential application in spintronic devices. Apart from the technological application, these Heusler alloys are very interesting because of their wide diversity in terms of magnetic property. These alloys may be ferromagnetic, ferrimagnetic, anti-ferromagnetic and also non-magnetic depending on the chemical composition. So it is of immense interest to have an in depth study on the magnetic interactions present in these systems. In most of the full Heusler alloys, A2 BC, B is the primary moment carrying atom. In many of the Heusler alloys, A2 BC, there is presence of a delocalized-like common d-band formed by the delectrons of the A and B atoms, which are both typically ﬁrst-row transition metal atoms [20]. Additionally, there is also an indirect RKKY-type exchange mechanism [21] between the B atoms, primarily mediated by the electrons of the C atoms, which also plays an important role in deﬁning the magnetic properties of these materials [20,22]. Staunton et al. [23] reported the role of RKKY interaction behind the origin of magnetic anisotropy of a system. For the magnetic shape memory alloys, magnetic anisotropy energy plays an important role. In this regard also, it will be interesting to study the RKKY interaction in detail in these systems. In a very recent paper, we have shown in detail the similarities and differences between the Heusler alloys which are likely to show shape memory alloy (SMA) property and which are not, in terms of the electronic, magnetic as well as mechanical prop-

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Fig. 1. (Color online.) Structure of conventional Heusler alloy (A2 BC). Blue, black and red balls represent A, B and C atoms respectively.

erties [14]. The main focus of our study here is to show how the magnetic exchange interactions of B atom between B atom itself, and with other magnetic atoms of the A2 BC systems, are evolving in going from the materials which are prone to martensite transition (which are generally metallic in nature) to the other class of Heusler alloys (which are typically half-metallic in nature) i.e. which do not show SMA property. Here we study about the nature of indirect type RKKY interaction and direct exchange interaction as well, for four sets of materials Ni2−x Cox MnGa, Ni2−x Cox FeGa, Pt2−x Cox MnGa, Pt2−x Cox MnSn as a function of x (x = 0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00). In all the cases the material is likely to show SMA property for x = 0.00 and is predicted to be half-metallic for x = 2.00. In the section following the methodology, the results of the work and the relevant discussion are presented. Finally, we summarize and conclude in the last section. 2. Method The Heusler alloys (A2 BC) studied here possess L21 structure that consists of four interpenetrating face-centered-cubic (fcc) sublattices 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). For the conventional Heusler alloy structure, the ﬁrst two sub-lattices are occupied by A atom and the third by B and fourth by C atom. In total, there are 16 atoms in the cell. Fig. 1 depicts the structure of a conventional Heusler alloy. While we study the Co substitution in A2 BC systems, the Co atom substitutes the A atom only. First we carry out full geometry optimization of the materials, of all the materials corresponding to x = 0.00, 0.25, 1.75, 2.00, using the 16 atom cell. For the geometry optimization, we employ the Vienna Ab Initio Simulation Package (VASP) [24] in combination with the projector augmented wave method [25]. We use an energy cutoff of minimum 500 eV for the planewave basis set. The calculations have been performed with a k mesh of 15 × 15 × 15. The energy and force tolerance used were 10 μeV and 10 meV/Å, respectively. After obtaining the equilibrium lattice constants of the four above-mentioned materials by using the VASP package, we plot the same. A linear variation of the lattice constant is observed. We deduce the lattice constants of the other materials, corresponding to x = 0.50, 0.75, 1.25, 1.50 by the method of interpolation. To gain insight into the magnetic interactions of these materials, we calculate and discuss their Heisenberg exchange coupling parameters. We use the spin-polarized-relativistic Korringa– Kohn–Rostoker method (SPR-KKR) to calculate the Heisenberg exchange coupling parameters, Ji j , as implemented in the SPR-KKR programme package [26]. The mesh of k points for the SCF cycles has been taken as 21 × 21 × 21 in the BZ. The angular momentum expansion for each atom is taken such that lmax = 3. The partial and total moments have also been calculated for all the materials studied. We use local density approximation (LDA) for exchange correlation functional [27]. For composition with x = 0 or 2 we use SPR-KKR method with coherent potential approximation.

Fig. 2. (Color online.) x dependence of magnetic moments for Ni2−x Cox MnGa, Ni2−x Cox FeGa, Pt2−x Cox MnSn, Pt2−x Cox MnGa. The line is only guide to the eyes.

The calculation of Heisenberg exchange coupling parameter (Ji j ) is based on the real space approach following the method proposed by Lichtenstein et al. [28]. It involves the magnetic force theorem to evaluate the Heisenberg exchange coupling parameter

Jij =

1 4π

E F

ij

ij

dE ImT r L {i τ↑ j τ↓ }

(1)

where τ is the scattering path operator. Difference in the inverse single site t matrices of up and down spin has been presented by . Tr L represents the trace of scattering matrices over the orbital indices L(l, m). The calculation of Ji j has been carried out with a cluster radius of 3a, a is the lattice parameter of the particular system. TC has been calculated from Heisenberg exchange coupling parameters using mean ﬁeld approximation [29]. Convergence of the TC has been tested with respect to cluster radius. For the calculation of density of states using SPR-KKR, we use following parameters NKTAB = 2500, NE = 550, ImE = 0.001 Ry [30]. 3. Results and discussion Total and partial moments As mentioned above, we studied here four sets of materials, Ni2−x Cox MnGa, Ni2−x Cox FeGa, Pt2−x Cox MnGa, Pt2−x Cox MnSn with x = 0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00. At the two ends of the composition, i.e. for x = 0.00 and x = 2.00, all the materials except Pt2 MnSn are already reported in the literature. We note that, all the materials, corresponding to x = 0.00, are likely to exhibit martensite transition. We predict here that Pt2 MnSn also possesses conventional Heusler alloy structure in its ground state and exhibits the martensite transition. All the studied materials here are taken to be ferromagnetic in nature. We observe from Fig. 2 that for the three sets of materials namely, Ni2−x Cox MnGa, Pt2−x Cox MnGa, Pt2−x Cox MnSn the variation of total moment (μ T ) follows the same trend, which is for lower value of x, μ T increases and then starts to fall at a higher x value, attaining a maximum value in between the range of

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Fig. 3. (Color online.) Variation of the total energy of (a) Co2 MnGa, Co2 FeGa, Co2 MnSn (b) Ni2 MnGa, Ni2 FeGa, Pt2 MnGa, Pt2 MnSn in their respective ground state magnetic conﬁgurations as a function of c/a. Energy E in the Y-axis signiﬁes the energy difference between the cubic and tetragonal phase. Some results of these ﬁgures are part of published literature [14,15]. Estimated values of T M for Ni2 MnGa, Ni2 FeGa, Pt2 MnGa, Pt2 MnSn are 70 K, 197 K, 817 K, 56 K respectively, taking cue from the literature [7]. The dashed line is only guide to the eyes.

x = 0.00 to x = 2.00. For Ni2−x Cox MnGa, the nature of variation of the moment as a function of x matches with the existing literature [31]. The variation of the total moment as a function of x, can be well understood from the variation of the partial moments for the respective systems. We ﬁnd that for Ni2−x Cox MnGa, Pt2−x Cox MnGa, Pt2−x Cox MnSn, the partial moment of Co and Mnatom decreases linearly as a function of x. This may be because, as we move towards the higher value of x, the lattice parameter of the systems decreases which leads to decrease of the Mn and Co partial moment. But as the absolute value of moment of Co-atom is much larger compared to that of Ni or Pt, the total moment increases initially with increasing value of x. However, this increasing factor has to compete with the continuous reduction of the partial moments of Co and Mn-atom, which dominates at higher value of x. This results in a fall of the total value of the moment. Because of these two competing factors, initially we get a maximum value of μ T and then it falls, ﬁnally reaches a value, very close to an integer following the Slater Pauling rule [18]. However, the total magnetic moment of Ni2−x Cox FeGa increases linearly as a function of x. This type of variation may be because of the almost constant partial moment of Fe and Co atom over the entire range of x. This is probably due to the fact that the lattice parameters for the two end materials Ni2 FeGa (a = 5.76 Å) and Co2 FeGa (a = 5.73 Å) are very close. Here the only controlling factor is the change of moment due to Ni substitution by Co-atom, which is always positive and proportional to the substitution and effectively results in a linear increase of the total moment of this system. Energy vs. c /a curve Heusler alloys may be used as shape memory device if they undergo a structural transition from high temperature cubic phase to low temperature non-cubic phase upon cooling. The alloys, which are likely to undergo this structural transition, must have the non-cubic phase with much lower energy compared to its cubic phase. We have applied a tetragonal distortion on the cubic phase of the stoichiometric material to probe

Fig. 4. (Color online.) Ji j of Mn atom with its neighbors as a function of normalized distance d/a for Ni2−x Cox MnGa system. a is the lattice parameter for x = 0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00.

whether they are prone to undergo tetragonal distortion or not. In the upper panel of Fig. 3, we ﬁnd that there is no lowering of energy under tetragonal distortion. For this set of materials, namely Co2 MnGa, Co2 MnSn, Co2 FeGa, the cubic phase is the lowest energy state (c /a = 1) and they are not likely to undergo martensite transition [13,14,32]. In the lower panel of Fig. 3 we observe that for all the materials shown here (i.e. Ni2 MnGa, Ni2 FeGa, Pt2 MnGa, Pt2 MnSn), energy of the systems is lowered under tetragonal distortion which indicates to a possibility of martensite transition in these materials. Except Pt2 MnSn, the other three materials, namely Ni2 MnGa, Ni2 FeGa, Pt2 MnGa, of the lower panel are already reported to undergo martensite transition [10,33,34]. Direct exchange interaction Now we plot (Fig. 4 to Fig. 7) the Heisenberg exchange coupling parameters (Ji j ), between Mn or Fe with other magnetic atoms of Ni2−x Cox MnGa, Ni2−x Cox FeGa, Pt2−x Cox MnGa, Pt2−x Cox MnSn (x = 0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00), as a function of interatomic spacing in the units of lattice parameter (a). We study the evolution of the magnetic exchange interaction in going from a material which shows the SMA property (x = 0.00) to one which does not show the SMA property (x = 2.00). The variation of Heisenberg exchange coupling constant between Mn or Fe atoms themselves as a function of Co-substitution will be discussed later. Here we will focus on the same between the B atom (Mn or Fe atom) and other magnetic atoms present (Pt or Ni and Co atom) i.e. inter sublattice exchange interaction present in the system. We know that for conventional Heusler alloy structure (A2 BC), A atom is the ﬁrst nearest neighbor of the B atom. Being the ﬁrst nearest neighbor the magnetic exchange interactions between A and B atom will be of direct type, which is already reported in the literature [20,22,35,36]. From Fig. 4 we observe that for the offstoichiometric systems where both Ni and Co are equidistant atoms from the Mn atom,

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Fig. 5. (Color online.) Ji j of Fe atom with its neighbors as a function of normalized distance d/a for Ni2−x Cox FeGa system. a is the lattice parameter for x = 0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 2.00.

the Heisenberg exchange coupling constant is much weaker between Mn and Ni compared to that of the Mn and Co atoms, as a direct consequence of lower magnetic moment of Ni atom with respect to Co atom. For Ni2−x Cox FeGa (Fig. 5), we observe the same, i.e. the direct exchange coupling constant between Co and Fe atom is much stronger compared to that between Ni and Fe. From Fig. 6 and Fig. 7 we observe that in case of Pt-based systems, the exchange coupling constant between Mn and Co is much higher compared to that of between Mn and Pt, as a consequence of much lower magnetic moment of Pt atom compared to Co atom. In case of direct exchange, the interaction energy is sizable only between the nearest neighbors. It dies very fast with distance. Fig. 8 shows how the nearest neighbor exchange interaction energy varies with the Co substitution. For Ni2−x Cox MnGa we observe that, the direct exchange coupling constant between Mn and Co atom is maximum (16.94 meV) at x = 0.25, then falls almost linearly to its minimum value (9.57 meV) at x = 2.00. But the exchange coupling constant between Mn and Ni atom increases linearly as a function of Co substitution for the same system. For the rest of the systems i.e. Ni2−x Cox FeGa, Pt2−x Cox MnSn, Pt2−x Cox MnGa, the direct exchange interaction energy between B and Co atom varies in the same way as found in case of Ni2−x Cox MnGa. In all the cases, the exchange coupling constant decreases almost linearly with increasing x. For Ni2−x Cox FeGa and Pt2−x Cox MnSn the nearest neighbor exchange interaction energy between B atom and Ni or Pt, (depending on the system) increases almost linearly as a function of Co substitution. To understand the trend of the variation of nearest neighbor direct exchange interaction, we show how the product of the magnetic moments of A and B atom varies as function of Co substitution, in Fig. 9. Here we ﬁnd that the product μ B ∗ μCo varies in the same way as that of exchange coupling energy between B and Co atom as shown in

Fig. 6. (Color online.) Ji j of Mn atom with its neighbors as a function of normalized distance d/a for Pt2−x Cox MnSn system. a is the lattice parameter for x = 0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00.

Fig. 7. (Color online.) Ji j of Mn atom with its neighbors as a function of normalized distance d/a for Pt2−x Cox MnGa system. a is the lattice parameter for x = 0.00, 0.25, 0.50, 0.75, 1.25, 1.50, 1.75, 2.00.

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Fig. 8. (Color online.) In panels from top to bottom, nearest neighbor direct exchange interaction, J1A B (A = Co, Ni or Pt; B = Mn or Fe depending on the system) as a function of Co doping, x for Ni2−x Cox MnGa, Ni2−x Cox FeGa, Pt2−x Cox MnSn, Pt2−x Cox MnGa, respectively. Dashed line is a guide to the eye.

the Fig. 8. This is also true for the Ni or Pt and Co atom. In case of Pt2−x Cox MnGa, we ﬁnd that the direct exchange interaction energy between Mn and Pt atom increases ﬁrst for lower value of x, then falls, after attaining a maximum value in between x = 0.00 and x = 2.00. The same trend has been observed for the product μMn ∗ μ P t . This observation shows there is a strong dependence of the direct exchange interaction between two magnetic moments on the product of the interacting moments. Curie temperature from Heisenberg coupling constant We have calculated the ferromagnetic transition temperature (TC ) for all the stoichiometric and off-stoichiometric materials as well, studied here. TC has been calculated from exchange coupling parameter’s as has been done earlier in the literature [29]. A stronger ferromagnetic coupling yields higher value of TC . Fig. 10 shows the variation of calculated TC as a function of Co substitution. For the stoichiometric cases (x = 0.00, x = 2.00), we compare our calculated TC , with the existing literature. The experimental values of TC are shown as black square in Fig. 10, wherever available. We observe that for the calculated and the experimentally measured TC values are quite close for the stoichiometric materials. We observe that with increasing percentage of Co doping, TC increases, which can be attributed to the much stronger direct exchange coupling between B and Co atoms compared to that of the B and Ni or Pt atoms. Kanomata et al. [31] have shown experimentally how TC varies with x for Ni2−x Cox MnGa. We notice there is a discrepancy between our calculated and experimental values [31] of TC in the nonstoichiometric region of Ni2−x Cox MnGa, which may be ¯ symmetry of the mabecause of our consideration of ideal Pm3m terials without the presence of any kinds of defects or disorder. RKKY type indirect exchange interaction RKKY type of interaction plays a very important role in the systems where the localized moments are far apart to have any direct exchange interaction.

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Fig. 9. (Color online.) In panels from top to bottom, x dependence of product of magnetic moments of A and B atom (μ A ∗ μ B ) for Ni2−x Cox MnGa, Ni2−x Cox FeGa, Pt2−x Cox MnSn, Pt2−x Cox MnGa, respectively. Dashed line is a guide to the eye.

Fig. 10. (Color online.) x dependence of Curie temperature (TC ) for (a) Ni2−x Cox MnGa, experimental values of TC for Ni2 MnGa and Co2 MnGa have been taken from Ref. [1] and Ref. [41], respectively (b) Ni2−x Cox FeGa, experimental values of TC for Ni2 FeGa and Co2 FeGa have been taken from Ref. [34] and Ref. [42], respectively (c) Pt2−x Cox MnSn, experimental value of TC for Co2 MnSn has been taken from Ref. [41] (d) Pt2−x Cox MnGa, experimental value of TC for Co2 MnGa has been taken from Ref. [41]. Dashed line is a guide to the eye.

There are extensive studies on the RKKY interactions in various dilute magnetic systems, where the magnetic atoms like Mn or Fe are present in a very low concentration in the nonmagnetic metallic host material [37]. The presence of RKKY interaction between localized-like moments (Mn or Fe) was reported. This interaction

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Fig. 11. (Color online.) J B − B (B = Mn or Fe depending on the systems) as a function of normalized distance d/a for (a) Ni2−x Cox MnGa (b) Ni2−x Cox FeGa (c) Pt2−x Cox MnSn (d) Pt2−x Cox MnGa. a is the lattice parameter for different values of x.

was via the conduction electrons of the host material, which may be Au, Ag, Mo, Zn etc. Not only in metallic system, RKKY interaction plays a crucial role in determining the magnetic property of dilute magnetic semiconductor also [38]. Experimentally the presence of long range oscillatory types of magnetic exchange interaction between the localized Mn moments of Ni2 MnSn and Pd2 MnSn was conﬁrmed by Noda et al. [39] using neutron spin wave scattering method. In this study all the systems contain Mn or Fe atom, and the magnetic moments are mainly conﬁned to these atoms. As all the systems studied here possess conventional Heusler alloy structure (A2 BC), the separation between the B atoms (Mn or Fe depending on the systems) is too large to have a direct exchange interaction between them. For this kind of Heusler alloy structure, B atom is surrounded by eight A atoms, which makes A atoms play a very important role in determining the magnetic exchange interactions between B atoms themselves. The localized nature of the magnetic moment of the B atoms spin-polarizes the free-like electrons present in the system and these spin-polarized conduction electrons effectively couple the B atoms [40]. Previously in literature [20] it was mentioned for A2 MnC systems (A = Cu, Pd; C = Al, In, Sb), that the conduction electrons of the C atom take part in the coupling between Mn atoms. In a recent study [22] the role of conduction electron of A atom has also been conﬁrmed for a number of Mn-based Heusler alloys. In our studied systems here, for a given series of materials, C -atom is ﬁxed which is Ga for Ni2−x Cox MnGa, Ni2−x Cox FeGa, Pt2−x Cox MnGa and Sn for Pt2−x Cox MnSn. However with substitution, nature of A atom changes. Here we will discuss only about the role of A-atom in the RKKY interaction between B atoms themselves. It is to be noted that the spin polarization of the conduction electron will depend on the value of partial magnetic moment of the B atom and the number of the conduction electrons present in the system. Now as we move from Ni2 MnGa to Co2 MnGa we are effectively reducing the number of conduction electrons of the system, as Ni has one more d-electron compared to Co-atom. This may cause a weaker coupling between the Mn atoms themselves. From Fig. 11(a) we

ﬁnd that for Ni2 MnGa (x = 0.00) the Mn–Mn interaction is the most oscillatory in nature (Heisenberg exchange coupling constant varies between 1.39 meV (d/a = 1) to −0.25 meV (d/a = 1.73)) whereas for Co2 MnGa the oscillation is minimal (varies between 0.2 meV (d/a = 0.71) to −0.02 meV (d/a = 1.58)). The oscillatory nature gets reduced gradually as we move from Ni2 MnGa to Co2 MnGa. For Ni2−x Cox FeGa system also, we observe same kind of variation for Mn–Mn interaction as we move from x = 0.00 to x = 2.00. For x = 0.00 i.e. for Ni2 FeGa the amplitude of Fe–Fe RKKY interaction varies between 1.96 meV (d/a = 0.71) and −0.76 meV (d/a = 1.41) which is the strongest among the Ni2−x Cox FeGa series. In going from Pt-based systems to Co-based systems (Fig. 11(c) and Fig. 11(d)) also, we are reducing the number of conduction electrons. One more factor which we must consider when we discus about Pt2−x Cox MnSn and Pt2−x Cox MnGa, is the change in lattice parameter between the compounds corresponding to x = 0.00 and 2.00. On the other hand, for Ni2−x Cox MnGa and Ni2−x Cox FeGa this change is very nominal as both Ni and Co has very close values of atomic radius. As we move from Pt2 MnSn (a = 6.46 Å) to Co2 MnSn (5.98 Å) there is a contraction of lattice parameter of about 0.48 Å. For Pt2 MnGa (a = 6.23 Å) to Co2 MnGa (a = 5.72 Å), a contraction of about 0.51 Å takes place. This larger lattice parameter for Pt-based systems causes more localization of Mn partial magnetic moment (3.97 μ B and 3.82 μ B in Pt2 MnSn and Pt2 MnGa respectively) compared to the values in Co-based system (3.19 μ B and 2.73 μ B in case of Co2 MnSn and Co2 MnGa respectively). In Ref. [36], Bose et al. have mentioned that the strength of exchange interaction between two interacting magnetic moments also depends on the values of the respective magnetic moments. Therefore, if we focus on Fig. 11(c) we observe that the Mn– Mn exchange interaction energy for x = 0.00 (Pt2 MnSn) oscillates between a maximum value of 1.21 meV (d/a = 1.00) and minimum value of −1.29 meV (d/a = 1.73) but oscillation becomes weaker gradually as we increase x and for Co2 MnSn it varies between 1.61 meV (d/a = 0.71) and 0.03 meV (d/a = 0.58). It means RKKY type of interaction is stronger in Pt2 MnSn compared to Co2 MnSn, which may be because of more localized-like Mnmoments in Pt2 MnSn. For Pt2−x Cox MnGa system also we ﬁnd that the for x = 0.00, RKKY type of interaction between Mn–Mn is the most oscillatory (for Pt2 MnGa it varies between −1.33 meV and 1.94 meV at d/a = 0.71 and 1.00 respectively) and gradually with increasing x, the interaction becomes less oscillatory in nature. Density of states for Ni2−x Cox MnGa and Pt2−x Cox MnSn Fig. 12, depicts the spin polarized and atom projected density of states (DOS) of Ni2−x Cox MnGa for x = 0.00, 0.50, 1.50, 2.00. We observe that the spin polarization at Fermi level is maximum for x = 2.00, i.e. for Co2 MnGa, as minority density of states almost vanishes there. Majority density of states is found to move towards the higher energy side with increasing Co substitution in place Ni and this well supports a rigid band model. For all cases, we observe a large exchange splitting in the Mn atom’s density of states. For Mn atom, the occupied density of states is dominated by the majority spin whereas the unoccupied region is dominated by the minority spin. In case of Ni2 MnGa (x = 0.00), the majority peak of DOS corresponding to the Mn atom is centered around −1.1 eV whereas its minority peak is centered around +1.3 eV. This exchange splitting for Mn atom decreases gradually with increasing value of x and becomes minimum for x = 2.00, where the majority peak is centered around −0.6 eV and minority peak is centered around +1.5 eV. It suggests that in going from Ni2 MnGa to Co2 MnGa, the localized nature of the Mn magnetic moment will be much more reduced, which effectively causes an weaker RKKY type of indirect coupling between them for the latter, which has been observed earlier.

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Fig. 12. (Color online.) Spin polarized and atom projected density of states of (a) Ni2−x Cox MnGa (b) Mn (c) Ni (d) Co, for x = 0.00, 0.50, 1.50, 2.00 (we have not shown the density of states for all values of x for better clarity of the ﬁgure).

From Fig. 12(a), we observe a peak in minority DOS, which is very close to the Fermi level. A detailed analysis shows that this peak is derived from Ni 3d e g level. In literature, the role of this peak in the possibility of martensite transition has been extensively discussed. Here we observe that, this peak diminishes gradually as we move towards Co2 MnGa end from Ni2 MnGa end, and the DOS of minority spin electrons gets reduced at Fermi level. Complete substitution of Ni by Co atom makes the system half metallic-like but the majority density of states of Ni atom changes hardly near Fermi level. This reduction of the minority carriers with increasing Co substitution at the Ni site effectively reduces the conduction electrons present in the system, causing a weaker exchange coupling between the localized Mn moments, which we observe in Fig. 11. For Co atom, DOS for both majority and minority spin moves to the lower binding energy side with increasing value of x. Fig. 13 shows spin polarized total and partial density of states for Pt2−x Cox MnSn for x = 0.00, 0.50, 1.50, 2.00. From Fig. 13(a), we observe that the systems become very close to half-metallic for x = 2.00, i.e. for Co2 MnSn, whereas Pt2 MnSn (x = 0.00) is a metallic system. We ﬁnd that there a clear shift of the majority density of states towards the higher binding energy with increasing value of x. Similar to the Ni2−x Cox MnGa case, the exchange splitting of the Mn atom is maximum for x = 0.00 and minimum in the case of x = 2.00. For x = 0.00 and x = 2.00, the peak corresponding to majority spin is centered around −1.9 eV and −1.0 eV, whereas the corresponding minority peaks are centered around +1.2 eV and +1.4 eV respectively. It suggests the magnetic moment of Mn atom is much more localized in case of Pt2 MnSn in comparison to the Co2 MnSn, which may be because of the much larger lattice parameter of the former one. Now, if we focus on the DOS of Co and Pt atom, we observe the increase in the majority DOS of Co atom with increasing value of x is almost counterbalanced by decrease of majority DOS of Pt at the Fermi level. But the minority DOS decreases gradually with increasing value of x for both Pt and Co atom at Fermi level. As a result the effective number of conduction electron of the system seems to decrease as we move from x = 0.00 to x = 2.00. In going from Pt2 MnSn to Co2 MnSn we observe a gradual decrease in the strength of RKKY type of inter-

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Fig. 13. (Color online.) Spin polarized and atom projected density of states of (a) Pt2−x Cox MnSn (b) Mn (c) Pt (d) Co, for x = 0.00, 0.50, 1.50, 2.00 (we have not shown the density of states for all values of x for better clarity of the ﬁgure).

action (Fig. 11(c)) mainly due to two facts: (a) magnetic moment of the Mn atom become more delocalized-like, (b) continuous reduction of the number of conduction electrons, with increasing value of x. For the other two sets of systems namely Ni2−x Cox FeGa and Pt2−x Cox MnGa, we observe the same kind of evolution of the electronic structure as a function of x as in case of the other two, where both of them seem to be metallic for x = 0.00 and become almost half-metal for x = 2.00, as the minority spin DOS diminishes gradually with increase of Co substitution. 4. Conclusion From density functional theory based calculations we study the effects of Co substitution in Ni and Pt-based Heusler alloys which are likely to show SMA. Our results suggest that there is a decrease in strength of the RKKY interaction as we increase the Co doping at Ni or Pt site. It indicates about the dominant role played by A atom’s d-electron in the formation of coupling between localizedlike moments of B atom in the A2 BC systems studied here. We also report the dependence of the strength of the RKKY interaction on the localization of B atom’s magnetic moment. Our study signiﬁes the implicit and important presence of RKKY interaction in the magnetic shape memory Heusler alloys. Acknowledgements Authors thank P. A. Naik and A. Banerjee for encouragement throughout the work. Authors thank S. R. Barman and C. Kamal for useful discussion. TR thanks RRCAT for ﬁnancial support. Computer centre, RRCAT is thanked for resources and help in installing the codes. References [1] P.J. Webster, K.R.A. Ziebeck, S.L. Town, M.S. Peak, Philos. Mag. B 49 (1984) 295. [2] K. Ullakko, J.K. Huang, C. Kantner, R.C. O’Handley, V.V. Kokorin, Appl. Phys. Lett. 69 (1996) 1966. [3] A. Sozinov, A.A. Likhachev, N. Lanska, K. Ullakko, Appl. Phys. Lett. 80 (2002) 1746.

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