Ab initio calculations of structural and magnetic properties of NiCo-Mn-Cr-Sn alloys

The composition dependences of crystal lattice parameters, bulk modulus, magnetic moments, magnetic exchange parameters in Ni2-yCoyMn1.5-xCrxSn0.5 (y = 0.2, 0.4; 0.0 x 0.4) Heusler alloys are investigated with the help of ab initio calculations. Our simulations have shown that crystal lattice parameter firstly increased and then decreased with Cr content (x) increasing. The strongest ferromagnetic interaction for Ni1.6Co0.4Mn1.4Cr0.1Sn0.5 is nearest-neighbor interaction between Co and Mn1 (on own sites). The strongest antiferromagnetic interaction is observed between nearest-neighbor Mn1-Cr atoms in the first coordination sphere and it is equal to -15meV. Total magnetic moment of Ni2-yCoyMn1.5-xCrxSn0.5 (y =0.2, 0.4; 0.0 x 0.4) takes value in region from 6.1 μB to 6.6 μB.


Introduction
Heusler alloys have attracted a great interest because of number of effects (such as shape memory, magnetocaloric effects (MCE), exchange bias and superelasticity) and their potential applications as intelligent functional materials [1,2].Recently experimentalists started to study Cr-doped Ni-Co-Mn-Z (Z = In, Sn, Sb) Heusler alloys.The idea of these investigations is that the addition of Cr lead to decreasing of magnetic moment at martensitic state.In this case the large difference between magnetization in austenite and martensite is observed, which lead to the large value of magnetocaloric effect [3,4].In [5][6][7] structural and magnetic properties of Ni-Co-Mn-Cr-In alloys with the help of first principles and Monte-Carlo simulations were investigated.Sokolovskiy et al. show that the substitution of 5% Co for Ni and 5% Cr for Mn results in a first-order magnetostructural transition from ferromagnetic (FM) austenite to antiferromagnetic (AFM) martensite, which is accompanied by a spin-flip transition upon cooling.As a result, a large magnetization drop and giant inverse MCE can be achieved of ǻT ad §10 K in the 2 T field across the magnetostructural phase transition [7].
In this work we present the results of ab initio investigations of structural and magnetic properties of austenite of Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 (y =0.2, 0.4; 0.0 x 0.4) Heusler alloys.

Calculation details
All the calculations were performed using the density functional theory as part of the spin-polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) package [8,9].This code is based on the KKR-Green's function formalism that makes use of the multiple-scattering theory, and the electronic structure is expressed in terms of the corresponding Green's function as opposed to Bloch wave functions and eigenvalues.In this code, chemical disorder is treated through the coherent potential approximation [9].The SPR-KKR was used to determine the optimized lattice parameters with L2 1 unit cell which consist of the 4-atoms.In [5][6][7][8] has been shown that austenitic phase in Cr-doped alloys has FM ground state.So, we considered FM phase also.In this state all magnetic moments of Ni, Co, Mn and Cr atoms are parallel (Sn atom has small magnetic moment and it has not been taken into account).For the optimized lattice parameter the Heisenberg's magnetic exchange coupling parameters J ij were calculated.The exchange coupling parameters J ij within a real-space approach were calculated using an expression proposed by Liechtenstein et al. [10].In our calculations we used the generalized gradient approximation for the exchange correlation functional in the formulation of Perdew, Burke and Ernzerhof [11].We used L2 1 structural phase with space group of Fm3m (225), which consists of four interpenetrating face-centered cubic sublattices.In this structure the Sn atoms occupy the sites (0; 0; 0); Mn occupy (1/2; 1/2; 1/2) ones; Ni and Co atoms are randomly located at the sites (1/4; 1/4; 1/4) and (3/4; 3/4; 3/4).The location of addition of Mn and Cr atoms was assumed to be at Sn-site [12].3 Results of the first principles calculations

Equilibrium lattice parameter
On the Table 1 we can see equilibrium lattice parameter of Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 alloys in austenite phase for different value of Co excess (y) and of Cr excess (x).
Equilibrium lattice constants firstly increases with Cr excess jumping from 0 to 0.1 and then decreases with increasing of Cr excess.Obtained values of lattice constant were used for further calculations of magnetic properties of Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 alloys.

Bulk modulus
We also calculated bulk modulus for Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 (0 x 0.4; y= 0.2, 0.4) alloys.In Table 2 we present the bulk modulus for different Cr excess (x) and Co excess (y).As we can see, for Co=0.2 bulk modulus has a minimum at Cr=0.3 and this minimum is equal to 140.7 GPa; for Co=0.4 bulk modulus has a minimum at Cr=0.2 and this minimum is equal to 143.5GPa.

Magnetic moment
The calculations of partial and total magnetic moments of Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 alloys have been done by means of the SPR-KKR package.Here and further we will use follow designations such as Mn 1 and Mn 2 .Here, Mn atoms, which are located at regular Mn sublattice, are denoted as Mn 1 , whereas Mn 2 designation corresponds to Mn atoms which occupied the Sn sublattice.In Fig. 1f we present the composition dependences of the total magnetic moment, whereas partial moments for each sort of atoms as function of Cr excess x is depicted in Fig. 1a -1e.As we can see from Fig. 1f for zero Cr excess total magnetic moment is 6.55 μ B and 6.65 μ B for alloy containing 0.

Magnetic exchange parameters
Fig. 2 displays the magnetic exchange parameters for Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 alloys in austenite phase as a function of a distance between pairs of atoms.Here and further, the positive exchange constants (J ij > 0) are presented a FM coupling, whereas the negative ones (J ij < 0) indicate an AFM coupling.The oscillating damped behavior of J ij can be observed.
The strongest FM interaction for Ni 1.6 Co 0.4 Mn 1.4 Cr 0.1 Sn 0.5 alloy is nearest-neighbor (NN) interaction between Co and Mn 1 and Co and Mn 2 (Fig. 2a).Note, that value of these interactions are approximately equal to 20 meV and 15 meV respectively.The strongest AFM interaction is observed between NN Mn 1 -Cr atoms in the first coordination sphere and it is equal to -15meV.This interaction becomes FM at second coordination sphere and then again turns to AFM at third.All other alloys show the same behavior (Fig. 2b, 2c, 2d).It worth mentioning that distance between values of Co-Mn 1 and Co-Mn 2 interactions decreases with increasing Cr excess.
Exchange parameters at the first coordination sphere in austenite state of Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 alloys as a function Cr excess (x) are depicted in Fig. 3.Most weak interactions from Fig. 2 are not shown.Substitution of Ni instead of Mn 1 or Mn 2 into Mn 1 -Mn 2 turns J ij from AFM state to FM.According to Fig. 3c, 3d, 3f magnetic exchange parameters at the first coordination sphere increases with increasing Co excess and decreases with Cr excess.Whereas, replacing by Co one of atoms from Mn 1 -Mn 2 changes behavioral pattern of J ij -it has bend.Mn 1 -Co has pronounced maximum 18.9 meV for Co=0.2 and 18.5 for Co=0.4 at Cr=0.1.

Summary
The composition dependence of crystal lattice parameters, bulk modulus, magnetic moments and magnetic exchange parameters in Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 (0 x 0.4; y= 0.2, 0.4) Heusler alloys are investigated with the help of first principles using the SPR-KKR method.Crystal lattice parameter firstly increased and then decreased with Cr content (x) increasing.The strongest FM interaction for Ni 1.6 Co 0.4 Mn 1.4 Cr 0.1 Sn 0.5 alloy is nearest-neighbor interaction between Co and Mn 1 .The strongest AFM interaction is observed between NN Mn 1 -Cr atoms in the first coordination sphere and it is equal to -15meV.Total magnetic moment takes value in region from 6.1 μ B to 6.6 μ B .It should be noted that for more complete understanding of crystal and magnetic structure it is necessary to investigate different magnetic structures.

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences, 2015

Figure 1 :
Figure 1: Calculated spin magnetic moment of Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 alloys in austenite phase for different value of Co excess (y) as a function of Cr excess (x).a) partial magnetic moment of Ni atoms; b) partial magnetic moment of Cr atoms; c) partial magnetic moment of Mn 2 atoms; d) partial magnetic moment of Mn 1 atoms; .e)partial magnetic moment of Co atoms; f) total magnetic moment.
2 and 0.4 of Co respectively.It particularly linearly decreases on 0.4 μ B with increasing Cr excess up to 0.4.Concentration dependence of partial magnetic moment of Cr does not depend on Co excessthe difference of Co magnetic moment for two alloys Ni 1.6 Co 0.4 Mn 1.5-x Cr x Sn 0.5 and Ni 1.8 Co 0.2 Mn 1.5-x Cr x Sn 0.5 distinguish for 1.5%.Partial magnetic moment of Ni increases with increasing of Co, but decreases with increasing of Cr.It's value approximately equal 0.44 μ B .Partial magnetic moment of Mn 1 linearly decreases with increasing of Cr content from 3.57 μ B to 3.54 μ B for Co=0.4.In contrast, partial magnetic moment of Mn 2 linearly increases with increasing of Cr.

Figure 2 :
Figure 2: Exchange couplings parameters in austenitic phase for Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 alloys as a function of a distance between pairs of atoms i and j (in units of the lattice constant a).a) Ni 1.6 Co 0.4 Mn 1.4 Cr 0.1 Sn 0.5 ; b) Ni 1.8 Co 0.2 Mn 1.4 Cr 0.1 Sn 0.5 ; c) Ni 1.6 Co 0.4 Mn 1.2 Cr 0.3 Sn 0.5 ; d) Ni 1.8 Co 0.2 Mn 1.2 Cr 0.3 Sn 0.5 .

Figure 3 :
Figure 3: The magnetic exchange parameters J ij in the first coordination sphere for Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 alloys in austenite phase as a function of Cr excess (x).a) Mn 1 -Co coupling; b) Mn 1 -Cr coupling; c) Mn 1 -Ni coupling; d) Mn 2 -Mn 1 coupling; e) Mn 2 -Co coupling; f) Mn 2 -Ni coupling.

Table 1 .
Calculated equilibrium lattice constants of Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 alloys in austenite state, Å

Table 2 .
Calculated bulk modulus of Ni 2-y Co y Mn 1.5-x Cr x Sn 0.5 in austenite state, GPa