Structural, Elastic, Electronic and Magnetic Properties of MnNbZ (Z=As, Sb) and FeNbZ (Z=Sn, Pb) Semi-Heusler Alloys

The study of structural, electronic, magnetic, and elastic properties of new series of semi-Heusler alloys MnNbZ (Z=As, Sb) and FeNbZ (Z=Sn, Pb) has been performed by density functional theory. The magnetic phase and hence the structural stability of the alloys were considered wherein ferromagnetic state is found to stable. The half-metallic states are observed from the density of states and band structure calculations. The total magnetic moments found for all studied compounds are 1 $\mu_B$/f.u., which obey Slating-Pauling rule for semi-Heusler with ferromagnetic behavior. The calculated elastic constant C$_{ij}$, cohesive energy, and formation energy confirmed that these materials are mechanically stable. Among the four system, MnNbAs is found to have the highest ductility while the remaining systems are found to be brittle in nature. These properties confirmed that among others, MnNbAs is one of the novel candidate for spintronic devices applications.


Introduction
The development of new half-metallic ferromagnets is a great interest due to their potential for technological application in electronic devices. 1,2 Half-metallic means one of the channel is semiconducting or insulating in nature and other channel is metallic character around Fermi level, which means 100% polarization at the Fermi level. Many half-metallic materials have been investigated theoretically with high spin polarization. [3][4][5][6] The different types of materials have been studied, like transition metal pnictides and metal-chalcogenides, 7,8 perovskites and double perovskite, [9][10][11][12][13] oxides 14,15 and Heusler alloys. 16 These magnetic materials are very important in terms of spintronics applications. [17][18][19] Among these structures Heusler alloys are the best candidate for electronic devices because of their high Curie temparature, which are used for industrial applications.
The first half-metal ferromagnetic material was discovered in 1983 by de Groot et al. in the NiMnSb semi-Heusler compound. 29 After that, researchers were focused on many halfmetallic Heusler compounds by experimental synthesis and computational investigations.
The aim of the study to investigate the theoretical calculations of structural, electronic, magnetic, and elastic properties of Nb-based compounds. These are computationally investigated so, to check stabiliy is very important thing, which was tested by elastic constant, cohesive energy and formation energy calculations. In this study, it has been shown that MnNbAs, MnNbSb, FeNbSn, and FeNbPb compounds are used in spintronic devices fabrication for future experiments.

Method of Calculations
We performed the density functional theory (DFT) calculations to investigate new semi-Heusler compounds using the full-potential linearized augmented plane wave (FP-LAPW) method as implemented in the WIEN2k code. 35 The standard generalized gradient approximation (PBE-GGA) 36 and Tran-Blaha modified Becke-Johnson (TB-mBJ) potential 37,38 were used for the exchange correlation correction. For mechanical stability, the elastic results were studied using the ElaStic-1.0 package. 39 and K max are the smallest atomic sphere radius and plane wave cutoff respectively. The G max and cut off energy values used for all these calculations were 10 and -6 respectively in order to get stability in calculations. For the convergence test, energy criterion and charge criterion were 10 −6 Ry and 10 −4 electron respectively, which gives reliable results for the semi-Heusler compounds. Using the tetrahedron method, full Brillouin Zone was sample by 5000 kpoints (17×17×17 k-mesh) for self consistent field (SCF) calculations.

Results and discussion
Structural optimization and elastic properties The cohesive and formation energies for all these compounds are calculated by using the formula given below to check the investigated compounds are thermodynamically stable or not.  Elastic properties calculation gives the mechanical stability of a solid structure against the arbitrary deformation and the physical properties by using elastic constants (C ij ). The strain was used to determine these elastic constants in such a way that the total volume of the system remains constant. Due to the symmetry of the cubic system reduces the total number of three independent elastic parameters, i.e. C 11 , C 12 and C 44 . These are the elements of elastic stiffness matrix of order 6×6, with 6 eigen values. On the basis these parameters mechanical stability of crystals has been studied. The Born and Huang mechanical stability conditions for cubical materials are given by the following equations: 42,43 From above equation if these criterion are not satisfied by elastic constants, then cubic crystals becomes unstable. The C 11 , C 12 and C 44 elastic constants were obtained by calculation, which are depicted in table (2). By analysing these calculated elastic constants, all these compounds satisfying the stability conditions. Therefore, MnNbAs, MnNbSb, FeNbSn and FeNbPb compounds are mechanically stable against deformation. The bulk modulus is a measure of resistance to volume change and the shear modulus can be explained as reversible deformations on shear stress. The bulk modulus (B) and the isotropic shear modulus (G) has been calculated using the Voigt-Reuss and Hill approximation. 44,45 Using C ij elastic constants, they can be calculated as:

Electronic structure and magnetic properties
The study of electronic structure for ferromagnetic semi-Heusler alloys has been studied earlier in analogous compounds. 40,48,49 To investgate the similar nature for our system, we According to figure (5), the greatest contribution to the total density of states is due to d-orbital. But, for As-atom p-orbital has more contribution than s and d-orbitals, whose   The Slater-Pauling (SP) rule is one of the methods to determine the total magnetic moment in Heusler compounds. This method can be used by subtracting 24 of the total valence electrons in full-Heusler compounds and 18 of the total valence electrons in semi-Heusler compounds. 52 The electronic configurations of these atoms are Mn=3d 5 4s 2 , Fe=3d 6 4s 2 , Nb=4d 4 5s 1 , As=3d 10 4s 2 4p 3 , Sb=4d 10 5s 2 5p 3 , Sn=4d 10 5s 2 5p 2 and Pb=4f 14 5d 10 6s 2 6p 2 .
The total number of valence electrons Z t of MnNbAs compound is 17. According to SP rule (M t =Z t -18), therefore the total magnetic moment of MnNbAs compound is 1.00 µ B /f.u.
Similarly, other remaining three compounds have also 17 valence electrons, so their magnetic moments are the same, i.e., 1.00 µ B /f.u., as shown in table (3). To check the magnetic stability of these compounds on the basis of SP rule and by applying strains with changing the lattice parameters but total magnetic moment remains same for all these compounds. Thus, our results are compatible with the SP rule. The total and atomic moments are plotted as a function of lattice parameters as shown in figure (7).
From table (3), the total magnetic moment obtained by using GGA and mBJ methods are 1.00 µ B /f.u for all compounds, which comes from Slater-Pauling and confirmed these are ferromagnetic analogous to the previously reported compounds. 40,48 The spin magnetic moment of transition elements Mn-3d, Fe-3d and Nb-4d atoms have increased after applying mBJ over GGA but total magnetic moments remain the same. This is because, 3d-elements show positive spin magnetic moment and 4d-element has a negative spin magnetic moment, which cancel each other and finally shows total magnetic remains the same under both approximations. While other p-orbital elements As, Sb, Sn, and Pb have very small magnetic moments, which is clear from table (3) and as shown in figures (5) and (7).

Conclusions
The ground state electronic and magnetic properties of newly semi-Heusler compounds MnNbZ (Z=As, Sb) and FeNbZ (Z=Sn, Pb) were investigated using the density functional theory under generalized gradient approximation(GGA) and modified Becke-Johnson (mBJ) potential. By analysing volume optimization plots, density of states and band structure, these are ferromagnetic with half-metallic character. The cohesive energy, formation energy and calculated independent elastic constants C ij confirmed these are stable and possible to synthesized experimentally. In addition, the total magnetic moments per formula unit for all half-metallic compounds were found to be 1.00 µ B which comes from Slating-Pauling rule for semi-Heusler. MnNbAs shows ductile and remaining three MnNbSb, FeNbSn and FeNbPb compounds are brittle materials. Since, there are no experimental studies of these compounds and their related properties. Thus, these studies are interesting in the future for an experimental researcher to synthesis and utilize for spintronic applications.