Mechanical Stability, Electronic, and Magnetic Properties of XZrAs (X = Cr, Mn, V) Half-Heusler Compounds

The structural, half-metallic, and elastic characteristics of XZrAs (X = Cr, Mn, and V) half-Heusler compounds were theoretically calculated using the WIEN2k code. For the term of the potential exchange and correlation (XC), we calculated structural, electronic, and magnetic properties using the generalized gradient approximation (GGA). The type 1 arrangement in ferromagnetic (FM) phases is more energetically stable than other type arrangements in all compounds. The spin-up electrons of both XZrAs (X = Cr, Mn) half-Heusler compounds in the type III structure are semiconducting with energy gaps, whereas the spin-dn electrons are metallic. Research has also been done on the VZrAs compound, which exhibits metallic characteristics in both type I and type III structures. XZrAs (X = Cr, Mn, and V) half-Heusler compounds are elastically stable and ductile, according to calculated Cij elastic constants. Finally, at equilibrium lattice constants a = 6.1196 A° for MnZrAs and 6.142 A° for CrZrAs, real half-metal ferromagnetic materials (HMF) were produced from XZrAs (X = Cr, Mn) half-Heusler compounds within 3 µB and 2µB, respectively.


Introduction
In physics, materials with half-metallic characteristics are extremely important. And this is after de Groot's theoretical realization in 1983, which ushered in a revolution [1]. Several efforts have been made to comprehend, study, and realize new half-metallic materials. One of the primary reasons for this contribution is the effective use of half metals in spintronic device applications [2,3], such as the magnetic random access memory (MRAM) [4]. Half metals have features in that they have metallic behavior for the majority-spin channel at Fermi level, but insulating/semiconducting behavior for the minority-spin channel, the total magnetic moment is an integer, which is another property of half metals. Halfmetallic materials are classified into two types: complete half metals with 100% spin polarization at the Fermi level and half-metallic materials with a limited density of states at the Fermi level and less than 100% spin polarization [5].
The fascinating physical characteristics of the Heusler compounds for applications in spintronic, optoelectronic, and thermoelectric systems have generated a great deal of interest since their discovery in 1903 [10].
Half-Heusler compounds are ternary intermetallic compounds with the general composition XYZ. In this class, X and Y stand for transition metals with d electrons, and Z stands for an element with sp electrons. Half-Heusler alloys are one of the most important centers of research for halfmetallic materials. Half-Heusler compounds were projected to be half metals in large numbers [11,12]. The structural, magnetic, electrical, and elastic properties of half-Heusler compounds draw a lot of attention. They play a crucial role in a variety of fields of technology due to their various physical features. The half-Heusler based on zirconium and arsenic is one of the most investigated [13,14]. In recent years, a variety of methods have been used to investigate Zr-based and As-based half-Heusler alloys. Some As-and Zr-based half-Heusler compounds have been theoretically and experimentally investigated for structural, electronic, thermoelectric, half-metallic, and elastic properties such as FeZrX (X = P, As, Sb, and Bi) [15] and MnZrX (X = In, Tl, C, Si, Ge, Sn, Pb, N, P, As, Sb, O, S, Se, Te) half-Heusler compounds [13], VZrAs and VZrSb half-Heusler compounds, and Al 1-x M x As (M = Co, Fe and x = 0.0625, 0.125, 0.25) diluted magnetic semiconductors [14].
We used density functional calculations to look into the physical properties of XZrAs (X = Cr, Mn, V) compounds, such as electronic, magnetic, and elastic properties. The rest of the paper is organized as follows: in Sect. 2, we describe the method and the calculations in detail. In Sect. 3, we present and discuss the results, followed by a conclusion in Sect. 4.

Calculation Method
The first-principles calculation with full potential linear augmented plane wave (FP-LAPW) [16,17] as implemented in the in the WIEN2K package [18] is used in our calculation. In this method, non-overlapping muffin-tin (MT) spheres are distributed in space that is separated by an interstitial region [19]. Each atom is encircled by a hypothetical sphere as the computations are being done, MT [19]. To do the potential calculations of atoms more precisely, the radii of the muffintin spheres should be properly determined [19]. Using the generalized gradient approximation with spin polarization (GGA) [20], the space is divided into non-overlapping MT spheres separated by an interstitial region.
The wave functions of the muffin-tin spheres were approximated by the spherical harmonic functions, while the wave functions of interstitial regions of the muffin-tin model were modeled with Fourier series. For the wave function expansion inside the muffin-tin spheres, the maximum value of angular momentum used in this method is l max = 10. To control the convergence of the basis set, we take a cutoff parameter R MT .K max = 7 (R MT is the smallest muffin-tin sphere radius and K max is the largest reciprocal lattice vector used in the plane wave expansion within interstitial region). The energy threshold between the core and the valence states and k points were set to −7 Ryd and 1000 respectively. The magnitude of largest vector in the charge density Fourier expansion used is G max = 14(a.u) −1 . The calculations continue to ensure good convergence until the energy deviation is less than 0.00001 eV/atom.

Results and Discussions a) Structural Properties
In this subsection, we present the results of the geometrical structure of the XZrAs (X = Cr, Mn, V) half-Heusler alloys as well as the lattice parameters, bulk modulus, the ground-state energies (E), and the formation energies (E f ). Compounds with the structure have the general formula XYZ and crystallize in non-centrosymmetric cubic MgAgAs (C1b) structure with Fm3m space group. There are five atomic arrangements in the conventional cubic unit cell: type1: 4a (0, 0, 0), 4b (0.5, 0.5, 0. Some experimental studies show that the atomic disorderness influences half-Heusler structures [21]; as a result, investigation of XYZ compounds in the five possible arrangements is required. The crystalline structure (C1b) of this type of material is also discussed well in literature [22]. Using optimization process, the structural properties of the of XZrAs (X = Cr, Mn, V) alloys are predicted. Figure 1 presents the variation of the unit cell energy versus volume for five atomic arrangements as shown in Table 1.
The minimum energy on the E-V curve is the unit cell's ground-state energy, and the volume corresponding to the minimum is the optimized volume. The configuration of the type 1 is energetically the most stable arrangement of atoms because of it lowest energy. The structural lattice parameters are evaluated using this unit cell volume. Table 2 shows the structural parameters such as lattice constant a (A°) and bulk modulus B at zero pressure using this unit cell volume. As it can be observed from Table 2, the lattice constant ɑ of XZrAs compounds increases with increasing of the atomic size of the X element in XZrAs alloys in the following sequence: ɑ (VZrAs) < ɑ (MnZrAs) < ɑ (CrZrAs). As Zr and As atoms are the same in the three compounds, this result can be easily explained by considering the atomic radii of Cr, Mn, and V: R (Cr) = 1.42 Å, R (Mn) = 1.40 Å, R (V) = 1.35 Å. Furthermore, there are many different types of X atoms, and their radii vary greatly. Changes in atom radius do not directly determine which structure type is more appropriate. The lattice constants and the volumes of the compounds increase as the radius of the atoms increases. In our research, we discovered that different compounds with different atom radiuses have the same type of structure.
However, the effects of factors such as covalent and ionic binding, interatomic interactions, and the number of valence electrons on crystal structure arrangements have previously been studied [23,24]. Meanwhile, the B values decrease in the following sequence: B (VZrAs) > B (MnZrAs) > B (CrZrAs) in reverse order to "a." According to the established connection between B and the lattice constants a: The stability of the hypothetical alloys can be refer by studying the formation energy (E F ). The energy formation expressed as: where E XZrAs tot is the ground-state total energy of XZrAs (X = Cr, Mn, and V) half-Heusler alloys, and E Bulk X , E Bulk Zr , and E Bulk As correspond to the total energy per atoms. The Variation of the total energy as a function of the unit cell volume of XZrAs (X = Cr, Mn and V) for five atomic arrangements Table 1 Sites occupied by atoms X, Y, and Z in five phase arrangements  Table 2. The formation energy was calculated for the five possible arrangements. The negative value of this energy for XZrAs (X = Cr, Mn, and V) half-Heusler alloys confirms that these compounds are thermodynamically stable and can be experimentally synthesized.

b) Electronic Properties
To predict the electronic nature of XZrAs (X = Cr, Mn, V) compounds, electronic band structure simulations are performed. Figures 2, 3, and 4 show the majority-and minority-spin (spin-up and spin-down) band structures for CrZrAs, MnZrAs, and VZrAs in type 1 and type 3 arrangements, along the high symmetry direction of the first Brillouin zone at the equilibrium lattice parameters. For type 3 arrangement, Fig. 2 and 3 show that CrZrAs and MnZrAs half-Heusler compounds had direct and indirect  As can be observed from these figures, for MnZrAs alloy the majority-spin states have a semiconductor character, whereas minority-spin states have a metallic nature. For MnZrAs, the band gap is produced by the maximum valence band (VBM) and the minimum conduction band (CBM) at Γ-point.  in half-Heusler compounds, they are expected to contribute the most to the density of states, while As elements are expected to contribute the least. The atomic total densities of states were drawn to see these contributions. As expected, the most contribution to DOS in regions near the Fermi energy level comes from the transition metals, while As elements contribute the most in regions far from the Fermi energy level. Additionally, the energy gap of the CrZrAs compound is indirect band gap (Γ-L) and that of MnCrAs has a direct band gap (Γ-Γ). It can be seen that the contributions in the conduction band come from the As element, whereas the main contributions in the valance band come from the Mn element. The transition metals Mn and Zr have d-orbitals as major carriers. The band gaps are explained as a result of d-d-orbitals in transition metals or covalent hybridizations between bonding and antibonding states.
For CrZrAs, the valence band maximum is −0.1394 eV, the conduction band minimum is 0.91181 eV, the band gap is 1.05121 eV, and the half-metallic band gaps is 0.1394 eV. For MnZrAs, the valence band maximum is −0.16205 eV, the conduction band minimum is 0.35697 eV, the band gap is 0.51902 eV, and the half-metallic band gaps is 0.16205 eV. iii) Magnetic Properties The calculated total and partial spin moments of XZrAs (X = Cr, Mn, V) are listed in Table 3. The three XZrAs (X = Cr, Mn, V) alloys with type 1 arrangement are all metallic, and have a total moment of 2.43 µ B , 2.68 µ B , and −1.18 µ B respectively. While the alloys with type 3 arrangement have an integer total moment, except for VZrAs, which has a total moment of 0.24 µ B . The total magnetic moment in the full-Heusler and half-Heusler alloys is submissive to the Slater-Pauling rule. For the full-Heusler alloy, it will be equal to M T = Z T -24 and for the half-Heusler alloy, it will be M T = Z T -18, where MT is the total magnetic moment per unit cell and ZT is the total number of valence electrons. The obtained values of the total magnetic moment of the CrZ-rAs and MnZrAs alloys by utilizing the GGA approximation are listed in Table 3. The CrZrAs and MnZrAs alloys have 15 and 16 valence electrons, respectively, which generate total magnetic moment of 3.00µ B and 2.00 µ B , respectively, according to the Slater-Pauling rule. The total magnetic moment of the CrZrAs alloy obtained is 2.00 μ B which is in excellent agreement with the value of the Slater-Pauling rule, whereas the metallic behavior of the VZrAs alloy generates a deviation of the total magnetic moment from the Slater-Pauling value. In addition, we noted that the total magnetic moment of the two alloys is mainly contributed by the Cr and Mn sites, where these contributions are due to the large exchange splitting in the Cr and Mn atoms for the majority-spin and minority-spin channels. The local magnetic moments of each site are also listed in Table 3, where the Zr and As atoms have a negligible local magnetic moment with opposite sign in comparison with the Cr and Mn elements. This indicates that the magnetic moment of the Cr and Mn sites interacts in anti-parallel behavior with those of the Zr atoms. iv) Mechanical Properties In order to investigate the mechanical stability of XZrAs (X = Cr, Mn, V) half-Heusler alloys, the obtained elastic constants of these alloys are presented in Table 4. From the elastic constants, the anisotropy factor A, Young's modulus Y, and Poisson's ratio v are calculated since these are the most interesting elastic properties of any materials. Moreover, for a cubic crystal, there are only three independent elastic constants, C 11 , C 12 , and C 44 . Hence, a set of three equations is needed to determine all these constants.
From symmetry, we have the following conditions: C 11 = C 22 = C 33 ; C 12 = C 23 = C 13 ; and C 44 = C 55 = C 66 . The necessary and sufficient mechanical stability conditions of the elastic constants for a cubic crystal are as follows [28][29][30]: For XZrAs (X = Cr, Mn, V) alloys, Table 4 shows that our elastic constants C ij are all positive and follow the Born criteria, indicating mechanical stability. The C 11 values (Table 4) of all compounds are higher than C 12 and C 44 , indicating a higher resistance to deformation by stress applied on the (100) plane with polarization in the < 100 > direction than the resistance to shape in the same plane with polarization in the < 010 > direction.
Our calculated elastic constants satisfy the above conditions and therefore, all XZrAs (X = Cr, Mn, V) compounds are elastically stable.
For all XZrAs (X = Cr, Mn, V) alloys, the Voigt approximation was used to calculate the bulk modulus and shear modulus (2) C 11 − C 12 > 0; C 44 > 0; C 11 + 2C 12 > 0  [31]. Bulk modulus B and Voigt shear modulus S v for cubic structure can be calculated from the following equations: The elastic anisotropy A is an important parameter to measure the degree of anisotropy of materials [32]; A can be calculated by the following formula: Mechanical parameters like B (bulk modulus), G (shear modulus), Y (Young's modulus), and v (Poisson's ratio) define a material's resistance to uniaxial stress and provide information on its stiffness degree. The computed Young's modulus of CrZrAs compound is greater than that of MnZ-rAs and VZrAs compounds, indicating that CrZrAs alloy has elastic stiffness. Furthermore, due to the difference in cohesive forces between the materials, the shear modulus and Young's modulus are larger. We can define the solid stability using the v (Poisson's ratio) value, for a stable solid this value ranging from −1.0 to 0.5 [33]. Table 4 reveals that the Poisson ratio values for XZrAs (X = Cr, Mn, V) alloys are 0.33, 0.32, and 0.39 respectively, indicating that all compounds are stable under shear stress. Furthermore, according to Frantsevich et al. [34], if this parameter's value is greater than 1/3, the material is ductile. Otherwise, it is brittle. We note that both CrZrAS and VZrAS compounds have a Poisson ratio more than 1/3, implying that the two alloys are ductile.

Conclusion
DFT calculations were used to investigate the electronic structure, magnetic, and elastic properties of XZrAs (X = Cr, Mn, V) half-Heusler compounds. According to these calculations, Therefore, these compounds are convenient materials for using in spintronic applications.
Author Contribution H. Mokhtari, M. Mokhtari, and D. Fethallah: collected the data, analysis tools, wrote the paper. L. Boumia, D. Mansour, and R. Khenata: collected the data, performed the analysis. All authors have been personally and actively involved in substantial work leading to the paper, and will take public responsibility for its content.

Availability of Data and Materials
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