Theoretical Investigations on the Mechanical, Magneto-Electronic Properties and Half-Metallic Characteristics of ZrRhTiZ (Z = Al, Ga) Quaternary Heusler Compounds

The electronic, magnetic, and mechanical properties were investigated for ZrRhTiZ (Z = Al, Ga) quaternary Heusler compounds by employing first-principles calculations framed fundamentally within density functional theory (DFT). The obtained electronic structures revealed that both compounds have half-metallic characteristics by showing 100% spin polarization near the Fermi level. The half-metallicity is robust to the tetragonal distortion and uniform strain of the lattice. The total magnetic moment is 2 μB per formula unit and obeys the Slater-Pauling rule, Mt = Zt − 18 (Mt and Zt represent for the total magnetic moment and the number of total valence electrons in per unit cell, respectively). The elastic constants, formation energy, and cohesive energy were also theoretically calculated to help understand the possibility of experimental synthesis and the mechanical properties of these two compounds.


Introduction
Half-metallic materials (HMMs) [1] can provide completely spin-polarized conducting electrons due to their unique electronic band structure which shows metallic characteristics in one spin channel and semiconducting/insulating properties in the other spin channel. Hence, HMMs are very much appreciated as key materials for providing high spin polarized carriers for spintronics devices.
As one of the important HMM families, Heusler compounds have a special importance due to their high Curie temperature, tunable electronic structure, and wide change of lattice constant [2,3]. Moreover, from the previous reports concerning pseudo-quaternary or ternary Heusler compounds with half-metallicity, it was found that the materials containing 4d or 5d transition metal elements generally demonstrate a wide band gap implying a robust half-metallicity in a quite uniform strain or tetragonal distortion, which is quite suitable for practical applications. Therefore, in recent years, the investigations on the Heusler compounds containing 4d or 5d transition metal elements have become the research focus [4][5][6][7][8][9]. Although many ternary Heusler compounds containing 4d transition metal elements have been reported in the literature, the quaternary Heusler compounds containing 4d or 5d transition metal elements, especially, the stoichiometric quaternary Heusler (SQH) compounds

Results and Discussions
The quaternary Heusler compounds have a LiMgPdSb-type structure, which belongs to the F43m (No.216) space group. There are four crystallographic sites in Heusler compounds. For quaternary Heusler compounds, there are three possible atomic arrangements depending on each crystallographic site occupation. In this present study, the atomic arrangement properties of ZrRhTiZ (Z = Al, Ga) compounds were investigated by calculating the total energy in ferromagnetic (FM) and non-ferromagnetic (NM) states. The calculated results show that for the ZrRhTiZ (Z = Al, Ga) compound, the most stable case is that Zr and Ti atoms with less valence electrons enter into the Wyckoff sites 4a(0, 0, 0) and 4c(0.25, 0.25, 0.25), Rh atoms with more valence electrons occupy the 4b(0.5, 0.5, 0.5) site, and the main group of atoms, Z atoms, tend to locate at the 4d(0.75, 0.75, 0.75) site. The corresponding simulated crystal structure is shown in Figure 1. The stable arrangement of ZrRhTiZ (Z = Al, Ga) Heusler compounds is like the combination of inverse Zr 2 RhZ [14] and Ti 2 RhZ (Z = Al, Ga) [15] Heusler compounds and found to be similar to the other Zr-based SQH compounds [16,17]. Figure 2 shows the dependence of the total energy on the lattice parameters in FM and NM states for ZrRhTiZ (Z = Al, Ga) compounds. One can see that both compounds are more stable in FM than NM states according to the viewpoint of the total energy minimization. The equilibrium lattice constants corresponding to the ground state energy are 6.47 Å and 6.45 Å for ZrRhTiAl and ZrRhTiGa, respectively.    and ZrRhTiGa compounds. From Figure 3, an indirect band gap can be clearly observed in the spin-down channel, and the Fermi level is located within the band gap. In the spin-up channel, the valence and conduction bands are overlapping, and the Fermi level is intersecting them for ZrRhTiZ (Z = Al, Ga) compounds, which indicates that both compounds are half-metallic materials. The band gap (E bg ) and HM band gap (E HM ) in the spin-down channel are listed in Table 1. The HM band gap is the minimum energy required to flip a minority of spin electrons from the valence band maximum (VBM) to the majority spin Fermi level. The E bg is 0.432 eV for ZrRhTiAl and 0.541 eV for ZrRhTiGa. Usually, the large E bg can be considered as evidence of the robustness of half-metallicity to lattice distortion. The Mulliken atomic populations quantify the charge transfer from one atom to another one and are listed in Table 2 for ZrRhTiZ (Z = Al, Ga) compounds. It is clear that the charge transfer from Ti to Rh is 0.69e and Zr to Al is 0.08e in ZrRhTiAl, and the charge transfer from Ti to Zr is 0.27e and Ga to Rh is 0.93e in ZrRhTiGa. The spin-projected band structures in the reduced Brillouin zone along the high symmetry directions were calculated under the equilibrium lattice constant and plotted in Figure 3 for ZrRhTiAl and ZrRhTiGa compounds. From Figure 3, an indirect band gap can be clearly observed in the spindown channel, and the Fermi level is located within the band gap. In the spin-up channel, the valence and conduction bands are overlapping, and the Fermi level is intersecting them for ZrRhTiZ (Z = Al, Ga) compounds, which indicates that both compounds are half-metallic materials. The band gap (Ebg) and HM band gap (EHM) in the spin-down channel are listed in Table 1. The HM band gap is the minimum energy required to flip a minority of spin electrons from the valence band maximum (VBM) to the majority spin Fermi level. The Ebg is 0.432 eV for ZrRhTiAl and 0.541 eV for ZrRhTiGa. Usually, the large Ebg can be considered as evidence of the robustness of half-metallicity to lattice distortion. The Mulliken atomic populations quantify the charge transfer from one atom to another one and are listed in Table 2 for ZrRhTiZ (Z = Al, Ga) compounds. It is clear that the charge transfer from Ti to Rh is 0.69e and Zr to Al is 0.08e in ZrRhTiAl, and the charge transfer from Ti to Zr is 0.27e and Ga to Rh is 0.93e in ZrRhTiGa.    Figure 4 shows the total density of state (TDOS) and partial density of state (PDOS) patterns. It is clear that the main contributions to the TDOS near the Fermi level are the strong d-d hybridization  Table 1. The equilibrium lattice constants (Å), total magnetic moments (µ B ), atomic magnetic moments (µ B ), valence band maximum (eV), conduction band minimum (eV), band gap (eV), half-metallic (HM) band gap (eV), and the number of valence electrons for ZrRhTiZ (Z = Al, Ga) compounds.  Figure 4 shows the total density of state (TDOS) and partial density of state (PDOS) patterns. It is clear that the main contributions to the TDOS near the Fermi level are the strong d-d hybridization among the Zr-4d state, Ti-3d state, and Rh-4d state. The spin polarization (P) at Fermi energy is 100% for ZrRhTiZ (Z = Al, Ga), for the Fermi level lies in a band gap in the spin-down channel. For Heusler compounds with half-metallicity, the origin of the half-metallic band gap is usually owed to the hybridization of the d electrons between the transition metal atoms. Through the classical molecular orbital approach, Zr-4d and Rh-4d hybridization within tetrahedral symmetry has been considered, as presented in Figure 5. In ZrRhTiZ (Z = Al, Ga) Heusler compounds, the 4d orbitals hybridization of Rh and Zr atoms generated five bonding bands (3t 2g and 2e g ) and five non-bonding bands (2e u and 3t 1u ). Then, the 3d orbitals of Ti atom hybridize with the five bonding 4d hybridized orbitals (3t 2g and 2e g ) of Rh and Zr atom, while the five non-bonding hybridized 4d orbitals (2e u and 3t 1u ) still hold with no hybridization. Finally, the distribution of the 15 d orbitals in the spin-down channel can be determined, i.e., 3 × t 2g , 2 × e g , 2 × e u , 3 × t 1u , 3 × t 2g , and 2×e g are arranged from high energy to low energy. The main-group of Z atoms generate 1 × s orbital and 3 × p orbitals which are distributed below the 15 hybridized d orbitals and totally occupied in ZrRhTiZ (Z = Al, Ga) compounds. The triple-degeneracy t 1u states are not occupied in the spin-down direction, and a band gap is formed between t 1u (non-bonding) and t 2g (bonding) orbitals in the spin-down channel of ZrRhTiZ (Z = Al, Ga) compounds. Similarly, from Table 1, it can be seen that the number of valence electrons is 20 per ZrRhTiZ (Z = Al, Ga) unit cell, and the calculated total magnetic moment is 2 µ B , which follows the Slater-Pauling rule M t = Z t − 18 and is also consistent with the reported Zr-based SQH materials [4][5][6][7][16][17][18]. among the Zr-4d state, Ti-3d state, and Rh-4d state. The spin polarization (P) at Fermi energy is 100% for ZrRhTiZ (Z = Al, Ga), for the Fermi level lies in a band gap in the spin-down channel. For Heusler compounds with half-metallicity, the origin of the half-metallic band gap is usually owed to the hybridization of the d electrons between the transition metal atoms. Through the classical molecular orbital approach, Zr-4d and Rh-4d hybridization within tetrahedral symmetry has been considered, as presented in Figure 5. In ZrRhTiZ (Z = Al, Ga) Heusler compounds, the 4d orbitals hybridization of Rh and Zr atoms generated five bonding bands (3t2g and 2eg) and five non-bonding bands (2eu and 3t1u). Then, the 3d orbitals of Ti atom hybridize with the five bonding 4d hybridized orbitals (3t2g and 2eg) of Rh and Zr atom, while the five non-bonding hybridized 4d orbitals (2eu and 3t1u) still hold with no hybridization. Finally, the distribution of the 15 d orbitals in the spin-down channel can be determined, i.e., 3 × t2g, 2 × eg, 2 × eu, 3 × t1u, 3 × t2g, and 2×eg are arranged from high energy to low energy. The main-group of Z atoms generate 1 × s orbital and 3 × p orbitals which are distributed below the 15 hybridized d orbitals and totally occupied in ZrRhTiZ (Z = Al, Ga) compounds. The triple-degeneracy t1u states are not occupied in the spin-down direction, and a band gap is formed between t1u (non-bonding) and t2g (bonding) orbitals in the spin-down channel of ZrRhTiZ (Z = Al, Ga) compounds. Similarly, from Table 1, it can be seen that the number of valence electrons is 20 per ZrRhTiZ (Z = Al, Ga) unit cell, and the calculated total magnetic moment is 2 μB, which follows the Slater-Pauling rule Mt = Zt − 18 and is also consistent with the reported Zr-based SQH materials [4][5][6][7][16][17][18].     Figure 6 displays total magnetic moment and atomic magnetic moments as a function of lattice constant for ZrRhTiAl and ZrRhTiGa compounds. It can be noted that the total magnetic moment is constant at 2μB over the range of the lattice constant, from 6.09 (6.10) to 6.61 (6.75) Å for ZrRhTiAl (ZrRhTiGa) compounds. The atomic magnetic moment is appreciably increasing with the lattice constant values for the Rh and Ti atoms, whereas the atomic magnetic moment of Zr as well as Al/Ga was not found to change appreciably over the wide range of lattice constants. The constant total magnetic moment can be attributed to the antiparallel arrangement of the magnetic moments of Rh and Ti atoms. The main contributors to the magnetic moment are Zr and Ti atoms. The Rh and Z atoms only carry small magnetic moments antiparallelly aligned to those of the Zr and Ti atoms.  Figure 6 displays total magnetic moment and atomic magnetic moments as a function of lattice constant for ZrRhTiAl and ZrRhTiGa compounds. It can be noted that the total magnetic moment is constant at 2µ B over the range of the lattice constant, from 6.09 (6.10) to 6.61 (6.75) Å for ZrRhTiAl (ZrRhTiGa) compounds. The atomic magnetic moment is appreciably increasing with the lattice constant values for the Rh and Ti atoms, whereas the atomic magnetic moment of Zr as well as Al/Ga was not found to change appreciably over the wide range of lattice constants. The constant total magnetic moment can be attributed to the antiparallel arrangement of the magnetic moments of Rh and Ti atoms. The main contributors to the magnetic moment are Zr and Ti atoms. The Rh and Z atoms only carry small magnetic moments antiparallelly aligned to those of the Zr and Ti atoms.
The half-metallicity is very much sensitive to the change of lattice constant. Therefore, it is very useful to acquire knowledge about the sensitivity of the half-metallicity to the uniform strain (US) for the ZrRhTiZ (Z = Al, Ga) compounds. Here, the CBM (conduction band minimum) and VBM in the spin-down channel at different lattice constants have been recorded to show the effects of the US on half-metallic behavior, as plotted in Figure 7. Obviously, the half-metallicity can be maintained in the lattice constant range of 6.09-6.61 Å for ZrRhTiAl and 6.10-6.75 Å for ZrRhTiGa. In addition, the effects of the US on the E bg and E HM are also shown in Figures 8 and 9 for the ZrRhTiZ (Z = Al, Ga) compounds, respectively. The obtained results revealed both E HM and E bg first increase and then decrease with the increasing lattice constant.  Figure 6 displays total magnetic moment and atomic magnetic moments as a function of lattice constant for ZrRhTiAl and ZrRhTiGa compounds. It can be noted that the total magnetic moment is constant at 2μB over the range of the lattice constant, from 6.09 (6.10) to 6.61 (6.75) Å for ZrRhTiAl (ZrRhTiGa) compounds. The atomic magnetic moment is appreciably increasing with the lattice constant values for the Rh and Ti atoms, whereas the atomic magnetic moment of Zr as well as Al/Ga was not found to change appreciably over the wide range of lattice constants. The constant total magnetic moment can be attributed to the antiparallel arrangement of the magnetic moments of Rh and Ti atoms. The main contributors to the magnetic moment are Zr and Ti atoms. The Rh and Z atoms only carry small magnetic moments antiparallelly aligned to those of the Zr and Ti atoms. The half-metallicity is very much sensitive to the change of lattice constant. Therefore, it is very useful to acquire knowledge about the sensitivity of the half-metallicity to the uniform strain (US) for the ZrRhTiZ (Z = Al, Ga) compounds. Here, the CBM (conduction band minimum) and VBM in the spin-down channel at different lattice constants have been recorded to show the effects of the US on half-metallic behavior, as plotted in Figure 7. Obviously, the half-metallicity can be maintained in the lattice constant range of 6.09-6.61 Å for ZrRhTiAl and 6.10-6.75 Å for ZrRhTiGa. In addition, the effects of the US on the Ebg and EHM are also shown in Figures 8 and 9 for the ZrRhTiZ (Z = Al, Ga) compounds, respectively. The obtained results revealed both EHM and Ebg first increase and then decrease with the increasing lattice constant.    The half-metallicity of most Heusler compounds will be influenced and even be broken by the tetragonal distortion (TD) of the lattice. To analyze the effect of TD on half-metallicity and magnetic moments, the unit-cell volume was fixed at the equilibrium bulk volume and then the c/a ratio changed. In Figures 10 and 11, the TD effects on the half-metallicity properties and magnetic moments are described by the magnetic moment and the CBM and VBM dependence on c/a ratio. It is clear that the total and atomic magnetic moments of ZrRhTiZ (Z = Al, Ga) are nearly unchanged, and the half-metallicity can be kept in the c/a ratio range of 0.96-1.06 for ZrRhTiAl and 0.92-1.12 for ZrRhTiGa, respectively. We further plotted the curves of Ebg and EHM as a function of c/a ratio, as shown in Figures 12 and 13, respectively. The Ebg and EHM have a similar variation tendency and showed a rise-fall characteristic with increasing c/a ratio. The largest Ebg and EHM occur in the cubic structure (c/a = 1) instead of tetragonal phase. The half-metallicity of most Heusler compounds will be influenced and even be broken by the tetragonal distortion (TD) of the lattice. To analyze the effect of TD on half-metallicity and magnetic moments, the unit-cell volume was fixed at the equilibrium bulk volume and then the c/a ratio changed. In Figures 10 and 11, the TD effects on the half-metallicity properties and magnetic moments are described by the magnetic moment and the CBM and VBM dependence on c/a ratio. It is clear that the total and atomic magnetic moments of ZrRhTiZ (Z = Al, Ga) are nearly unchanged, and the half-metallicity can be kept in the c/a ratio range of 0.96-1.06 for ZrRhTiAl and 0.92-1.12 for ZrRhTiGa, respectively. We further plotted the curves of E bg and E HM as a function of c/a ratio, as shown in Figures 12 and 13, respectively. The E bg and E HM have a similar variation tendency and showed a rise-fall characteristic with increasing c/a ratio. The largest E bg and E HM occur in the cubic structure (c/a = 1) instead of tetragonal phase.
is clear that the total and atomic magnetic moments of ZrRhTiZ (Z = Al, Ga) are nearly unchanged, and the half-metallicity can be kept in the c/a ratio range of 0.96-1.06 for ZrRhTiAl and 0.92-1.12 for ZrRhTiGa, respectively. We further plotted the curves of Ebg and EHM as a function of c/a ratio, as shown in Figures 12 and 13, respectively. The Ebg and EHM have a similar variation tendency and showed a rise-fall characteristic with increasing c/a ratio. The largest Ebg and EHM occur in the cubic structure (c/a = 1) instead of tetragonal phase.     Figure 13. The half-metallic band-gaps as functions of the c/a ratio (TD) for ZrRhTiAl and ZrRhTiGa.
Next, the elastic properties were investigated for ZrRhTiZ (Z = Al, Ga) compounds by calculating the single crystal elastic constants Cij. Heusler compounds belong to a cubic structure which needs only three independent elastic constants (C11, C12, and C44) for the complete description of their elastic properties.
The following expressions are used to depict the elastic behavior of a material given in Reference [19]: ) -( 5 12 11 44 2 C Figure 13. The half-metallic band-gaps as functions of the c/a ratio (TD) for ZrRhTiAl and ZrRhTiGa.
Next, the elastic properties were investigated for ZrRhTiZ (Z = Al, Ga) compounds by calculating the single crystal elastic constants C ij . Heusler compounds belong to a cubic structure which needs only three independent elastic constants (C 11 , C 12 , and C 44 ) for the complete description of their elastic properties.
The following expressions are used to depict the elastic behavior of a material given in Reference [19]: (6) where B, G, E, and A stands for bulk modulus, shear modulus, Young's modulus, and anisotropy factor, respectively. The estimated elastic constants and the mechanical properties for ZrRhTiAl and ZrRhTiGa are given in Table 3. For cubic structures, the following criteria for the mechanical stability must be satisfied [20]: Therefore, from Table 3, one can easily see that ZrRhTiZ (Z = Al, Ga) compounds are mechanically stable. The calculated results of Young's modulus (E = 57.9 and 47.8 for ZrRhTiAl and ZrRhTiGa, respectively) indicate that ZrRhTiAl is stiffer than ZrRhTiGa. The Pugh's index (B/G) and Cauchy pressure (C 12 − C 44 ) are usually used to characterize the brittleness or ductility of a compound [21][22][23]. For our materials, the Pugh's index (B/G) is 2.92 and 4.01, and the Cauchy pressures (C12 − C44) are 41.91 and 37.91 for the ZrRhTiAl and ZrRhTiGa compounds, respectively. Elastic anisotropy (A) is also a very important parameter for engineering applications. An A value of 1.0 shows the isotropic nature of the crystal, whereas values other than 1.0 indicate anisotropy. As shown in Table 3, for the ZrRhTiAl and ZrRhTiGa compounds, the value of A = 1, demonstrating that they are elastically anisotropic under ambient conditions. Finally, we have computed the formation and cohesive energies for ZrRhTiZ (Z = Al, Ga) based on the following expressions [24][25][26][27][28]: The calculated E c and E f for ZrRhTiZ (Z = Al, Ga) compounds have been listed in Table 3. The E c is larger than 20 eV, which indicates that ZrRhTiZ (Z = Al, Ga) compounds are stable due to energetic chemical bonding. The E f is smaller than 0 eV, showing the compounds are stable against separation into the four pure elements. Table 3. Calculated elastic constants C ij , bulk modulus B, shear modulus G, Young's modulus E (GPa), Pugh's ratio B/G, anisotropy factor A, and formation and cohesive energies (eV) for the stoichiometric quaternary Heusler (SQH) compounds ZrRhTiAl and ZrRhTiGa.

Summary
In this paper, the electronic structures, magnetic structures and moments, elastic constants, formation energies, and cohesive energies are calculated for ZrRhTiZ (Z = Al, Ga) quaternary Heusler compounds. Based on the obtained results, we predicted that ZrRhTiZ (Z = Al, Ga) quaternary Heusler compounds are half-metallic materials. These two compounds are among the rare SQH compounds with half-metallicity that contain 4d transition metal elements. The investigations on the effects of US and TD show that the half-metallicity is robust to US and TD. The half-metallicity can be sustained over a wide lattice constant range of 6.09 Å-6.61 Å for ZrRhTiAl and 6.03 Å-6.75 Å for ZrRhTiGa. Under TD, the range of c/a ratio maintaining the half-metallicity is 0.96-1.06 for ZrRhTiAl and 0.92-1.12 for ZrRhTiGa, respectively. The calculated elastic parameters are listed in this paper for ZrRhTiZ (Z = Al, Ga) compounds to help understand their mechanical properties.