Origin of the half-metallic band-gap in newly designed quaternary Heusler compounds ZrVTiZ (Z = Al, Ga)

Abstract

In this work, first-principles calculations have been used to investigate the electronic structures, magnetic properties, and half-metallic nature of the newly designed quaternary Heusler compounds ZrVTiAl and ZrVTiGa. The calculated results reveal that these two compounds are half-metallic ferrimagnets with a total magnetic moment (M_t) of 2 μ_B, and the M_t is in line with the Slater–Pauling curve of M_t = 18 − Z_t, where Z_t is the total number of valence electrons. Furthermore, via a schematic diagram of the possible d–d hybridization between the transition-metal elements Zr, V, and Ti, we discuss the origin of the half-metallic band gap in the majority spin channel. Also, we have investigated the half-metallic states versus the lattice parameter and the structural stability, i.e., the cohesion energy and formation energy of ZrVTiAl and ZrVTiGa compounds. We hope that our work may trigger Zr-based Heusler compounds for application in future spintronics devices.

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1. Introduction

To improve the performance of spintronic devices, e.g. the spin filter and the spin valve,¹ we need to search for new magnetic materials with high spin polarization. Among them, half-metallic Heusler compounds are an ideal choice of high-spin-polarization materials due to their 100% spin polarized charge carriers at the Fermi level: namely, one spin channel shows metallic behavior, and the other one shows semiconducting behavior.

It is well known that Heusler compounds have very high Curie temperatures (T_C),² compatible lattice structure to the widely used semiconductors, and easy fabrication, which make them one class of the most promising candidate materials for spintronics devices. The family of Heusler compounds has become a topic of research focus since the concept of half-metallicity arose from the band-structure calculations by de Groot et al. for half-Heusler-based NiMnSb compound in 1983.³ After that, many Heusler based compounds were predicted to be half-metals from the band-structure calculations,^4–10 and this was verified experimentally.^4,5

Very recently, the scope of the half-metals with Heusler structure has been extended to the compounds containing 4d elements,^11–27 especially Zr-based Heusler compounds. Zr₂CoZ (Z = Al, Ga, In, Si, Ge, Sn, Pb, Sb),^13–16 Zr₂CrZ (Z = Ga, In),¹⁷ Zr₂IrZ (Z = Al, Ga, In),¹⁸ Zr₂RhZ (Z = Al, Ga, In),¹⁹ Zr₂MnZ (Z = Al, Ga, In),²⁰ and Zr₂VZ (Z = Al, Ga, In, Si, Ge, Sn, Pb)^21–23 compounds with a Hg₂CuTi-type Heusler structure have been found to be half-metals based on first-principles calculations, which has greatly enriched the potential applications of Heusler compounds. Also, the quaternary Heusler compounds ZrCoTiZ (Z = Si, Ge, Ga, Al),²⁴ ZrFeTiZ (Z = Al, Si, Ge),²⁵ ZrNiTiAl,²⁵ ZrMnVZ (Z = Si, Ge),²⁶ ZrCoFeZ (Z = Si, Ge, P),²⁶ ZrCoVIn, ZrCoCrBe, ZrFeCrZ (Z = Ga, In), and ZrFeVGe²⁷ have been reported to be half metals. Among these quaternary Heusler compounds mentioned above, ZrCoVIn, ZrCoCrBe, ZrFeCrIn, and ZrCoFeP compounds²⁷ can even be regarded as spin-gapless semiconductors.²⁸ To the best of our knowledge, a few quaternary compounds containing 4d elements, such as CoRhMnGa and CoRhMnSn,^29,30 have been realized experimentally, but a large part of the current research is focused on theoretically predicting half-metallic materials containing 4d elements.

In this manuscript, we mainly focus on the electronic, magnetic, and half-metallic properties of the equiatomic quaternary Heusler compounds ZrVTiAl and ZrVTiGa. These two compounds can be regarded as combinations of Zr₂VZ (Z = Al, Ga)^19,20 and Ti₂VZ (Z = Al, Ga)³¹ compounds. Although Zr₂VZ (Z = Al, Ga) and Ti₂VZ (Z = Al, Ga) with a Hg₂CuTi-type Heusler structure have already been predicted by Gao et al. and Galehgirian et al., based on first-principles calculations, ZrVTiZ (Z = Al, Ga) compounds have not been investigated, either experimentally or theoretically. As is well known, the half-metallic band gap is the defining characteristic for the half-metals ZrVTiZ (Z = Al, Ga), and thus, we will discuss the origins of the band gap via a simple schematic diagram of possible d–d hybridizations between the transition-metal elements Zr, V, and Ti. Moreover, we will test the structural stability and the evolution of half-metallic states under two types of strain, i.e., hydrostatic strain and tetragonal strain.

2. Computational details

The full-Heusler alloys represent a class of ternary intermetallic compounds with the general formula X₂YZ, where X and Y are transition metals and Z is a main group element. When one of the two X atoms is replaced by a different transition metal M, a new quaternary Heusler alloy with the formula XMYZ is obtained. Normally, Zr based equiatomic quaternary Heusler compounds crystallize in the LiMgPdSn-type crystal structure, including ZrCoTiZ, ZrFeTiZ, and ZrNiTiAl compounds.^24,25,32 The resulting structure has F43̄m symmetry with Wyckoff positions X: 4a (000), M: 4d (0.5, 0.5, 0.5), Y: 4c (0.25, 0.25, 0.25), Z: 4b (0.75, 0.75, 0.75). The crystal structures of the ZrVTiZ (Z = Al, Ga) compounds in our current work are plotted in Fig. 1.

Fig. 1 Crystal structure of quaternary Heusler compounds ZrVTiZ (Z = Al, Ga). Generally, ZrVTiZ quaternary Heusler compounds crystallize in the LiMgPdSn-type crystal structure. In the Wyckoff coordinates of the LiMgPdSn-type crystal structure, the Zr atoms occupy the 4a (0,0,0) site, V atoms occupy the 4c site, Ti atoms occupy the 4b site, and Z atoms occupy the 4d site.

The Cambridge Serial Total Energy Package (CASTEP) code is an effective ab initio program based on quantum mechanics. It can precisely simulate the ground state structure, bands, optical properties, magnetic properties, etc. In this work, all calculations were performed by using CASTEP code based on the pseudopotential method with a plane-wave basis set.^33,34 Moreover, the valence electron configurations of Zr (4d²5s²), V (3d³4s²), Ti (3d²4s²), Al (3s²3p¹), and Ga (4s²4p¹) for ZrVTiZ (Z = Al, Ga) compounds were selected. The interactions between the atomic core and the valence electrons were described by ultrasoft pseudopotentials.³⁵ The generalized gradient approximation (GGA)³⁶ was selected for the exchange–correction functional. For all cases, a plane-wave basis set cut-off of 450 eV was used. A mesh of 12 × 12 × 12 k-points was used for Brillouin zone integrations. These parameters ensured good convergence for total energy and the convergence tolerance for the calculations was selected as a difference in total energy within 1 × 10⁻⁶ eV per atom.

3. Results and discussion

Firstly, to present the ground state properties, we calculated the total energy as a function of the lattice constant around the equilibrium lattice constant for each compound in both the non-magnetic (NM) and ferromagnetic (FM) configurations. The results on the lattice constant versus total energy for the quaternary Heusler ZrVTiZ (Z = Al, Ga) compounds are shown in Fig. 2. We can see that the FM state is energetically the most stable magnetic configuration for ZrVTiZ (Z = Al, Ga) compounds, and the obtained equilibrium lattice constants of these two compounds are listed in Table 1, i.e., 6.54 Å and 6.5 Å, respectively. Furthermore, we need to point out that the FM state is still stable compared to the NM state under uniform strain and tetragonal distortion.

Fig. 2 Calculated total energy as a function of lattice constant for quaternary Heusler compounds ZrVTiZ (Z = Al, Ga). NM and FM correspond to nonmagnetic and ferromagnetic calculations.

Table 1 Optimized lattice constants (a), calculated total and atomic magnetic moments (M_t) per formula unit, size of the band gap (E_bg) and the spin-flip gap (E_sfg), and the cohesive (E_c) and formation (E_f) energies for the hypothetical quaternary Heusler compounds ZrVTiZ (Z = Al and Ga)

Compounds	a (Å)	Mtot (μB)	MZr	MV	MTi	MZ	Ebg (eV)	Esfg (eV)	Eformation	Ec
ZrVTiAl	6.54	2.00	−0.72	2.48	0.36	−0.12	0.49	0.19	−0.26	25.33
ZrVTiGa	6.5	2.00	−0.62	2.40	0.38	−0.16	0.56	0.19	−0.69	24.63

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To the best of our knowledge, there are no previous experimental or theoretical data on ZrVTiZ (Z = Al, Ga) compounds. Thus, we need to investigate the structural stability of these two newly designed compounds from the aspects of cohesion energy and formation energy. The cohesion energy E_c was calculated according to the formula: E_c = E^iso_Zr + E^iso_V + E^iso_Ti + E^iso_Z − E^total_{ZrVTiZ(Z = Al,Ga)}, where E^total_ZrVTiZ is the equilibrium total energy achieved by first principles calculations of the ZrVTiZ (Z = Al, Ga) compounds per formula unit and E^iso_Zr, E^iso_Ti, E^iso_V, and E^iso_Z are the energies of isolated Zr, Ti, V, and Z atoms, respectively. As is well known, the value of E_c is a measure of the strength of the force that binds atoms together in the solid state, which is correlated with the structural stability in the ground state. The calculated values obtained for the cohesion energy at the equilibrium lattice constant are 25.33 eV for ZrVTiAl and 24.63 eV for ZrVTiGa compounds, respectively. Obviously, all the values of E_c for ZrVTiZ (Z = Al, Ga) compounds are greater than 20 eV. Such high cohesion energies indicate that ZrVTiZ (Z = Al, Ga) compound crystals should be expected to be stable due to the high energy of their chemical bonds.

Formation energy refers to the stability of the compound against decomposition into its bulk constituents. The formation energy is calculated using the formula: E_f = E^total_{ZrVTiZ(Z = Al,Ga)} − [E^bulk_Zr + E^bulk_V + E^bulk_Ti + E^bulk_Z], where E^bulk_Zr, E^bulk_V, E^bulk_Ti, and E^bulk_Z correspond to the total energy per atom for the Zr, V, Ti, and Z atoms, respectively. The values obtained for the formation energy are −0.26 eV, and −0.69 eV, respectively, and their negative formation energies also indicate the thermodynamic stability of these compounds. We also calculated the phonon spectra of ZrVTiZ (Z = Al, Ga) compounds because phonon spectrum calculations³⁷ are always used as another method to examine the stability of compounds. The results showed, however, that the spectra have imaginary frequencies for these two compounds. Although the calculated phonon spectra indicate that these two compounds are not stable, the negative formation energies and the high energy of their chemical bonds can be used as important evidence for the structural stability of these two compounds.

With the GGA – Perdew–Burke–Ernzerhof (PBE) method, we calculated the band structures of ZrVTiZ (Z = Al, Ga) compounds along the main symmetry directions in the irreducible Brillouin zone, based on the obtained equilibrium lattice constants. The calculated results are shown in Fig. 3. For these two compounds, we can see that the bands overlap with the Fermi level in the minority spin channel. In the majority spin channel, however, a relatively wide band gap can be observed, and the Fermi level is located in the gap, reflecting its semiconducting nature. Therefore, ZrTiVAl and ZrTiVGa compounds can be regarded as half-metallic materials.

Fig. 3 ZrVTiZ (Z = Al, Ga) band structures at their equilibrium lattice constants. (The blue and red lines denote the majority and minority spin band structures, respectively.)

The band gaps of ZrVTiZ (Z = Al, Ga) compounds were calculated by using the highest occupied and the lowest unoccupied bands in the majority spin channel. The results show that the band gaps of these two compounds are quite large, with values of 0.49 eV for ZrVTiAl and 0.56 eV for ZrVTiGa. Furthermore, the spin-flip gap, which is defined as the minimum between the lowest energy of the majority (minority) spin conduction bands with respect to the Fermi level and the absolute value of the highest energy of the majority (minority) spin valence bands, is 0.19 eV for ZrVTiZ (Z = Al, Ga) compounds. These spin-flip gaps are comparable to the large spin-flip gaps of a quaternary Heusler compound without a 4d transition-element, CoFeCrAl (about 0.16 eV).³⁸ Normally, the half-metallic behaviour of 3d transition-element-based Heusler compounds is not stable and is always destroyed when the lattice is slightly strained, mainly because of the small spin-flip gap. Therefore, ZrVTiZ (Z = Al, Ga) compounds can be regarded as good candidates for practical applications as spintronic devices due to their excellent half-metallic behaviour with large spin-flip gaps.

It is well accepted that the origin of the band gap is very important for the design of new half-metallic materials. Normally, the explanations for the origin of the band gap for half-metallic materials can be divided into three types: (i) the covalent band gap; (ii) the d–d orbital hybridization band gap; and (iii) the charge transfer band gap.³⁹ Based on the works of Skaftouros' group and Zhang et al.,^40,41 we start the discussion with a possible d–d hybridization between the transition-metal atoms. Through the classical molecular orbital approach, Zr-4d and V-3d hybridization within octahedral symmetry has been considered, as presented in Fig. 4(a). Obviously, the Zr–V bonding creates five bonds and five anti-bonds, i.e., 2e_u, 3t_1u, 3e_g, and 3t_2g from top to bottom. Then, the Zr–V orbits hybridize with those of Ti-3d atoms. The e_g and t_2g orbitals of the Zr–V system hybridize with the d_xy, d_yx, d_zx, d_z², and d_x²−y² orbitals of the Ti-3d atom, resulting in a lower energy bonding orbital, 3t_2g/2e_g, and a higher energy anti-bonding orbital, 2e_u/3t_1u. To further understand the mechanism for the origin of the band gap, we selected ZrVTiAl compound as an example and show the band structures in both spin channels based on the schematic diagram of d–d hybridization between the Zr, V, and Ti atoms in Fig. 4(b) and (c), and we find that the complex exchange splitting of Z–V and Zr–Ti will shift the Fermi level to the appropriate position (between the t_1u and t_2g bands) in the majority spin channel, and it may be responsible for the origin of the gap denoted as the t_1u–t_2g band gap. Fig. 5 presents the spin-projected total and atomic density of states (DOS) for ZrVTiAl and ZrVTiGa compounds. Obviously, the main contributions in the energy regions between −4 eV and −2 eV come from the p–d hybridization between the Z-p and the Zr-4d, Ti-3d, and V-3d states. The states over the range from −2 eV to 5 eV can mainly be attributed to the d–d hybridization among the Zr-4d, Ti-3d, and V-3d states, which is in good agreement with our above discussion. The p–d hybridization between the Al/Ga and Zr/V/Ti atoms is not very obvious, however, in our current calculations, as shown in Fig. 5.

Fig. 4 (a) Schematic diagram of possible d–d hybridization between the transition-metal elements Zr-4d, V-3d, and Ti-3d in the ZrVTiZ (Z = Al, Ga) quaternary Heusler compounds. The main group elements Z (Z = Al, Ga) are not taken into account because the sp-bands are located at deep energy levels and barely contribute to the gap formation. (b) Majority and (c) minority spin band structures of the ZrVTiAl compound.

Fig. 5 Spin-polarized total and atom-resolved DOSs of the quaternary Heusler compounds ZrVTiZ (Z = Al, Ga) at their equilibrium lattice constants: (a) ZrVTiAl and (b) ZrVTiGa.

To test the stability of the half-metallic states against change in the lattice parameters for ZrVTiAl and ZrVTiGa compounds, we calculated a series of band structures for ZrVTiAl and ZrVTiGa compounds with different lattice parameters. The valence band maximum (VBM) and the conduction band minimum (CBM) in the majority spin channel are used to characterize the half-metallic properties because the band structures are quite similar under different degrees of lattice compression/expansion except for the band gap. Fig. 6(a) and (b) shows the energies of the CBM and VBM versus the lattice parameters. Obviously, the ZrVTiAl and ZrVTiGa compounds can keep their half-metallic properties at lattice constant values of 6.14–6.91 Å and 6.06–6.86 Å, respectively. At the lattice constants a = 6.13 Å (6.05 Å) and a = 6.92 Å (6.87 Å), the VBM of ZrVTiAl (ZrVTiGa) overlaps with the Fermi level, and the half-metallic behavior disappears, and thus, the ZrVTiAl (ZrVTiGa) compound becomes a common magnet.

Fig. 6 (a) and (b) CBM and VBM in the minority spin direction as a function of the lattice constant for ZrVTiZ (Z = Al, Ga) compounds; (c) and (d) CBM and VBM in the minority spin direction as a function of the c/a ratio for ZrVTiZ (Z = Al, Ga) compounds.

It is well accepted the half-metallic properties of most Heusler compounds will be influenced and even be broken by the effect of tetragonal deformation. Therefore, we will investigate the stability of the half-metallic states under lattice tetragonalization. In order to do this, we fix the unit-cell volume at the bulk equilibrium value, i.e., V = a³, and then we change the c/a ratio. The curves of the energies of the CBM and VBM as functions of the lattice constant have been plotted in Fig. 6(a) and (b) for ZrVTiAl and ZrVTiGa, respectively, and the c/a ratio for the ZrVTiAl and ZrVTiGa compounds has been plotted in Fig. 6(c) and (d), respectively. The half-metallic properties of the ZrVTiAl and ZrVTiGa compounds exhibit a low sensitivity to tetragonalization. In detail, ZrVTiAl and ZrVTiGa compounds can maintain their half-metallic properties when their c/a ratios are changed in the range of 0.92–1.12 and 0.91–1.14, respectively.

Finally, we come to the magnetic properties of ZrVTiAl and ZrVTiGa compounds. We give the values of the total magnetic moments and atomic magnetic moments of ZrVTiAl and ZrVTiGa compounds in Table 1. The total spin magnetic moments are an integral value of 2 μ_B, and the atomic magnetic moments of Z (Z = Al, Ga) atoms are very small and only make a small contribution to the total magnetic moment. For these two compounds, the total magnetic moment mainly comes from the Zr, V, and Ti atoms. The atomic magnetic moments of Zr are −0.72 μ_B, −0.62 μ_B, the spin magnetic moments of V atoms are 2.48 μ_B, 2.40 μ_B, and the spin magnetic moments of Ti atoms are 0.36 μ_B, 0.36 μ_B, respectively. Comparing the atomic magnetic moments of the Zr and V/Ti atoms, we find that the atomic magnetic moments of these atoms are in antiparallel alignment, and thus, the ZrVTiAl and ZrVTiGa compounds are half-metallic ferrimagnets. The magnetic moments of the Zr and V atoms are in agreement with the theoretical data on Zr₂VZ compounds.^21,22 Furthermore, we know that the total magnetic moment (M_t) of the half-metallic Heusler compounds Zr₂VZ and Ti₂VZ^21,22,31,42 scales linearly with the number of valence electrons (Z_t), according to M_t = 18 − Z_t. As mentioned above, the total magnetic moment is 2 μ_B for ZrVTiAl and ZrVTiGa compounds, which have 16 valence electrons per unit cell. Therefore, the Slater–Pauling rule M_t = 18 − Z_t is suitable for these two quaternary Heusler compounds in our current work.

4. Conclusions

First-principles calculations have been used to investigate the electronic structures, and the magnetic and half-metallic properties of the LiMgPdSn-type quaternary Heusler compounds ZrVTiAl and ZrVTiGa. The results show that these two equiatomic quaternary Heusler compounds are half-metallic ferrimagnets with large band gaps in the majority spin channel of 0.49 eV for ZrVTiAl and 0.56 eV for ZrVTiGa. By the classical molecular orbital approach, we find that the majority spin channel band gaps mainly arise from t_1u–t_2g splitting. The spin-flip gap of ZrVTiZ (Z = Al, Ga) is 0.19 eV, which is comparable to the large spin-flip gap of CoFeCrAl Heusler compound. The calculated total magnetic moment of ZrVTiZ (Z = Al, Ga) is 2 μ_B, and the value agrees well with the Slater–Pauling curve based on 18 minus the total number of valence electrons. ZrVTiZ (Z = Al, Ga) compounds are stable from the aspects of cohesion energy and formation energy, and thus, these compounds may be synthesized experimentally. Moreover, the half-metallic states could be maintained as the lattice constants changed from 6.14–6.91 Å for ZrVTiAl and 6.06–6.86 Å for ZrVTiGa, and the c/a ratio ranged from 0.92–1.12 for ZrVTiAl and from 0.91–1.14 for ZrVTiGa, respectively.

Acknowledgements

Many thanks are owed to Dr Tania Silver for critical reading of the manuscript. Z. X. Cheng thanks the Australian Research Council for support. G. D. Liu acknowledges financial support from the Natural Science Foundation of Hebei Province (No. E2016202383), the Hebei Province Higher Education Science and Technology Research Foundation for Young Scholars (No. Q2012008), the Project for Scientific Research for High Level Talent in Colleges and Universities of Hebei Province (No. GCC2014042), and the Hebei Province Program for Top Young Talents.

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