Site preference and electronic structure of Mn₂RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb): a theoretical study

Heusler alloy has a structure of four interpenetrating face-center-cubic (fcc) lattices and a stoichiometric composition of X₂YZ, where X and Y atoms are transition metal elements, and Z atom is an element of group III, IV or V. Generally, Heusler alloy has two possible structures: Hg₂CuTi-type and Cu₂MnAl-type. In Hg₂CuTi-type structure, the X atoms occupy the A (0, 0, 0) and B (1/4, 1/4, 1/4) sites, and the Y atom enters the C (1/2, 1/2, 1/2) site, with Z atom located at the D (3/4, 3/4, 3/4) site in the Wyckoff coordinates. While in Cu₂MnAl-type structure, the X atoms occupy the A and C sites, Y atom enters the B site, leaving the D site to Z atom.

For 3d transition metal elements, the preference of transition metal elements is determined by the number of their valence electrons. And the elements with less electrons prefer to occupy the B sites, while others prefer to enter the (A, C) sites [29]. According to the previous studies on Mn₂RhZ (Z = Al, Ga, Sn), Felser et al. studied the tetragonal distortion based on the cubic Heusler alloys in Hg₂CuTi-type structure [30]. Thus, we considered Mn₂RhZ (Z = Al, Ga, In, Si, Ge, Sn and Sb) with Hg₂CuTi-type structure in this paper, and performed calculations for two cases of different initial spin moments for Mn atoms: antiparallel and parallel.

All these calculations concerning total energy as a function of lattice parameters, except for Mn₂RhIn alloy, are completely identical in both cases, and the calculated equilibrium lattice constants are listed in Table 1. In Table 1, the equilibrium lattice constants increase with increasing atomic number of Z atoms from the same column of the periodic table. The theoretical lattice constant of Mn₂RhAl is smaller than that of Mn₂RhGa because the Al atoms have a smaller atomic radius than Ga atoms. The radius of In atom is much larger than those of Al and Ga atoms. Thus, the lattice constant of Mn₂RhIn is the largest in the series of Mn₂RhZ (Z = Al, Ga and In). Similarly, Mn₂RhSn has the largest lattice constant among Mn₂RhZ (Z = Si, Ge and Sn) compounds and the lattice constants increase with the rising of the atomic number of Si → Ge → Sn.

The electronic structure of the Mn₂RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb) alloys has been studied by first-principles calculations. The calculated total and partial densities of states (DOSs) of Mn₂RhZ are shown in Fig. 1. In Fig. 1, we chose the majority-spin states as positive values and the minority-spin states as negative values. The states between –5 eV and –6 eV are mainly from the p electrons of Z atoms in the occupied valence states, which hybridize with the d electrons of Mn and Rh atoms and determine the degree of occupation of the p-d orbitals. The total DOSs of –4 eV to +2 eV arise from d states of transition metal atoms. The widely spread d states are mainly from covalent hybridization between Mn and Rh atoms. The covalent hybridization between the lower-energy d states of the higher-valence transition metal atom, like Rh, and the higher-energy d states of the lowervalence transition metal atom, like Mn, can be strong and lead to the formation of bonding and antibonding bands [31].

Fig. 1

The total DOSs of Mn₂RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb) shown in Fig. 2 can be divided into three groups according to valence electrons of Z atoms: Al, Ga and In alloys belonging to the first group, Si, Ge and Sn alloys attributed to the second group, and Mn₂RhSb as the third group. All total DOSs of the three-group compounds are similar in shape. The low energy states are mainly contributed by s electrons of Z atoms, and the s states are separated from the d states by a gap in the DOSs in both spin directions. Thus, we have ignored the effect of s electrons on the d-d interaction and have not presented s states in Fig. 1. Similar behavior has also been observed in Mn₂FeZ (Z = Al, Ga, Si, Ge, Sb) [32] alloys.

Fig. 2

In order to clarify the variation trend of the DOSs of the three groups, we analyzed the DOSs of Mn₂RhIn, Mn₂RhSn and Mn₂RhSb alloys, which are shown in the Fig. 2 (the 3^rd, 6^th and 7^th panel). The DOSs obviously move to the lower energy as the number of valence electrons increases. Sn atom has one more p state valence electron in comparison to In atom, and Mn₂RhSb also has an extra valence electron compared with Mn₂RhSn. The p electrons of the main group element hybridize with the d states of Mn and Rh, determining the degree of occupation of p-d orbitals and influencing the position of the majority spin states relative to the Fermi level due to the extra electrons from Z atoms filling in only majority spin states [33]. The majority states move to a lower energy as Z element is varying from In to Sn. The Fermi level locates at a deep DOS valley in the Mn₂RhSi, and moves to a large DOS peak in Mn₂RhSn. However, the change in minority states is not so obvious. The minority states are slightly shifted to a lower energy with respect to the Fermi level. Similar results have also been found in Mn₂FeAl and Mn₂FeSi alloys [32]. In the same way, the majority states of Mn₂RhSb shift to a lower energy with respect to the Fermi level. Meanwhile, the extra charge leads to a change in their spin moments. Mn (A) has four Z atoms as nearest neighbors, so the hybridization between Mn (A) and Z is strong. Thus, Mn (A) spin moment increases intensely.

It is worth noting that Sb atom has the largest number of valence electrons among the main group atoms Z (Z = Al, Ga, In, Si, Ge, Sn, Sb), and the majority DOS of Mn₂RhSb moves to a lower energy relative to the other Mn₂RhZ (Z = Al, Ga, In, Si, Ge, Sn) due to the extra electrons filling in only majority spin states. Mn (A) is the nearest neighbor atom of Sb atom, and the magnetic moment of Mn (A) reveals the most obvious change in Table 1. The moments of Mn (A) and Mn (B) for Mn₂RhSb are parallel instead of antiparallel ones for the other Mn₂RhZ (Z = Al, Ga, In, Si, Ge, Sn). Thus, the magnetic structure of Mn₂RhSb is ferromagnetic.

The total DOS of Mn₂RhAl compound is shown in Fig. 1a. A three-peak structure is in the majority spin states. The peak above E_F is mainly composed of the antibonding peak of Mn (A) and the occupied bonding peak results from the d states of Rh atom. It was reported that the bonding hybrids are localized mainly at the higher-valence transition metal atom site, such as Rh, and the unoccupied antibonding states are mainly from the lower-valence transition metal atom site, such as Mn [31]. A two-peak structure (bonding and antibonding peak), which is separated by an energy gap at the E_F, can be observed in the minority spin states. The PDOS shapes of Mn (A) and Mn (B) are opposite to each other. In the majority states of Mn (B), the two peaks are occupied and lie below the Fermi level. In the minority of PDOS, the exchange splitting moves the antibonding peak high above E_F, resulting in a large local moment of Mn (B). Large Mn moment has also been observed in other Heusler alloys, such as Mn₂FeZ (Z = Al, Ga, Si, Ge, Sb) [32]. In contrast, the partial DOS of Mn (A) lies below E_F in the minority spin states, and above E_F in the majority spin states. Therefore, the contributions to the total DOS from Mn (A) and Mn (B) are opposite to each other, indicating an antiparallel configuration of their spin moments [34]. With the help of the calculated total and partial magnetic moments in Table 1, it can be inferred that the spin splitting of Mn (B) is large, resulting in a large local moment of Mn (B). The atomic energy level E_(A;C) < E_(B) has been reported by Bansil et al. [34]. They stated that the spin down bonding states preferentially reside at Rh (C) site and produce a negative spin density in the occupied states, resulting in strong bonding states in the minority spin states of Rh and causing a small exchange splitting. As a result, Rh shows only a small magnetic moment. The total moment of Mn₂RhAl is 2.00 μ_B, which is mainly determined by the antiparallel aligned Mn (A) and Mn (B) spin moments.

The DOSs of Mn₂RhGa and Mn₂RhIn are shown in Fig. 1b and Fig. 1c, and are similar in shape to the one of Mn₂RhAl. The two-peak structures are in the minority spin states and are separated by the energy gaps around E_F. In the majority spin states, the Fermi level falls in a pseudogap, dividing the majority states into bonding and antibonding parts. The PDOSs of Mn (A) and Mn (B) contribute to both the bonding and antibonding parts in the total DOS. Mn atoms show spin splitting in both sites, and a different trend in their PDOS is observed. In Fig. 2, we gathered the calculated total DOS of Mn₂RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb). Mn(A) is the nearest neighbor atom of the main group atoms (Al, Ga, In), thus, the influence of Al ! In on the Mn(A) atom is the largest. The DOSs of Mn₂RhZ (Z = Al, Ga, In) move to a lower energy as Z is varying from Al to In because of the p-d hybridization between Z and Mn (A) atoms. From the total DOSs of Mn₂RhZ (Z = Si, Ge, Sn) in Fig. 1d, Fig. 1e and Fig. 1f, an energy gap is in the minority states at the E_F. And in the majority spin states, the antibonding peak mainly from Mn (A), is just located at the Fermi level. The energy gap variation trend, which is shown in Fig. 2 (the 4^th to 6^th panel), is similar to the trend in Mn₂RhAl, Mn₂RhGa and Mn₂RhIn. The DOS moves to a lower energy as Z varies from Si to Sn, and the E_F falls throughout the energy gap. In contrast, the bonding peak in the majority states moves to a higher energy with the increase in atomic number of Si → Ge → Sn. The Z atom plays an important role in Heusler alloys, which causes s-p states to hybridize with d electrons and determines the degree of the p-d orbital occupation [31]. The splitting of Z atom p states electrons occurs at higher energy when Z varies from Si to Sn (–2.9 eV for Si, –2.8 eV for Ge and –2.5 eV for Sn). Thus, the bonding hybridization between p and d electrons can be observed at the higher energy. The total magnetic moment of the Mn₂RhZ (Z = Si, Ge, Sn) alloys is 3.00 μ_B, which is in line with the s-p rule of M_t = Z_t – 24. Mn (B) spin moment is large and increases with increasing atomic number of Z atoms. Mn (A) and Rh atom moments are small and contribute little to the total moment.

The calculated total and partial DOS for Mn₂RhSb is shown in Fig. 1g. There are three-peak structures in both spin directions. The bonding peak around –3.2 eV in the minority spin states is mainly determined by the Rh atom. And in the majority spin states, the bonding peak around –3 eV mainly arises from the hybridization of Mn and Rh atoms. The majority states are similar to the one of Mn₂RhSi, the Fermi level is also located on the antibonding part. In this paper, Mn₂RhSb is a ferromagnet with a small positive magnetic moment at the Mn (A) site, different from the ferrimagnet of the others. The magnet moment strongly depends on the atomic order of the nearest neighbor shell. The moment of Mn (B) is nearly constant because the Mn (A) and Rh atoms are the nearest atoms to Mn (B). In contrast, the moment of Mn (A) atom changes considerably as the nearest atom of Sb provides p electrons to hybridize with d states and determines the degree of p-d orbital occupation [30]. The PDOS of Mn (A) moves to a lower energy, resulting in the spin moment parallel to that of Mn (B).

In the total DOSs of Mn₂RhZ, the energy gap can be obtained in the minority spin around the Fermi level. However, in the majority states, the DOSs are quite large, indicating a high spin polarization at the E_F. The spin polarization ratios of Mn₂RhZ (Z = Al, Ge, Sb) are 100%due to the fact that the alloys do not have DOSs at E_F in the minority gap. Here, the spin polarization ratio is calculated as the value of (N_↑ – N_↓)/(N_↑+ N_↓), where N_↑ and N_↓ are the majority-spin and minorityspin DOSs at the E_F, respectively. So, Mn₂RhAl, Mn₂RhGe and Mn₂RhSb are “true” half-metals.

With the help of the band structure, the majority-spin and minority-spin band structures under the theoretical equilibrium lattice parameters, 5.80 Å for Mn₂RhAl, 5.79 Å for Mn₂RhGe and 6.02 Å for Mn₂RhSb, are shown in Fig. 3. In the majority-spin states, the top of valence band overlaps with the E_F, exhibiting metallic character at the E_F. However, the minority-spin states exhibit an indirect G-X gap, and the E_F is exactly located in the band gap, indicating semiconducting nature. The values of the band gap are shown in Table 1. In summary, Mn₂RhAl, Mn₂RhGe and Mn₂RhSb alloys have an ideal half-metallic band structure.

Fig. 3

In the half-metallic Heusler alloys, the total moments M_t are integral values and follow the Slater-Pauling rule of M_t = Z_t– 24, where Z_t is the number of valence electrons [33]. Mn₂RhAl, Mn₂RhGe and Mn₂RhSb have 26, 27 and 28 valence electrons, respectively. The total magnetic moment should be 2.00 μ_B, 3.00 μ_B and 4.00 μ_B, respectively, which agrees with our results in Table 1 quite well.

The calculations for all the cases have been performed under their theoretical equilibrium lattice constants and Mn₂RhZ (Z = Al, Ge, Sb) alloys exhibit half-metallic characters. However, a small change in the lattice parameter may shift the E_F with respect to the minority gap, which obviously affects the half-metallic character [35]. In experimental studies, such as melt spinning or ball milling, the strain may be quite large and will make the lattice constant deviate from the ideal one. Furthermore, the growth of thin film materials becomes possible by modern techniques. The lattice constant of the thin films is strongly influenced by the relationship between the half-metallic character and the lattice constant for a given material from both theoretical and technical point of view. Therefore, it is meaningful to study the dependence of the magnetic properties and spin polarization on the lattice parameters for Mn₂RhAl, Mn₂RhGe and Mn₂RhSb.

Here, we have investigated the relationship between the dependence of the magnetic moments and lattice parameters for the three half-metals. The total magnetic moments as functions of the lattice parameter for Mn₂RhZ (Z = Al, Ge, Sb) are shown in Fig. 4. A moderate variation in the lattice parameter does not significantly change the total magnetic moments in these compounds as the movement of the Fermi level usually takes place within the minority gap. The half-metallic properties of Mn₂RhZ (Z = Al, Ge, Sb) can be obtained within the range of 5.56 Å to 5.96 Å, 5.69 Å to 6.06 Å, 5.81 Å to 6.49 Å, respectively. Thus, Mn₂RhAl, Mn₂RhGe and Mn₂RhSb can be considered as new candidates for spintronics applications.

Fig. 4

Furthermore, we have shown the influence of tetragonal distortion on the half-metallicity of Mn₂RhAl as an example of the later research of tetragonal distortion in the Mn₂RhZ. We have investigated the relative stability of the unit cell geometry by calculating the total energy as a function of the c/a ratio at a constant volume and the respective graph is shown in Fig. 5. The lowest total energy can be observed at the point of c/a = 1, indicating the stability of cubic phase in the structure.

Fig. 5