Spin Hall Conductivity and Anomalous Hall Conductivity in Full Heusler compounds

The non-zero SHC can be obtained for all the energy levels below and above the Fermi level normally. We can plot the SHC as a function of the Fermi level. However, only the electrons with energy around the Fermi level are relevant in most spin transport phenomena. We first show the values of $\sigma_{xy}^{z}$ at the Fermi level. They are plotted with respect to the number of the valence electron as shown in Fig.2(a). The chemical formula of the compounds with large SHC are shown in the figure. All the data can be found in the Supplemented Materials. We can see from the figure that the value reaches its maximum when the number of the valence electrons $N_{v}$ is 26. The largest value in the nonmagnetic compounds is about 400 $\hbar/e$($\Omega^{-1}$cm^-1) in Ni₂LiSn. In the ferromagnetic compounds, the maximum value is more than 500 $\hbar/e$($\Omega^{-1}$cm^-1) in Co₂MnGa and in Cu₂CoSn, with the magnetic moment about 4.1 $\mu_{B}$/f.u. and 1.1 $\mu_{B}$/f.u., respectively. The magnetic moment projected on Co and Mn is about 0.67 and 3.0 $\mu_{B}$ in Co₂MnGa, and on Co is 1.13$\mu_{B}$ in Cu₂CoSn. The non-integer magnetic moment indicates the partial filling of the d-orbitals and results in a low mobility of conduction electrons. The valence electron number dependent SHC can be partly understand from the electronic structure where the different orbitals are occupied as discussed in the next sections. From the data, we see that they are all below 2000 $\hbar/e$($\Omega^{-1}$cm^-1), which is the value typically obtained in Pt[10]. We see that for the nonmagnetic compounds, for example, the SHC in Ni₂LiSn can reach as large as 400 $\hbar/e$($\Omega^{-1}$cm^-1) while in the ferromagnetic ordered Co₂MnGa and Cu₂CoSn, it is about 600 $\hbar/e$($\Omega^{-1}$cm^-1). The relatively small value of the SHC is rooted in the position of the Fermi level. It is not located at the place where the SOC mixes the states and opens a gap, which can enhance the Berry curvature. Thus the position of the Fermi level is crucial. However, in the experiments by Leiva et al. [22], the resistivity of Co₂MnGa thin film is about 220 $\mu\Omega^{-1}cm^{-1}$, which is ten times larger than that of Pt thin films. When taken into account of this smaller electric conductivity in these compounds[7], the spin Hall angle, which measures the efficiency of the spin to charge conversion efficiency, can be significantly larger than that in pure noble metals.

The AHC at the Fermi level is also shown in Fig.2(b). We can see that although the maximum of AHC obtained is in Rh₂MnAl. The same group of the compounds, Co₂MnGa, Co₂MnAl, and Rh₂MnAl can show significant high AHC and SHC among others with the same N_v. The overall distributions of both conductivities with respect to the number of valence electrons are similar. This similarity is rooted on the same mechanism of the both effects. In a simplified two band conducting model, if the spin-non-conserving contributions are overlooked [23, 24], $\sigma_{xy}^{A}=\sigma_{xy}^{\uparrow}+\sigma_{xy}^{\downarrow}$ and $\sigma_{xy}^{S}=(\sigma_{xy}^{\uparrow}-\sigma_{xy}^{\downarrow})\frac{2\hbar}{e}$. The relative contributions of the spin channels can be measured by spin polarization $p^{H}$ of the Hall current written as

where $\sigma_{xy}^{\uparrow}$ and $\sigma_{xy}^{\downarrow}$ are the spin-up and spin-down Hall conductivities, respectively. We listed the total magnetic moment, spin and Hall current polarization, SHC and AHC of the five compounds in Tab. 1. We read that except Rh₂MnAl, all others show absolute value of $p^{D}$ being around -1.0. The Co-based Heusler compounds were explored by Tung and Guo[24], in which Co₂MnAl and Co₂MnGa were highly proposed to be used in spintronics due to their simultaneously high AHC and SHC. According to our present results, Cu₂CoSn should be added to this material family of high Hall conductivity. We noticed the discrepancies of the values reported by the different works. It may come from the slightly different different lattice constants we used during the calculations. We uses the lattice constants determined by the experiments which also lead to the reduction of the magnetic moment.

The data do not show any systematic variation with the atomic numbers, the atomic weights and the valence numbers which have been explored by us. This is due to the fact that the SHC are determined by the mixing of the states due to the SOC, which is dependent on the orbitals and the dispersions alike as shown in Equ.2.

In order to get insight of the mechanism of the large SHC and AHC, we show the electronic structure of three selected compounds: Ni₂LiSn (N_v=26), Cu₂CoSn ($N_{v}$=35) and Co₂MnGa ($N_{v}$=28). These compounds have large SHC among the ones with the same number of valence electrons. The electronic structures of the Heusler compounds from the DFT calculations are explored in the published literatures. The basic characters of them can be understood from the molecular orbital model[25], including the crystal field splitting, the bonding of the atoms and the exchange splittings. For the nonmagnetic cases, it is schematically shown in Fig. 3, in which we take Ni₂LiSn and Cu₂NiSn as examples. In a summary, we group the elements into X-Y and X-Z subgroups. For most compounds, where the X and Y element are from the TM group, the hybridization the sp-orbitals from X and main group element Z produces bonding and antibonding $a_{1}$ and $t_{2}$ representation of cubic crystal field as shown in Fig. 3 (a) and (b), while those of d-orbitals from X and Y form $t_{2g}$ and $e_{g}$ representations as in Fig. 3(d). There are non-bonding $d$-orbitals. However, when Li instead of TM elements is at the Y site, ionic bonding are formed due to the large difference of the electron negativity of the elements. There are no additional nonbonding $d$-states. The cases $N_{v}=24$ and $N_{v}=35$ settle the Fermi level in the $e_{g}$ and $t_{2g}$ manifold respectively.

We show in Fig. 4 the electronic bands of the three compounds with relatively high SHC from the ab initio calculations. At the same time, we also show the weights from different atomic $d$-orbitals. In Ni₂LiSn, as shown in Fig. 4(a), the Fermi level is among the $e_{g}$ states from Ni because of the crystal field as analyzed in the MO orbital model in Fig.3(c). The $t_{2g}$ states located mainly between -1.0 eV and -3.0 eV. When Y is the TM elements, as Cu₂NiSn and Co₂MnGa shown in Fig. 4(b) and 4(c), the $e_{g}$ and $t_{2g}$ states from the two subgroups form bonding and antibonding states. The nonbonding $d$-states in Cu₂NiSn are around -3 eV. The Fermi level crosses the antibonding states from Cu-Ni. It has the $t_{2g}$ characters. The exchange splitting makes the situation a little more complicated. It shifts the occupations of the states in Cu₂MnGa as shown in Fig. 4(c). We see that in Cu₂MnGa the states near the Fermi level are coming from the 3d states of Mn. It is among the top of the t_2g state of the majority spin and among the dip of the states of the minority spins.

The SHC is a function of the Fermi level. As can be seen in Fig.4(a), the maximum of the SHC in Ni₂LiSn is about -1000 $\hbar/e$($\Omega^{-1}$cm^-1) and locates about 1.8 eV below the Fermi level. We plot the maximum of the SHC absolute values within $\pm 2.0$eV around the Fermi level achievable in the candidate compounds in Fig.5 (a). We see that in the figure the maximum is likely to be achieved in the magnetic compounds with Y = Mn and also in the nonmagnetic Ni₂YZ. According to our results in Fig. 5(a), the SHC in nonmagnetic compounds reaches its maximum about 1600 $\hbar/e$($\Omega^{-1}$cm^-1) in Ni₂SnCu. The values in Ni₂ScAl and Ni₂CuSb are a little bit smaller. We noticed that these compounds contains no elements such as Pt, W, or Bi, with large atomic SOC, However, the SHC is still comparable with that in materials contains heavy element. The value of AHC can exceed 2000 $\Omega^{-1}$cm^-1 in Co₂MnSb(Sn) and Rh₂MnGe as shown in 5(b). Both values are plotted as scattered circles filled with different colors for different Y elements as shown in 5(c). Generally, a larger AHC also leads to a larger SHC. We fitted the data for Y = Fe and Mn respectively. We noticed that when Y = Fe, the data are less scattered, because of the near half-metal characteristics observed in the DOS. This will minimize the spin-non-conserving contributions. For the other groups of the compounds, the data are too scattered. The straight line fitting is given just for guiding sight.

It was argued by Banerjee et al. that the mixing of the pairs of {$d_{xz},d_{yz}$} or {$d_{xy},d_{x^{2}-y^{2}}$} orbitals due to the SOC is the main underlying mechanism for the anomalous velocity in TMO[26]. $d_{xy}$ and $d_{x^{2}-y^{2}}$ orbitals belong to the different representation basis of the cubic group. According to the molecular orbital picture of the Heusler compounds, the hybridization of the X and Y orbitals and then to the orbitals of $Z$ and the second $X$ orbitals leads to the formation of $a_{1}$-$t_{2}$, $e$-$t_{2g}$ manifold due to the cubic symmetry of the crystal field. The hybridization of the orbitals with $m=\pm 1$ is possible only within the $t_{2g}$ symmetry. It is anticipated that the Fermi level is located among the $t_{2g}$ manifold in order to obtain a larger effect. Accordingly, for the nonmagnetic compounds, significant SHC can be found in the compounds when the Fermi level crosses the $t_{2}$ states or the region when $e_{g}$ and $t_{2g}$ manifold overlaps with $N_{v}$ less than 24, or larger than 32. Unfortunately, in quite some Heusler compounds, the Fermi level is located in the gaps formed due to the crystal field[25]. It is not expected that the Fermi level crosses the region of the band mixing, so that the SHC is not large. It is very occasionally that the Fermi level is in the vicinity where the SOC mixes the states and opens a gap. We may conclude that tuning the position of the Fermi level is the key to tune the SHC. Although the SHC at the Fermi level is not large, as we argued in the above section, it is due to the position of the Fermi level. When we change the Fermi level, the SHC can be significantly enhanced as shown in Fig. 4(a). In the nonmagnetic states, the $t_{2g}$ states in Ni₂LiSn from Ni $d_{yz}$ and $d_{xz}$ orbitals are located mainly in the energy range between -1.0 eV and -2.0 eV. It shows very pronounced SHC in the energy window. Hybridizations of the two orbitals occurs at the W-point, which is in agreement with scenarios proposed by Jadaun et al.[26] Therefore, as already commented by Kübler et al., the AHC does not only determined by the strength of atomic SOC, but also related to the details of the electronic structure, namely, the interactions of the atomic orbitals. From the data obtained so far, we cannot draw any conclusions of the principles to obtain large SHE in Heusler compounds, although the nodal lines are proposed as a character of the large SHC/AHC[15]. Actually, the values depend on the electronic band structures, and the wave functions as well. We can only figure out the places in the band structures responsible for the SHC/AHC peaks[24]. Thus a high-throughput calculation is a necessary and efficient way to figure out compounds with large SHC[15].