Spin-orbit-induced longitudinal spin-polarized currents in nonmagnetic solids

For certain nonmagnetic solids with low symmetry the occurrence of spin-polarized longitudinal currents is predicted. These arise due to an interplay of spin-orbit interaction and the particular crystal symmetry. This result is derived using a group-theoretical scheme that allows investigating the symmetry properties of any linear response tensor relevant to the field of spintronics. For the spin conductivity tensor it is shown that only the magnetic Laue group has to be considered in this context. Within the introduced general scheme also the spin Hall and additional related transverse effects emerge without making reference to the two-current model. Numerical studies confirm these findings and demonstrate for ${\left({\text{Au}}_{1-x}{\text{Pt}}_x\right)}_4\text{Sc}$ that the longitudinal spin conductivity may be on the same order of magnitude as the conventional transverse one. The presented formalism only relies on the magnetic space group and therefore is universally applicable to any type of magnetic order.

Figure 1

Top: Longitudinal conductivity ${\sigma }_{ii}$ for ${\left({\text{Au}}_{1-x}{\text{Pt}}_x\right)}_4\text{Sc}$ as a function of the concentration $x$ calculated without (NV) and with (VC) the vertex corrections. Middle: Transverse spin conductivities ${\sigma }_{ij}^{\mathrm{x}}$. Bottom: Transverse and longitudinal spin conductivity ${\sigma }_{\mathrm{xy}}^{\mathrm{z}}$ and ${\sigma }_{\mathrm{xx}}^{\mathrm{z}}$, respectively.

Figure 2

Top: Energy-dependent component-($\alpha$)-resolved DOS $n_{\alpha }\left(E\right)$ for (${\text{Au}}_{0.5}{\text{Pt}}_{0.5}{)}_4\text{Sc}$. Bottom: Component-resolved DOS $n_{\alpha }\left(E_{\mathrm{F}}\right)$ at the Fermi energy $E_{\mathrm{F}}$ for ${\left({\text{Au}}_{1-x}{\text{Pt}}_x\right)}_4\text{Sc}$ as a function of the concentration $x$.