Extrinsic Spin Hall Effect Due to Transition-Metal Impurities

We investigate the extrinsic spin Hall effect in the electron gas model due to transition-metal impurities based on the single-impurity Anderson model with orbital degrees of freedom. Both the skew scattering and side jump mechanisms are analyzed in a unified way, and the significant role of orbital degrees of freedom are clarified. The obtained spin Hall conductivities are in proportion to the spin-orbit polarization at the Fermi level ($R = <l \cdot s>$) as is the case with the intrinsic spin Hall effect: skew scattering term $\sigma^{ss} ~ R \delta_1/\rho$, and side jump term $\sigma^{sj} ~ R$, where $\delta_1$ is the phase shift for $p$ ($l=1$) partial wave. Furthermore, the present study indicates the existence of a nontrivial close relationship between the intrinsic term $\sigma^{int}$ and the extrinsic side jump term $\sigma^{sj}$.