Intrinsic and Extrinsic Spin Hall Effects of Dirac Electrons

Abstract: We investigate the spin Hall effect (SHE) of electrons described by the Dirac equation, which is used as an effective model near the $L$-points in bismuth. By considering short-range nonmagnetic impurities, we calculate the extrinsic as well as intrinsic contributions on an equal footing. The vertex corrections are taken into account within the ladder type and the so-called skew-scattering type. The intrinsic SHE which we obtain is consistent with that of Fuseya et al. [J. Phys. Soc. Jpn. 81, 93704 (2012)]. It is found that the extrinsic contribution dominates the intrinsic one when the system is (semi)metallic. The extrinsic SHE due to the skew scattering is proportional to $\Delta / n_{\rm i} u$, where $2\Delta$ is the band gap, $n_{\rm i}$ is the impurity concentration, and $u$ is the strength of the impurity potential.