A kinetic equation approach to the spin Hall effect in $p$-type bulk semiconductors

A two-band kinetic equation method is employed here to investigate the spin-Hall effect (SHE) in a $p$-type Luttinger semiconductor. We find that the previously predicted solely intrinsic SHE arises from a dc-field induced polarization directly, or in other words, a stationary Rabi process, associated with all hole states below the Fermi surface. The total SHE, encompassing a mix of both extrinsic and intrinsic features, results essentially from a related polarization process involving electric field excitations between heavy- and light-hole bands, both directly and indirectly through scattering. In particular, we examine effects of long-range disorder on spin-Hall current (SHC) within the self-consistent Born approximation. In contrast to the vanishing SHC contribution due to short-range disorder, we show that long-range impurity scattering produces a nonvanishing SHC, independent of impurity density, having its sign opposite to that of the intrinsic SHC contribution and leading to a significant reduction of the total SHC. This disorder-generated SHE arises from a disorder-mediated interband polarization, related only to hole states near the Fermi surface. We analyze the hole density dependencies of SHC and spin mobility numerically: with increasing hole density, the SHC first increases and then decreases, while the spin mobility decreases monotonically.