Dependence of the intrinsic spin-Hall effect on spin-orbit interaction character

We report on a comparative numerical study of the spin-Hall conductivity in two dimensions for three different spin-orbit interaction models; the standard k-linear Rashba model, the k-cubic Rashba model that describes two-dimensional hole systems, and a modified k-linear Rashba model in which the spin-orbit coupling strength is energy dependent. Numerical finite-size Kubo formula results indicate that the spin-Hall conductivity of the k-linear Rashba model vanishes for frequency ω much smaller than the scattering rate τ-1, with first-order relative fluctuations surviving out to large system sizes. For the k-cubic Rashba model case, the spin-Hall conductivity does not depend noticeably on ωτ and is finite in the dc limit, in agreement with experiment. For the modified k-linear Rashba model the spin-Hall conductivity is noticeably ωτ dependent but approaches a finite value in the dc limit. We discuss these results in the light of a spectral decomposition of the spin-Hall conductivity and associated sum rules, and in relation to a proposed separation of the spin-Hall conductivity into skew-scattering, intrinsic, and interband vertex correction contributions.