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 $\omega$ much smaller than the scattering rate $\tau^{-1}$, with order one relative fluctuations surviving out to large system sizes. For the k-cubic Rashba model case, the spin Hall conductivity does not depend noticeably on $\omega \tau$ and is finite in the {\em dc} limit, in agreement with experiment. For the modified k-linear Rashba model the spin Hall conductivity is noticeably $\omega \tau$ dependent but approaches a finite value in the {\em 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.