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 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 $\omega \tau$ and is finite in the 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 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.