## Spin Hall effect and edge-spin generation

Spin currents that flow perpendicular to the electric field direction are generic in metals and doped semiconductors with spin-orbit coupling.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.

### Figures

Spin Hall conductivity as a function of the disorder strength and the spin-orbit coupling

### References

Dependence of the intrinsic spin Hall effect on spin-orbit interaction charecter
K. Nomura, J. Sinova, N.A. Sinitsyn, and A. H. MacDonald
Phys. Rev. B. 72 165316 (2005) cond-mat/0506180

Edge-spin accumulation in semiconductor two-dimensional hole gases
K. Nomura, J. Wunderlich, J. Sinova, B. Kaestner, A.H. MacDonald, T. Jungwirth
Phys. Rev. B. 72 245330 (2005) cond-mat/0508532