`Spin-Spin' Hall Effect in Two-Dimensional Electron Systems with Spin-Orbit Interaction

We calculate spin Hall conductivities in the two dimensional electron systems with Rashba spin-orbit interaction. The salient feature is that, apart from the usual spin-Hall conductivity \sigma^z_{xy} which corresponds to the induction of out-of-plane spin-current due to the application of transverse charge current, there is a novel spin-Hall conductivity \sigma^\perp_{xy} which arises due to the induction of transverse spin-polarized current in the transverse direction by the application of in-plane spin-polarized current. This phenomenon which we call as `spin-spin' Hall effect is a spin analog of conventional Hall effect, but with no magnetic field. This contribution may be understood through the spin-diffusive equation.