Orbital ac Spin-Hall Effect in the Hopping Regime

The Rashba and Dresselhaus spin-orbit interactions are both shown to yield the low temperature spin-Hall effect for strongly localized electrons coupled to phonons. A frequency-dependent electric field E(ω) generates a spin-polarization current, normal to E, due to interference of hopping paths. At zero temperature the corresponding spin-Hall conductivity is real and is proportional to ω². At nonzero temperatures the coupling to the phonons yields an imaginary term proportional to ω. The interference also yields persistent spin currents at thermal equilibrium, at E=0. The contributions from the Dresselhaus and Rashba interactions to the interference oppose each other.