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 $\mathbf{E}(\omega )$ generates a spin-polarization current, normal to $\mathbf{E}$, due to interference of hopping paths. At zero temperature the corresponding spin-Hall conductivity is real and is proportional to ${\omega }^2$. At nonzero temperatures the coupling to the phonons yields an imaginary term proportional to $\omega$. The interference also yields persistent spin currents at thermal equilibrium, at $\mathbf{E}=0$. The contributions from the Dresselhaus and Rashba interactions to the interference oppose each other.