Variational approach to the stationary spin-Hall effect

The Kirchhoff-Helmholtz principle of least heat dissipation is applied in order to take into account the electrostatic screening in the derivation of the stationary states of the spin-Hall effect. Spin-orbit interaction, spin-flip relaxation, and charge accumulation are treated on an equal footing in the variational approach. It is shown that the electrostatic screening modifies the steady-state continuity equation due to the presence of surface currents, and that the stationary state defined by a linear spin-accumulation potential and zero pure spin current corresponds to the minimum power dissipated in the device.