Variational approach to the stationary spin-Hall effect

TL;DR

This paper introduces a variational method based on the Kirchhoff-Helmholtz principle to model the stationary spin-Hall effect, accounting for spin-orbit interaction, relaxation, and electrostatics, resulting in a nonlinear differential equation.

Contribution

It presents a novel variational framework that unifies various interactions in the spin-Hall effect and derives a comprehensive nonlinear differential equation for the stationary state.

Findings

Stationary state characterized by linear spin-accumulation potential when spin-flip length is small.

Zero pure spin-current in the stationary state under certain conditions.

Unified treatment of surface and bulk currents with spin-dependent potentials.

Abstract

The Kirchhoff-Helmholtz principle of least heat dissipation is applied in order to derive the stationary state of the spin-Hall effect. Spin-accumulation due to spin-orbit interaction, spin-flip relaxation, and electrostatic interaction due to charge accumulation are treated on an equal footing. A nonlinear differential equation is derived, that describes both surface and bulk currents and spin-dependent chemical potentials. It is shown that if the ratio of the spin-flip relaxation length over the Debye-Fermi length is small, the stationary state is defined by a linear spin-accumulation potential and zero pure spin-current.