Description of spin transport and precession in spin-orbit coupling systems and a general equation of continuity

TL;DR

This paper derives a general continuity equation for spin and charge in systems with arbitrary spin-orbit coupling, introducing a matrix formalism that captures spin precession and nonlinear effects.

Contribution

It generalizes the current density to a matrix form, providing a unified description of charge and spin currents in complex spin-orbit systems with new expressions for current and torque densities.

Findings

Derived a matrix continuity equation valid for arbitrary spin-orbit systems.

Introduced a new expression for spin current including nonlinear spin-orbit effects.

Showed that particle current must be modified for Hamiltonians with degree ≥ 3 to conserve particle number.

Abstract

By generalizing the usual current density to a matrix with respect to spin variables, a general equation of continuity satisfied by the density matrix and current density matrix has been derived. This equation holds in arbitrary spin-orbit coupling systems as long as its Hamiltonian can be expressed in terms of a power series in momentum. Thereby, the expressions of the current density matrix and a torque density matrix are obtained. The current density matrix completely describes both the usual current and spin current as well; while the torque density matrix describes the spin precession caused by a total effective magnetic field, which may include a realistic and an effective one due to the spin-orbit coupling. In contrast to the conventional definition of spin current, this expression contains an additional term if the Hamiltonian includes nonlinear spin-orbit couplings. Moreover, if the degree of the full Hamiltonian $\geq3$, then the particle current must also be modified in order to satisfy the local conservation law of number.