Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz-Gilbert equation of spin dynamics

TL;DR

This paper derives a relativistic theory for spin relaxation in magnetic solids, extending the Landau-Lifshitz-Gilbert equation to include anisotropic damping, spin-orbit effects, and optical spin torques from fundamental principles.

Contribution

It provides a rigorous derivation of a general spin relaxation tensor including anisotropic and chiral contributions from first principles.

Findings

Derived a general expression for the anisotropic damping tensor.

Identified contributions from electronic interband and intraband transitions.

Showed that electromagnetic fields with spin angular momentum induce optical spin torques.

Abstract

Starting from the Dirac-Kohn-Sham equation we derive the relativistic equation of motion of spin angular momentum in a magnetic solid under an external electromagnetic field. This equation of motion can be written in the form of the well-known Landau-Lifshitz-Gilbert equation for a harmonic external magnetic field, and leads to a more general magnetization dynamics equation for a general time-dependent magnetic field. In both cases with an electronic spin-relaxation term which stems from the spin-orbit interaction. We thus rigorously derive, from fundamental principles, a general expression for the anisotropic damping tensor which is shown to contain an isotropic Gilbert contribution as well as an anisotropic Ising-like and a chiral, Dzyaloshinskii-Moriya-like contribution. The expression for the spin relaxation tensor comprises furthermore both electronic interband and intraband transitions. We also show that when the externally applied electromagnetic field possesses spin angular momentum, this will lead to an optical spin torque exerted on the spin moment.