Generalisation of Gilbert damping and magnetic inertia parameter as a series of higher-order relativistic terms

TL;DR

This paper derives a series of higher-order relativistic correction terms for Gilbert damping and magnetic inertia parameters starting from the Dirac equation, providing analytical expressions for these parameters as functions of external field frequency.

Contribution

It introduces a systematic derivation of relativistic corrections to magnetisation dynamics parameters from the Dirac equation, extending previous first- and second-order models.

Findings

Series of higher-order relativistic correction terms derived

Closed-form analytical expressions for damping and inertia parameters obtained

Parameters expressed as functions of external field frequency

Abstract

The phenomenological Landau-Lifshitz-Gilbert (LLG) equation of motion remains as the cornerstone of contemporary magnetisation dynamics studies, wherein the Gilbert damping parameter has been attributed to first-order relativistic effects. To include magnetic inertial effects the LLG equation has previously been extended with a supplemental inertia term and the arising inertial dynamics has been related to second-order relativistic effects. Here we start from the relativistic Dirac equation and, performing a Foldy-Wouthuysen transformation, derive a generalised Pauli spin Hamiltonian that contains relativistic correction terms to any higher order. Using the Heisenberg equation of spin motion we derive general relativistic expressions for the tensorial Gilbert damping and magnetic inertia parameters, and show that these tensors can be expressed as series of higher-order relativistic correction terms. We further show that, in the case of a harmonic external driving field, these series can be summed and we provide closed analytical expressions for the Gilbert and inertial parameters that are functions of the frequency of the driving field.