The fractional Landau-Lifshitz-Gilbert equation

TL;DR

This paper introduces a fractional Landau-Lifshitz-Gilbert equation incorporating non-Ohmic damping, derived from a Caldeira-Leggett model, with potential experimental determination of the fractional order s.

Contribution

It derives a novel fractional LLG equation with non-Ohmic damping from a microscopic model, extending the traditional phenomenological approach.

Findings

Fractional damping parameter s can be experimentally determined.

Resonance frequency and linewidth exhibit non-linear scaling with effective field.

The model generalizes the classical LLG equation to include fractional derivatives.

Abstract

The dynamics of a magnetic moment or spin are of high interest to applications in technology. Dissipation in these systems is therefore of importance for improvement of efficiency of devices, such as the ones proposed in spintronics. A large spin in a magnetic field is widely assumed to be described by the Landau-Lifshitz-Gilbert (LLG) equation, which includes a phenomenological Gilbert damping. Here, we couple a large spin to a bath and derive a generic (non-)Ohmic damping term for the low-frequency range using a Caldeira-Leggett model. This leads to a fractional LLG equation, where the first-order derivative Gilbert damping is replaced by a fractional derivative of order $s \ge 0$. We show that the parameter $s$ can be determined from a ferromagnetic resonance experiment, where the resonance frequency and linewidth no longer scale linearly with the effective field strength.