Magnetization Dynamics, Gyromagnetic Relation, and Inertial Effects

TL;DR

This paper revisits the gyromagnetic relation and inertial effects in magnetization dynamics, deriving a generalized equation that incorporates inertia and recovers classical models in the appropriate limit.

Contribution

It introduces a generalized dynamic equation for magnetization that includes inertial effects using a Lagrangian approach with an arbitrary inertial tensor.

Findings

Derivation of a new inertial magnetization dynamics equation.

Recovery of classical Landau-Lifshitz-Gilbert equation at long time scales.

Identification of inertial effects relevant at short time scales.

Abstract

The gyromagnetic relation - i.e. the proportionality between the angular momentum $\vec L$ (defined by an inertial tensor) and the magnetization $\vec M$ - is evidence of the intimate connections between the magnetic properties and the inertial properties of ferromagnetic bodies. However, inertia is absent from the dynamics of a magnetic dipole (the Landau-Lifshitz equation, the Gilbert equation and the Bloch equation contain only the first derivative of the magnetization with respect to time). In order to investigate this paradoxical situation, the lagrangian approach (proposed originally by T. H. Gilbert) is revisited keeping an arbitrary nonzero inertial tensor. A dynamic equation generalized to the inertial regime is obtained. It is shown how both the usual gyromagnetic relation and the well-known Landau-Lifshitz-Gilbert equation are recovered at the kinetic limit, i.e. for time scales above the relaxation time $\tau$ of the angular momentum.