Generalised form of the magnetic anisotropy field in micromagnetic and atomistic spin models

TL;DR

This paper introduces a comprehensive method for deriving the effective magnetic anisotropy field applicable to various models, including higher order anisotropies, with implications for magnetic materials and spintronics.

Contribution

It provides a unified framework to calculate the effective anisotropy field for all orders of anisotropies using spherical harmonics, applicable in micromagnetic and atomistic models.

Findings

Derived effective anisotropy fields up to sixth order.

Unified framework for higher order anisotropies.

Applicable to functional magnetic materials and spintronics.

Abstract

We present a general approach to the derivation of the effective anisotropy field which determines the dynamical behaviour of magnetic spins according to the Landau-Lifshitz-Gilbert equation. The approach is based on the gradient in spherical polar coordinates with the final results being expressed in Cartesian coordinates as usually applied in atomistic and micromagnetic model calculations. The approach is generally valid for all orders of anisotropies including higher order combinations of azimuthal and rotational anisotropies often found in functional magnetic materials such as permanent magnets and an emerging class of antiferromagnetic materials with applications in spintronics. Anisotropies are represented in terms of spherical harmonics which have the important property of rational temperature scaling. Effective field vectors are given for anisotropies up to sixth order, presenting a unified framework for implementing higher order magnetic anisotropies in numerical simulations.