Mathematical models in micromagnetism: an introduction

Abstract

The modelization of ferromagnetic materials obeys the micromagnetic theory proposed by W. F. Brown [4] for static configurations and the Landau-Lifchitz (or Landau-Lifchitz-Gilbert) equations for the dynamics. In this paper, we briefly present these underlying models. Difficulties including physical relevance, numerical simulations and analytic properties of the equations that one wants to keep at the discrete level are discussed. Résumé. La modélisation de matériaux ferromagnétiques suit la théorie du micromagnétisme proposée par Brown pour les configurations statiques (d'équilibre) et leséquations de Landau-Lifchitz (ou Landau-Lifchitz-Gilbert) pour la dynamique. Dans ce papier, nous décrivons rapidement les principaux modèles utilisés. Les difficultés incluant la pertinence physique, les simulations numériques ainsi que le respect au niveau discret des propriétés analytiques du modèle sont précisées.

Numerical simulations of ferromagnetic materials

“…The first term is the exchange contribution and takes into account microscopic spin-spin interactions on the crystal network. The last three terms modelize respectively the stray energy (h d (m) is the stray field), anisotropy and external energies (see [1] for a more detailed description). Minimizers of this energy are physically relevant configurations.…”

“…The model usually used for modelizing ferromagnetic materials is the micromagnetism, introduced by W.-F. Brown in 60'. This model, based upon a thermodynamic analysis of the ferromagnetic phenomenon, is quickly recalled in section 1 (see also [1,3,5,6]). In a second section, we focus on the discretization of the stray field (more details are given in [4,8] while its use is described in [7,10]).…”