Minimal time of magnetization switching in small ferromagnetic ellipsoidal samples

TL;DR

This paper investigates the minimal time required to flip the magnetic moment in small ellipsoidal ferromagnetic samples using constrained external magnetic fields, providing precise characterizations and special case analyses.

Contribution

It establishes the existence of a minimal control threshold for magnetic switching and characterizes this threshold, including solutions for symmetric geometries without control constraints.

Findings

Existence of a minimal control amplitude for switching.

Precise characterization of the minimal control threshold.

Control in symmetric geometries without amplitude restrictions.

Abstract

In this paper, we consider a ferromagnetic material of ellipsoidal shape. The associated magnetic moment then has two asymptotically stable opposite equilibria, of the form $\pm\overline{m}$. In order to use these materials for memory storage purposes, it is necessary to know how to control the magnetic moment. We use as a control variable a spatially uniform external magnetic field and consider the question of flipping the magnetic moment, i.e., changing it from the $+\overline{m}$ configuration to the $-\overline{m}$ one, in minimal time. Of course, it is necessary to impose restrictions on the external magnetic field used. We therefore include a constraint on the $L^\infty$ norm of the controls, assumed to be less than a threshold value $U$. We show that, generically with respect to the dimensions of the ellipsoid, there is a minimal value of $U$ for this problem to have a solution. We then characterize it precisely. Finally, we investigate some particular configurations associated to geometries enjoying symmetries properties and show that in this case the magnetic moment can be controlled in minimal time without imposing a threshold condition on $U$.