Magnetization switching in small ferromagnetic ellipsoidal samples

Abstract

Abstract. The study of small magnetic particles has become a very important topic, in particular for the development of technological devices such as those used for magnetic recording. In this field, switching the magnetization inside the magnetic sample is of particular relevance. We here investigate mathematically this problem by considering the full partial differential model of Landau-Lifschitz equations triggered by a uniform (in space) external magnetic field.Mathematics Subject Classification.

Optimal Shape of Isolated Ferromagnetic Domains

“…In many papers related to our setting, a perfect ellipsoid shape is assumed [1,2,4,7,10,11,22]. This is a common simplification due to the fact that the self-demagnetization of ellipsoids can be treated analytically.…”

“…for some constant µ > 0, thus minimizing (3) can be reduced to (1). In [41] it has been shown that minimizers of our energy functional (1) with volume constraint (4) exist in all dimensions n and for all prescribed masses µ ≥ 0. For 2 ≤ n ≤ 7, (regular and local) minimizer are open and bounded with smooth boundary.…”

Hysteresis for ferromagnetism: asymptotics of some two-scale Landau–Lifshitz model

“…Global solutions and equilibria. In [1,Th. 4.3], in the case of ellipsoidal domains Ω ⊂ R 3 and under a smallness assumption (on h ext L ∞ and ∆m 0 L 2 ), Alouges and Beauchard construct global smooth solutions to (2.1).…”

“…Standard energy estimates ensure local-in-time existence and uniqueness of solutions continuous in time, with values in H 2 (Ω)) (with an existence time depending on ε): see for example [1] or [4]. By the usual continuation argument, we simply need to bound the H 2 norm of m ε to ensure existence up to time T .…”

“…This is enough to pass to the limit in (4.4), so that m ε is solution to (4.2). With ε fixed, showing that m ε is continuous in time with values in H 2 is standard, as well as uniqueness and stability properties: see [4], or [1].…”