Micromagnetics of curved thin films
- PDF / 560,583 Bytes
- 19 Pages / 547.087 x 737.008 pts Page_size
- 114 Downloads / 198 Views
Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Micromagnetics of curved thin films Giovanni Di Fratta Abstract. In this paper, we aim at a reduced 2d-model describing the observable states of the magnetization in curved thin films. Under some technical assumptions on the geometry of the thin-film, it is well-known that the demagnetizing field behaves like the projection of the magnetization on the normal to the thin film. We remove these assumptions and show that the result holds for a broader class of surfaces; in particular, for compact surfaces. We treat both the stationary case, governed by the micromagnetic energy functional, and the time-dependent case driven by the Landau–Lifshitz–Gilbert equation. Mathematics Subject Classification. 82D40, 49S05, 35C20. Keywords. Micromagnetics, Landau–Lifshitz–Gilbert equation, Curved thin films, Γ -Convergence.
1. Introduction and physical motivations According to the Landau–Lifshitz theory of ferromagnetism (cf. [6,8,9,25,29,35]), the observable states of a rigid ferromagnetic body, occupying a region Ω ⊆ R3 , are described by its magnetization M : Ω → R3 , a vector field verifying the fundamental constraint of micromagnetism: there exists a material-dependent positive constant Ms ∈ R+ such that |M | = Ms in Ω. In general, the spontaneous magnetization Ms :=Ms (T ) depends on the temperature T and vanishes above a critical value Tc , characteristic of each crystal type, known as the Curie temperature. Assuming the specimen at a fixed temperature well below Tc , the value of Ms can be considered constant in Ω. Therefore, the magnetization can be expressed in the form M :=Ms m where m : Ω → S2 is a vector field taking values in the unit sphere S2 of R3 . Although the modulus of m is constant in space, in general, it is not the case for its direction. For single crystal ferromagnets (cf. [1,3]), the observable magnetization states correspond to the local minimizers of the micromagnetic energy functional which, after a suitable normalization, reads as (cf. [35, p. 22] or [25, p. 138]) 1 1 2 |∇m| + ϕan (m) − hd [mχΩ ] · m −μ0 ha · m. (1) GΩ (m) := 2 2 Ω =:E(m)
Ω
=:A(m)
Ω
=:W(m)
Ω =:Z(m)
with m ∈ H 1 (Ω, S2 ) and mχΩ the extension of m by zero outside Ω. The variational problem (1) is non-convex, non-local, and contains multiple length scales. The first term, E (m), is the exchange energy and penalizes nonuniformities in the orientation of the magnetization. It summarizes the effects of short-range exchange interactions among neighbor spins. The magnetocrystalline anisotropy energy, A (m), models the existence of preferred directions of the magnetization; the energy density ϕan : S2 → R+ is a nonnegative function that vanishes only on a finite set of directions, the so-called easy axes. The quantity W (m) represents the magnetostatic self-energy and describes the energy due to the demagnetizing field (stray field) hd [mχΩ ] generated by mχΩ ∈ L2 (R3 , R3 ). 0123456789().: V,-vol
111
Page 2 of 19
G. Di Fratta
ZAMP
The operator hd : m → hd [m] is, for every m
Data Loading...
