Abstract. We consider an asymptotic regime for two-dimensional ferromagnetic films that is consistent with the formation of transition layers, called Néel walls. We first establish compactness of S 2 -valued magnetizations in the energetic regime of Néel walls and characterize the set of accumulation points. We then prove that Néel walls are asymptotically the unique energy minimizing configurations. We finally study the corresponding dynamical issues, namely the compactness properties of the magnetizations under the flow of the Landau-Lifshitz-Gilbert equation. The purpose of this paper is to study an asymptotic regime for two-dimensional ferromagnetic thin films allowing for the occurrence and persistence of special transition layers called Néel walls. We will prove compactness, optimality and energy concentration of Néel walls, together with dynamical properties driven by the Landau-Lifschitz-Gilbert equation. 1.1. A two-dimensional model for thin-film micromagnetics. We will focus on the following 2D model for thin ferromagnetic films. For that, let Ω = R × T with T = R/Z, be a two-dimensional horizontal section of a magnetic sample that is infinite in x 1 -direction and periodic in x 2 -direction. The admissible magnetizations are vector fields that are periodic in x 2 -direction (this condition is imposed in order to rule out lateral surface charges) and connect two macroscopic directions forming an angle, i.e., for a fixed We will consider the following micromagnetic energy approximation in a thin-film regime that is written in the absence of crystalline anisotropy and external magnetic fields (see e.g. [5] , [14] ): (2) is called the exchange energy, while the other two terms stand for the stray field energy created by the surface charges m 3 at the top and bottom of the sample and by the volume charges ∇ · m ′ in the interior of the sample. More precisely, the stray-field h(m ′ ) : Ω × R → R 3 generated only by the volume charges is defined as the unique that is x 2 -periodic and is determined by static Maxwell's equation in the weak sense 1 : For Explicitly solving (3) by use 2 of the Fourier transform F(·), the stray-field energy can be equivalently expressed in terms of the homogeneousḢ where so that the gradient of the energy E δ (m) is given by Here and in the following, we denote planar coordinates by x = (x 1 , x 2 ), (x 1 , x 2 ) ⊥ = (−x 2 , x 1 ), the vertical coordinate by z and furthermore, we write (∇, In this model, we expect two types of singular patterns: Néel walls and vortices (so-called Bloch lines in micromagnetic jargon). These patterns result from the competition between the different contributions in the total energy E δ (m) and the nonconvex constraint |m| = 1. We explain these structures in the following and compare their respective energies (for more details, see DeSimone, Kohn, Müller and Otto [6]). Néel walls. The Néel wall is a dominant transition layer in thin ferromagnetic films. It is characterized by a one-dimensional in-plane rotation connecting two directions (1) of the magnetization. More precisely, it is a one-dimensional transition m = (m 1 , m 2 ) : R → S 1 that minimizes the energy under the boundary constraint (1): 1 In other words, h(m ′ ) is the Helmholtz projection of the vector measure 2 Given a function ζ : Ω → R which is 1-periodic in x2, we introduce the combination of Fourier transformation in x1 and Fourier series in x2 by F(ζ)(ξ) = e −|ξ| |z| F(∇ · m ′ )(ξ) for ξ = 0 and z ∈ R. It follows that the minimizer is a two length scale object: it has a small core with fast varying rotation and two logarithmically decaying tails. 4 As δ → 0, the scale of the Néel core is given by |x 1 | w core = O(δ) (up to a logarithmic scale in δ) while the two logarithmic decaying tails scale as w core |x 1 | w tail = O(1). The energetic cost (by unit length) of a Néel wall is given by Micromagnetic vortex. A vortex point corresponds in our model to a topological singularity at the microscopic level where the magnetization points out-of-plane. The prototype of a vortex configuration is given by a vector field m : B 2 → S 2 defined in a unit disk Ω = B 2 of a thin film that satisfies: and minimizes the energy (2): Since the magnetization turns in-plane at the boundary of the disk B 2 (so, deg(m ′ , ∂Ω) = 1), a localized region is created, that is the core of the vortex of size ε, where the magnetization becomes indeed perpendicular to the horizontal plane. Remark that the energy E δ controls the Ginzburg-Landau energy, i.e., Regime. We will focus on an energetic regime allowing for Néel walls, but excluding vortices. More precisely, we will assume that δ → 0 and ε = ε(δ) → 0 such that and we will consider families of magnetization {m δ } 0<δ<1/2 satisfying the energy bound (7) sup δ→0 δ| log δ|E δ (m δ ) < +∞. In particular, (6) implies that the size ε of the vortex core is exponentially smaller than the size of the Néel wall core δ, i.e., ε = O(e − 1 δ| log δ| ). Compactness of Néel walls. We first show that the energetic regime (7) is indeed favorable for the formation of Néel walls. We start by proving a compactness result for 4 In our model, the tails are contained by the system thanks to the confining mechanism of steric interaction with the sample edges placed at x1 = ±1. 5 In our model, the parameter ε = ε(δ) > 0 is related to δ by the regime (6).
Estimates up to the boundary of a rectangle were obtained in REF Appendix] in the regime M ε , N ε ≪ | log ε|.
A thin-film limit in the Landau-Lifshitz-Gilbert equation relevant for the formation of N\'eel walls
Journal of Fixed Point Theory and Applications
Mathematics