A compactness result in thin-film micromagnetics and the optimality of the Néel wall

Ignat, Radu; Otto, Félix

doi:10.4171/jems/135

Cited by 28 publications

(51 citation statements)

References 14 publications

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“…Stability of geometrically constrained one-dimensional Néel walls with respect to large two-dimensional perturbations in soft materials was demonstrated asymptotically in [24]. More recently, Γ-convergence studies of the one-dimensional wall energy in the limit of very soft films and in the presence of an applied in-plane field normal to the easy axis were undertaken in [25,26], and a rigorous derivation of the effective magnetization dynamics driven by the reduced thin film energy introduced in [23] from the full three-dimensional Landau-Lifshitz-Gilbert equation was presented in [27].…”

Section: Introductionmentioning

confidence: 99%

One-dimensional Néel walls under applied external fields

Chermisi

Muratov

2013

Nonlinearity

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Abstract. We present a detailed analysis of one-dimensional Néel walls in thin uniaxial ferromagnetic films in the presence of an in-plane applied external field in the direction normal to the easy axis. Within the reduced one-dimensional thin film model, we formulate a non-local variational problem whose minimizers are given by one-dimensional Néel wall profiles. We prove existence, uniqueness (up to translations and reflections), regularity, strict monotonicity and the precise asymptotics of the decay of the minimizers in the considered variational problem.Mathematics Subject Classification: 78A30, 35Q60, 82D40

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Section: Introductionmentioning

confidence: 99%

One-dimensional Néel walls under applied external fields

Chermisi

Muratov

2013

Nonlinearity

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“…S 2 -magnetizations in the regime (6) and (7) that is reminiscent to the compactness results of Ignat and Otto in [12] and [13].…”

Section: 1mentioning

confidence: 77%

A thin-film limit in the Landau–Lifshitz–Gilbert equation relevant for the formation of Néel walls

Côte

Ignat

Miot

2014

J. Fixed Point Theory Appl.

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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. Introduction and main resultsThe 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 fieldsthat 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 fixedWe 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 uniquethat is x 2 -periodic and is determined by static Maxwell's equation in the weak sense 1 : ForExplicitly solving (3) by use 2 of the Fourier transform F(·), the stray-field energy can be equivalently expressed in terms of the homogeneousḢwhereso that the gradient of the energy E δ (m) is given byHere 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 dir...

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“…In order to achieve a characterization for less rigid functionals, methods need to be developed that do not use this trace condition. A related but different micro-magnetic functional E was studied by Ignat and Otto [15]. They also achieved a characterization of minimizers E showing that minimizers converge to Neel Walls, the focus of E was to provide a two dimensional approximation of the micro-magnetic energy in the absence of an external field and crystal anisotropy.…”

Section: Theorem 11 ([18]) Let ω Be a Convex Set With Diameter 2 Cmentioning

confidence: 99%

“…For subset S ⊂ R n let |S| denote the Lebesgue n-measure of S. Now if we run an ODE X(0) = y 0 , dX dt (s) = Du(X(s)) between 0 and t then taking v = 3) can in effect be justified. It is worth noting that the idea of following integral curves of the vector field given by Du (where u is the limit of a sequence of functions whose Aviles Giga energy tends to zero) was used by [16] and a similar idea later by [15].…”

Section: Now By Fubini and (24) We Havementioning

confidence: 99%