Ferromagnetic thin multi-structures (original) (raw)

Bounds on the micromagnetic energy of a uniaxial ferromagnet

Communications on Pure and Applied Mathematics, 1998

We identify the optimal scaling law for a nonconvex, nonlocal variational problem representing the magnetic energy of a uniaxial ferromagnet. Our analysis is restricted to a certain parameter regime, in which the surface tension is sufficiently small relative to the other parameters of the problem.

3D-2D Asymptotic Analysis for Micromagnetic Thin Films

ESAIM: Control, Optimisation and Calculus of Variations, 2001

Γ-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness ε approaches zero of a ferromagnetic thin structure Ωε = ω × (−ε, ε), ω ⊂ R 2 , whose energy is given by Eε(m) = 1 ε Z Ωε W (m, ∇m) + 1 2 ∇u • m dx subject to div(−∇u + mχΩ ε) = 0 on R 3 , and to the constraint |m| = 1 on Ωε, where W is any continuous function satisfying p-growth assumptions with p > 1. Partial results are also obtained in the case p = 1, under an additional assumption on W .

Junction of ferromagnetic thin films

Calculus of Variations and Partial Differential Equations, 2010

In this paper, starting from the classical 3D micromagnetic energy, we determine, via an asymptotic analysis, the free energy of two joined ferromagnetic thin films. We distinguish different regimes depending on the limit of the ratio between the small thicknesses of the two films.

A reduced theory for thin-film micromagnetics

Communications on Pure and Applied Mathematics, 2002

Micromagnetics is a nonlocal, nonconvex variational problem. Its minimizer represents the ground state magnetization pattern of a ferromagnetic body under a speci ed external eld. This paper identi es a physically relevant thin lm limit, and shows that the limiting behavior is described by a certain \reduced" variational problem. Our main result is the ;{convergence of suitably scaled 3D micromagnetic problems to a 2D reduced problem this implies, in particular, convergence of minimizers for any value of the external eld. The reduced problem is degenerate but convex as a result it determines some (but not all) features of the ground state magnetization pattern in the associated thin lm limit.

Asymptotic Analysis for micromagnetics on thin films governed by indefinite material coefficient

2010

International audienceAbstract.In this paper, a class of minimization problems, associated with the micromagnetics of thin films, is dealt with. Each minimization problem is distinguished by the thickness of the thin film, denoted by 0 < h <1, and it is considered under spatial indefinite and degenerative setting of the material coefficients. On the basis of the fundamental studies of the governing energy functionals, the existence of minimizers, for every 0< h <1, and the 3D-2Dasymptotic analysis for the observing minimization problems, as h tends to 0, will be demonstrated in the main theorem of this pape

Asymptotic analysis for micromagnetics of thin films governed by indefinite material coefficients

Communications on Pure and Applied Analysis, 2010

In this paper, a class of minimization problems, associated with the micromagnetics of thin films, is dealt with. Each minimization problem is distinguished by the thickness of the thin film, denoted by 0 < h < 1, and it is considered under spatial indefinite and degenerative setting of the material coefficients. On the basis of the fundamental studies of the governing energy functionals, the existence of minimizers, for every 0 < h < 1, and the 3D-2D asymptotic analysis for the observing minimization problems, as h ց 0, will be demonstrated in the main theorem of this paper.

Singularities in thin magnetic films

We study a particular regime for thin rectangular micromagnetic films. Following an idea of Ignat and Kurzke we reduce the full three-dimensional non-local vector-valued model to a two-dimensional local scalar model and study some of the properties of the latter. We then recover some results for the full model. As part of our study we study a particular PDE in a quadrant, for which we prove uniqueness and regularity results. The general structure follows the ideas of Ignat and Kurzke who studied more regular domains but we need to make some crucial modifications to deal with the presence of corners. We study boundary vortices and we prove rigorously that the so-called S state in a rectangle is minimizing in the aforementioned regime. In the last chapter we study a different problem about critical points of a family of scalar functionals related to micromagnetics. We extend a previously known result about minimizers: we show that critical points of the energy for which the penalty te...

Microstructure evolution model in micromagnetics

Zeitschrift für angewandte Mathematik und Physik, 2004

The proposed model combines tendency for minimization of Gibbs' magnetic energy with the rate-independent maximum-dissipation mechanism that reflects the macroscopical quantity of energy required to change one pole of a magnet to another. The microstructure is described on a "mesoscopical" level in terms of Young measures. Such mesoscopical, distributedparameter model is formulated (and, after a suitable regularization), analyzed, discretized, implemented, and eventually tested computationally on a uni-axial magnet. The desired hysteresis "macroscopical" response is demonstrated together with the influence of material properties.

Convergence of a ferromagnetic film model

Comptes Rendus Mathematique, 2007

In this note, we present a Γ-convergence type result for ferromagnetic films. We propose a model of films for which we could ensure the strong convergence of minimizers when the exchange parameter vanishes. In this model, the plate thickness is kept constant and the magnetization stays constant in the thickness of the film. Résumé Convergence pour un modèle de film mince en ferromagnétisme. Dans cette note, nous présentons un résultat de Γ-convergence pour les films de matériaux ferromagnétiques. Nous proposons un modèle pour lequel il est possible d'assurer une convergence forte des minimiseurs quand le paramètre d'échange tend vers zéro. Dans ce modèle, l'épaisseur du film est considérée comme constante et l'aimantation est contrainteà rester constante dans l'épaisseur du film.