Thin-film Micromagnetics Research Articles

An effective model for boundary vortices in thin-film micromagnetics

Ferromagnetic materials are governed by a variational principle which is nonlocal, nonconvex and multiscale. The main object is given by a unit-length three-dimensional vector field, the magnetization, that corresponds to the stable states of the micromagnetic energy. Our aim is to analyze a thin film regime that captures the asymptotic behavior of boundary vortices generated by the magnetization and their interaction energy. This study is based on the notion of “global Jacobian” detecting the topological defects that a priori could be located in the interior and at the boundary of the film. A major difficulty consists in estimating the nonlocal part of the micromagnetic energy in order to isolate the exact terms corresponding to the topological defects. We prove the concentration of the energy around boundary vortices via a [Formula: see text]-convergence expansion at the second order. The second-order term is the renormalized energy that represents the interaction between the boundary vortices and governs their optimal position. We compute the expression of the renormalized energy for which we prove the existence of minimizers having two boundary vortices of multiplicity [Formula: see text]. Compactness results are also shown for the magnetization and the corresponding global Jacobian.

Micromagnetics of thin films in the presence of Dzyaloshinskii–Moriya interaction

In this paper, we study the thin-film limit of the micromagnetic energy functional in the presence of bulk Dzyaloshinskii–Moriya interaction (DMI). Our analysis includes both a stationary [Formula: see text]-convergence result for the micromagnetic energy, as well as the identification of the asymptotic behavior of the associated Landau–Lifshitz–Gilbert equation. In particular, we prove that, in the limiting model, part of the DMI term behaves like the projection of the magnetic moment onto the normal to the film, contributing this way to an increase in the shape anisotropy arising from the magnetostatic self-energy. Finally, we discuss a convergent finite element approach for the approximation of the time-dependent case and use it to numerically compare the original three-dimensional (3D) model with the 2D thin-film limit.

Numerical quadratic energy minimization bound to convex constraints in thin-film micromagnetics

We analyze the reduced model for thin-film devices in stationary micromagnetics proposed in DeSimone et al. (R Soc Lond Proc Ser A Math Phys Eng Sci 457(2016):2983–2991, 2001). We introduce an appropriate functional analytic framework and prove well-posedness of the model in that setting. The scheme for the numerical approximation of solutions consists of two ingredients: The energy space is discretized in a conforming way using Raviart–Thomas finite elements; the non-linear but convex side constraint is treated with a penalty method. This strategy yields a convergent sequence of approximations as discretization and penalty parameter vanish. The proof generalizes to a large class of minimization problems and is of interest beyond the scope of thin-film micromagnetics.

Entropy method for line-energies

In this paper we analyze energy functionals concentrated on the discontinuity lines of unit-length, divergence-free 2D vector fields. The motivation comes from thin-film micromagnetics where these functionals correspond to mesoscopic wall-energies. A natural issue consists in characterizing the line-energy densities for which the functionals are lower semicontinuous for a relevant topology. In fact, this is a necessary condition for being the Γ-limit of a family of energies. A key point in our study is the use of the notion of entropy production borrowed from the field of conservation laws. With this tool, we build a large class of lower semicontinuous line-energies. In particular, we prove that certain power functions lead to such line-energy functionals as conjectured in Ambrosio et al. (Calc Var Partial Differ Equ 9(4):327–355, 1999). We also deduce compactness properties for these functionals leading to the existence of minimizers for their lower semicontinuous envelopes. Another natural question is whether the viscosity solution is a minimizing configuration. We show that the answer is in general negative by exhibiting some special nonconvex domains as counterexamples. However, we establish positive results for some special domains (stadium, ellipse and union of two discs). The case of general convex domains is still open.

A compactness result for Landau state in thin-film micromagnetics

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters ε and η and defined over vector fields m:Ω⊂R2→S2 that are tangent at the boundary ∂Ω. We are interested in the behavior of minimizers as ε,η→0. They tend to be in-plane away from a region of length scale ε (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that S1-transition layers of length scale η (Néel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of Néel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields {mε,η}ε,η↓0 of energies close to the Landau state in the regime where a vortex is energetically more expensive than a Néel wall. Our method uses techniques developed for the Ginzburg–Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of S2-vector fields by S1-vector fields away from the vortex balls.

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

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 degenerativesetting 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 analysisfor the observing minimization problems, as $ h \to 0 $, will be demonstrated inthe main theorem of this paper.

Solutions for the fractional Landau–Lifshitz equation

This article considers the dynamic equation of a reduced model for thin-film micromagnetics deduced by A. DeSimone, R.V. Kohn and F. Otto in [A. DeSimone, R.V. Kohn, F. Otto, A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math. 55 (2002) 1–53]. To derive the existence of weak solutions under periodical boundary condition, the authors first prove the existence of smooth solutions for the approximating equation, then prove the convergence of the viscosity solution when the viscosity term vanishes, which implies the existence of solutions for the original equation.

A Γ-convergence result for Néel walls in micromagnetics

Three reduced models are considered for Neel walls which are dominant transition layers in thin-film micromagnetics. Each model comes as a nonlocal and nonconvex variational principle for one-dimensional magnetizations and it depends on a small parameter e > 0. Our aim is to study the Γ-convergence of these models as \({\varepsilon \downarrow 0}\) . We prove that the limiting magnetization patterns are piecewise constant functions that correspond to a finite number of walls of the same angle. The Γ-limit energy is proportional to the number of walls of these configurations and the energetic cost of each wall is quartic for small wall angles.

A Two-Dimensional Landau-Lifshitz Model in Studying Thin Film Micromagnetics

The paper is concerned with a two-dimensional Landau-Lifshitz equation which was first raised by A. DeSimone and F. Otto, and so fourth, when studying thin film micromagnetics. We get the existence of a local weak solution by approximating it with a higher-order equation. Penalty approximation and semigroup theory are employed to deal with the higher-order equation.

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

We study a model for the magnetization in thin ferromagnetic films. It comes as a vari- ational problem forS 1 -valued mapsm 0 (the magnetization) of two variablesx 0 :

A dual approach to regularity in thin film micromagnetics

We prove a regularity result in the two-dimensional theory of soft ferromagnetic films. The associated Euler–Lagrange equation is given by a nonlocal degenerate variational inequality involving fractional derivatives. A difference quotient type argument based on a dual formulation in terms of magnetostatic potentials yields a Holder estimate for the uniquely determined gradient projection of the magnetization field.

A landau-lifshitz-type equation in studying thin film micromagnetics

The paper is concerned with a Landau-Lifshitz-type equation which was first raised by DeSimone and Otto, etc., when studying thin film micromagnetics. We get the existence of a continuous solution using penalty approximation, compensated compactness and commutator estimate.

Optimal grid-based methods for thin film micromagnetics simulations

Thin film micromagnetics are a broad class of materials with many technological applications, primarily in magnetic memory. The dynamics of the magnetization distribution in these materials is traditionally modeled by the Landau–Lifshitz–Gilbert (LLG) equation. Numerical simulations of the LLG equation are complicated by the need to compute the stray field due to the inhomogeneities in the magnetization which presents the chief bottleneck for the simulation speed. Here, we introduce a new method for computing the stray field in a sample for a reduced model of ultra-thin film micromagnetics. The method uses a recently proposed idea of optimal finite difference grids for approximating Neumann-to-Dirichlet maps and has an advantage of being able to use non-uniform discretization in the film plane, as well as an efficient way of dealing with the boundary conditions at infinity for the stray field. We present several examples of the method’s implementation and give a detailed comparison of its performance for studying domain wall structures compared to the conventional FFT-based methods.

Multiscale Finite Element Modelling of Pattern Formation in Magnetostrictive Composite Thin Film

A multiscale finite element model with subgrid-scale spatial homogenization and stability estimate for time discretization in the context of magnetostrictive composite thin-film micromagnetics are reported in this paper. Developments of the phenomenological microscopic constitutive model and the subsequent multiscale approach are aimed at analyzing the formation of magnetic domains in Terfenol-D/epoxy thin film under transverse magnetic (TM) mode of excitation. The phenomenological constitutive model is based on the density of domain switching (DDS) of an ellipsoidal inclusion in a unit cell. The subgrid-scale spatial homogenization works as a method of upwinding the small-scale micromagnetism and magnetostriction to the larger length scale. Numerical results indicate complex features of this thin-film dynamics, such as the formation of connected domains.

A reduced theory for thin‐film micromagnetics

AbstractMicromagnetics is a nonlocal, nonconvex variational problem. Its minimizer represents the ground‐state magnetization pattern of a ferromagnetic body under a specified external field. This paper identifies a physically relevant thin‐film limit and shows that the limiting behavior is described by a certain “reduced” variational problem. Our main result is the Γ‐convergence of suitably scaled three‐dimensional micromagnetic problems to a two‐dimensional reduced problem; this implies, in particular, convergence of minimizers for any value of the external field. 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‐film limit. © 2002 Wiley Periodicals, Inc.

Characterization of magneto-optical recording media in terms of domain boundary jaggedness

Noise in magneto-optical recording devices can be classified as system-related noise and media noise. Media noise is rooted in the magnetic and magneto-optical properties of the recording media. To investigate media noise and its relation to microstructure and micromagnetics of thin films, the jaggedness of magnetic domain boundaries is characterized using static domain observations in which the fractal dimension D is measured for the domain boundaries. Samples of TbFeCo deposited under similar conditions, but with slightly different compositions, exhibited different amounts of jaggedness, and hence, slightly different values of D, i.e., 1.05≤D≤1.20. Temperature dependent measurements performed on a TbFe sample showed increasing D with increasing temperature, 1.19≤D≤1.28 for 300 K ≤T≤360 K. Possible sources of jaggedness include structural/magnetic inhomogeneities as well as competition between domain-wall energy and demagnetization.