Vortex Motion for the Landau–Lifshitz–Gilbert Equation with Spin-Transfer Torque

Kurzke, Matthias; Melcher, Christof; Moser, Roger

doi:10.1137/100806965

Cited by 15 publications

(23 citation statements)

References 40 publications

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“…Similar bounds with errors of the form o ε (1) can be found in [18,30,42]. Our method of proof is inspired by the choice of test function found in the proof of Theorem 3 in [25]. THEOREM 14.…”

Section: Quantitative Bounds On the Kinetic Energymentioning

confidence: 72%

Vortex Liquids and the Ginzburg–landau Equation

Kurzke

Spirn

2014

Forum math. Sigma

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We establish vortex dynamics for the time-dependent Ginzburg-Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids, we prove that sequences of solutions converge to the hydrodynamic limit.

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Section: Quantitative Bounds On the Kinetic Energymentioning

confidence: 72%

Vortex Liquids and the Ginzburg–landau Equation

Kurzke

Spirn

2014

Forum math. Sigma

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“…where the dissipative tensor D ∈ R 2×2 is given by [14] and references therein. The proof follows from an adaption of the continuation argument in [5], using the implicit function theorem on Hilbert manifolds with the spin velocity v ∈ R 2 serving as a perturbation parameter.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Stability of axisymmetric chiral skyrmions

Li¹,

Melcher²

2018

Journal of Functional Analysis

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We examine topological solitons in a minimal variational model for a chiral magnet, so-called chiral skyrmions. In the regime of large background fields, we prove linear stability of axisymmetric chiral skyrmions under arbitrary perturbations in the energy space, a long-standing open question in physics literature. Moreover, we show strict local minimality of axisymmetric chiral skyrmions and nearby existence of moving soliton solution for the Landau-Lifshitz-Gilbert equation driven by a small spin transfer torque.

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“…In this paper, we consider a more general form of the Landau-Lifshitz-Gilbert equation including additional drift terms, which has been derived in a related setting in [22,21] (see also [15]):…”

Section: In This Case We Have the Concentration Of The Measures Definmentioning

confidence: 99%

“…The more general equation (LLG) (in the absence of the non-local energy term) was studied by Kurzke, Melcher and Moser in [15] where they derived rigorously the motion law of point vortices in a different regime, namely ε → 0 and δ = +∞. We highlight that in those papers, the parameter δ > 0 is kept fixed or large yielding a uniform H 1 bound via the energy; it is far beyond the grasp of (6).…”

Section: In This Case We Have the Concentration Of The Measures Definmentioning

confidence: 99%

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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Vortex Motion for the Landau–Lifshitz–Gilbert Equation with Spin-Transfer Torque

Cited by 15 publications

References 40 publications

Vortex Liquids and the Ginzburg–landau Equation

Vortex Liquids and the Ginzburg–landau Equation

Stability of axisymmetric chiral skyrmions

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

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