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Renormalised energy between boundary vortices in thin-film micromagnetics with Dzyaloshinskii-Moriya interaction

Radu Ignat, François L'Official

Abstract

We consider a three-dimensional micromagnetic model with Dzyaloshinskii-Moriya interaction in a thin-film regime for boundary vortices. In this regime, we prove a dimension reduction result: the nonlocal three-dimensional model reduces to a local two-dimensional Ginzburg-Landau type model in terms of the averaged magnetization in the thickness of the film. This reduced model captures the interaction between boundary vortices (so-called renormalised energy), that we determine by a $Γ$-convergence result at the second order and then we analyse its minimisers. They nucleate two boundary vortices whose position depends on the Dzyaloshinskii-Moriya interaction.

Renormalised energy between boundary vortices in thin-film micromagnetics with Dzyaloshinskii-Moriya interaction

Abstract

We consider a three-dimensional micromagnetic model with Dzyaloshinskii-Moriya interaction in a thin-film regime for boundary vortices. In this regime, we prove a dimension reduction result: the nonlocal three-dimensional model reduces to a local two-dimensional Ginzburg-Landau type model in terms of the averaged magnetization in the thickness of the film. This reduced model captures the interaction between boundary vortices (so-called renormalised energy), that we determine by a -convergence result at the second order and then we analyse its minimisers. They nucleate two boundary vortices whose position depends on the Dzyaloshinskii-Moriya interaction.
Paper Structure (19 sections, 26 theorems, 241 equations, 1 figure)

This paper contains 19 sections, 26 theorems, 241 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Nondimensionalisation
  3. Thin-film regime for boundary vortices
  4. Dimension reduction
  5. Gamma-convergence of the two-dimensional reduced functional $E_{\varepsilon,\eta}^\delta$
  6. Minimisation of the renormalised energy
  7. Gamma-convergence of the three-dimensional energy $E_h$
  8. Related models
  9. Outline of the paper
  10. Two-dimensional reduced model for maps $v \colon \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^2$
  11. Approximation by $\mathbb{S}^1$-valued maps
  12. Lifting
  13. Gamma-convergence in terms of liftings
  14. Gamma-convergence for vector-valued maps
  15. Minimisers of the renormalised energy
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Omega_h=\Omega \times (0,h)$ with $\Omega \subset \mathbb{R}^2$ a bounded, simply connected $C^{1,1}$ domain. In the regime DEF_regime3D, consider a family of magnetizations $\left\lbrace m_h \colon \Omega_h \rightarrow \mathbb{S}^2 \right\rbrace_{h \downarrow 0}$ that satisfies and let $\overline{m}_h=(\overline{m}'_h,\overline{m}_{h,3}) \colon \Omega \rightarrow \mathbb{R}^3$ be the avera

Figures (1)

  • Figure 1: Minimising pair $(a_1^\ast,a_2^\ast)$ for $\delta=\frac{1}{10} e^{i\pi/8}$ (left) and $\delta=10e^{i\pi/8}$ (right). The distance between the boundary vortices $a_1^\ast$ and $a_2^\ast$ becomes smaller as $\vert\delta\vert$ gets larger.

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2: Compactness at the boundary and lower bound at the first order
  • Definition 1.3
  • Theorem 1.4: Compactness in the interior and lower bound at the second order
  • Theorem 1.5: Upper bound
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 44 more