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Time-periodic oscillating Néel walls in ferromagnetic thin films

Antonio Capella, Valentin Linse, Christof Melcher, Lauro Morales, Ramón G. Plaza

Abstract

This paper studies the existence, the structure and the spectral stability of time-periodic oscillating 180-degree Néel walls in ferromagnetic thin films. It is proved that time-periodic coherent structures do exist as solutions to the reduced model for the in-plane magnetization proposed by Capella, Melcher, and Otto (Nonlinearity 20 (2007), no. 11, 2519--2537) when a weak and $T$-periodic external magnetic field is applied in the direction of the easy axes of the film, perturbing in this fashion the well-known static 180-degree Néel wall. The linearization around this time-periodic Néel wall is constituted by a family of linear operators, parametrized by the time variable, which generates an evolution system of generators (or propagator) for the linear problem. Profiting from the stability of the static Néel wall, it is shown that the Floquet spectrum of the monodromy map for the propagator is contained in the complex unit circle, proving stability of the oscillating solution at least at a linear level.

Time-periodic oscillating Néel walls in ferromagnetic thin films

Abstract

This paper studies the existence, the structure and the spectral stability of time-periodic oscillating 180-degree Néel walls in ferromagnetic thin films. It is proved that time-periodic coherent structures do exist as solutions to the reduced model for the in-plane magnetization proposed by Capella, Melcher, and Otto (Nonlinearity 20 (2007), no. 11, 2519--2537) when a weak and -periodic external magnetic field is applied in the direction of the easy axes of the film, perturbing in this fashion the well-known static 180-degree Néel wall. The linearization around this time-periodic Néel wall is constituted by a family of linear operators, parametrized by the time variable, which generates an evolution system of generators (or propagator) for the linear problem. Profiting from the stability of the static Néel wall, it is shown that the Floquet spectrum of the monodromy map for the propagator is contained in the complex unit circle, proving stability of the oscillating solution at least at a linear level.
Paper Structure (16 sections, 35 theorems, 166 equations, 1 figure)

This paper contains 16 sections, 35 theorems, 166 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Néel walls in ferromagnetic thin films
  3. Notation and preliminaries
  4. Micromagnetics, Néel walls and the thin film limit
  5. The static Néel wall and its stability properties
  6. Existence and structure of time-periodic Néel walls
  7. The linearized problem
  8. The nonlinear problem
  9. Proof of Theorem \ref{['theorem_existence']}
  10. Linearization and generation of the evolution system
  11. Linearized equation for perturbations
  12. The evolution system
  13. Translation invariance
  14. Spectral stability
  15. The Floquet spectrum
  16. ...and 1 more sections

Key Result

Proposition 2.1

There exists a static Néel wall solution with phase $\theta_0 = \theta_0(x)$, $\theta_0 : \mathbb{R} \to (-\pi/2,\pi/2)$, satisfying the following:

Figures (1)

  • Figure 1: Contours $\gamma_0$ (blue) and $\gamma_1$ (red) in the complex plane, encircling $\sigma({\mathcal{M}}_0)\backslash \{ 1 \}$ and $\{1\}$, respectively. The unit circle is represented by a dotted line. By taking $0 < \varepsilon \ll 1$ sufficiently small, the interiors of these contours also contain $\sigma({\mathcal{M}}_\varepsilon)\backslash \{ 1 \}$ and $\mu(\varepsilon) = 1$, respectively. It is to be observed that $\mu(\varepsilon) = 1$ persists as an eigenvalue of ${\mathcal{M}}_\varepsilon$ for $\varepsilon$ sufficiently small (color online).

Theorems & Definitions (84)

  • Proposition 2.1: properties of the static Néel wall's phase CMMP24CMO07Melc03
  • proof
  • Remark 2.2
  • Proposition 2.3: spectral properties of ${\mathcal{L}}_0$ CMO07CMMP24
  • proof
  • Remark 2.4
  • Proposition 2.5: spectral properties of ${\mathcal{A}}_0$ CMMP24CMMP25
  • proof
  • Remark 3.1
  • Definition 3.2
  • ...and 74 more