Reduced energies for thin ferromagnetic films with perpendicular anisotropy

Giovanni Di Fratta, Cyrill B. Muratov, Valeriy V. Slastikov

Research output: Contribution to journalArticlepeer-review

Abstract

We derive four reduced two-dimensional models that describe, at different spatial scales, the micromagnetics of ultrathin ferromagnetic materials of finite spatial extent featuring perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. Starting with a microscopic model that regularizes the stray field near the material's lateral edges, we carry out an asymptotic analysis of the energy by means of Γ-convergence. Depending on the scaling assumptions on the size of the material domain versus the strength of dipolar interaction, we obtain a hierarchy of the limit energies that exhibit progressively stronger stray field effects of the material edges. These limit energies feature, respectively, a renormalization of the out-of-plane anisotropy, an additional local boundary penalty term forcing out-of-plane alignment of the magnetization at the edge, a pinned magnetization at the edge, and, finally, a pinned magnetization and an additional field-like term that blows up at the edge, as the sample's lateral size is increased. The pinning of the magnetization at the edge restores the topological protection and enables the existence of magnetic skyrmions in bounded samples.

Original languageEnglish (US)
Pages (from-to)1861-1904
Number of pages44
JournalMathematical Models and Methods in Applied Sciences
Volume34
Issue number10
DOIs
StatePublished - Sep 1 2024
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Applied Mathematics

Keywords

  • Dzyaloshinskii-Moriya interaction
  • Micromagnetics
  • magnetic thin films
  • Γ -convergence

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