A Quantitative Description of Skyrmions in Ultrathin Ferromagnetic Films and Rigidity of Degree ±1 Harmonic Maps from R2 to S2

Anne Bernand-Mantel, Cyrill B. Muratov, Theresa M. Simon

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We characterize skyrmions in ultrathin ferromagnetic films as local minimizers of a reduced micromagnetic energy appropriate for quasi two-dimensional materials with perpendicular magnetic anisotropy and interfacial Dzyaloshinskii–Moriya interaction. The minimization is carried out in a suitable class of two-dimensional magnetization configurations that prevents the energy from going to negative infinity, while not imposing any restrictions on the spatial scale of the configuration. We first demonstrate the existence of minimizers for an explicit range of the model parameters when the energy is dominated by the exchange energy. We then investigate the conformal limit, in which only the exchange energy survives and identify the asymptotic profiles of the skyrmions as degree 1 harmonic maps from the plane to the sphere, together with their radii, angles and energies. A byproduct of our analysis is a quantitative rigidity result for degree ±1 harmonic maps from the two-dimensional sphere to itself.

Original languageEnglish (US)
Pages (from-to)219-299
Number of pages81
JournalArchive for Rational Mechanics and Analysis
Volume239
Issue number1
DOIs
StatePublished - Jan 2021

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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