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

Document Type

Article

Publication Date

1-1-2021

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.

Identifier

85091726843 (Scopus)

Publication Title

Archive for Rational Mechanics and Analysis

External Full Text Location

https://doi.org/10.1007/s00205-020-01575-7

e-ISSN

14320673

ISSN

00039527

First Page

219

Last Page

299

Issue

1

Volume

239

Grant

DMS-1614948

Fund Ref

National Science Foundation

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