Magnetic skyrmions under confinement

TL;DR

This paper develops a mathematical framework to analyze confined magnetic skyrmions in ultrathin ferromagnetic films, revealing their stable, particle-like behavior and boundary repulsion through variational and asymptotic methods.

Contribution

It introduces a variational approach to characterize skyrmions under confinement, deriving their limiting profiles and stability properties as DMI strength diminishes.

Findings

Skyrmions are shown to be stable and repelled from boundaries.

Explicit formulas for the renormalized energy in various geometries.

Existence of topologically non-trivial energy minimizers under Dirichlet conditions.

Abstract

We present a variational treatment of confined magnetic skyrmions in a minimal micromagnetic model of ultrathin ferromagnetic films with interfacial Dzylashinksii-Moriya interaction (DMI) in competition with the exchange energy, with a possible addition of perpendicular magnetic anisotropy. Under Dirichlet boundary conditions that are motivated by the asymptotic treatment of the stray field energy in the thin film limit we prove existence of topologically non-trivial energy minimizers that concentrate on points in the domain as the DMI strength parameter tends to zero. Furthermore, we derive the leading order non-trivial term in the $\Gamma$-expansion of the energy in the limit of vanishing DMI strength that allows us to completely characterize the limiting magnetization profiles and interpret them as particle-like states whose radius and position are determined by minimizing a renormalized energy functional. In particular, we show that in our setting the skyrmions are strongly repelled from the domain boundaries, which imparts them with stability that is highly desirable for applications. We provide explicit calculations of the renormalized energy for a number of basic domain geometries.