A quantitative description of skyrmions in ultrathin ferromagnetic films and rigidity of degree $\pm 1$ harmonic maps from ${\mathbb{R}}^2$ to ${\mathbb{S}}^2$

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 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 $\pm 1$ harmonic maps from the two-dimensional sphere to itself.

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Cited by in corpus (15)

• A micromagnetic theory of skyrmion lifetime in ultrathin ferromagnetic films
• Meron configurations in easy-plane chiral magnets
• Chiral magnetic skyrmions across length scales
• Topological invariants of vortices, merons, skyrmions, and their combinations in continuous and discrete systems
• Reduced energies for thin ferromagnetic films with perpendicular anisotropy
• Magnetic skyrmions under confinement
• Rigidity estimates for isometric and conformal maps from ${\mathbb{S}}^{n-1}$ to ${\mathbb{R}}^n$
• Theory of magnetic field-stabilized compact skyrmions in thin film ferromagnets
• Lattice solutions in a Ginzburg-Landau model for a chiral magnet
• A rigidity estimate for maps from $S^2$ to $S^2$ via the harmonic map flow
• Continuous in time bubble decomposition for the harmonic map heat flow
• Non-degeneracy and quantitative stability of half-harmonic maps from $\mathbb{R}$ to $\mathbb{S}$
• Skyrmion stacking in stray field-coupled ultrathin ferromagnetic multilayers
• Quantitative stability of harmonic maps from ${\mathbb{R}}^2$ to ${\mathbb{S}}^2$ with higher degree
• A note on a rigidity estimate for degree $\pm 1$ conformal maps on ${\mathbb{S}}^2$