Solitons in Real Space: Domain Walls, Vortices, Hedgehogs, and Skyrmions

Abstract

Recent years have seen tremendous progress in the understanding of topological phenomena in magnetism, in particular at the nanoscale. In this overview, we consider smooth topological textures such as smooth domain walls, meron or vortices, and most importantly skyrmions. These structures derive their topological stability from the fact that they cannot be undone without violating the continuity of the magnetization field, similar to a knot in a rope. Owing to their topological stability, domain walls and skyrmions are prominent candidates in racetrack-type memories introduced by Parkin and co-workers. These smooth textures should be contrasted with singular topological point defects where the magnetization field is forced to vanish in a submanifold. Such point defects include Ising domain walls, vortices of easy-plane spins, and 3D Bloch points, ‘hedgehogs’, or ‘monopoles’. As domain walls, vortices, and skyrmions including their dynamical versions will be discussed in detail in later chapters by Thiaville and Miltat, Behncke and Meier, Chen, Bauer et al., and Akerman, we give analytical arguments how domain walls emerge in quasi 1D nanowires, how magnetization reverses via nucleation, and why skyrmions exist in thin films. A variational ansatz for skyrmions that is derived from an exact \(2\pi \) domain wall profile provides an excellent approximation to numerical and experimental observations in films that include Dzyaloshinskii-Moriya interaction (DMI) and dipolar interactions. In systems of vanishing DMI, the two helical states of a skyrmion are degenerate, and switching between the two helicities occurs in a topologically allowed fashion. This mechanism is closely related to domain wall nucleation in nanowires. Finally we show that dynamical skyrmions may be regarded as 2D siblings of domain wall breathers, and can be described by the same variational ansatz inspired from \(2\pi \) domain walls as static skyrmions in thin films.