Theory of Néel-Bloch transition for compact magnetic skyrmions

Abstract

Magnetic skyrmions are a prime example of topologically non-trivial spin textures observed in a variety of magnetic materials. They emerge when the exchange and anisotropy energies promoting parallel alignment of spins in a ferromagnet enter in competition with energies favoring non-collinear alignment of spins such as the Dzyaloshinskii-Moriya interaction (DMI), the long-range dipolar interaction or higher-order exchange interactions. The orthodox theory of skyrmions in ultrathin ferromagnetic layers with interfacial DMI relies on a model that accounts for the dipolar interaction through an effective anisotropy term, neglecting long-range effects. At the same time, in single ferromagnetic layers with interfacial DMI, large chiral skyrmions, also called skyrmionic bubbles have been observed, suggesting a non-trivial interplay between DMI and long-range dipolar effects[1]. The competition between these two energies also leads to the formation of skyrmions exhibiting spin rotations with intermediate angles between Neel and Bloch, a phenomenon also present in domain walls. In addition, there is a growing body of theoretical evidence that points to a need to take into account the long-range dipolar energy in the models describing magnetic skyrmions. The above considerations put into question the validity of the commonly used assumption that the long-range contribution of the dipolar interaction is negligible.Here we use rigorous mathematical analysis to develop a skyrmion theory that takes into account the full dipolar energy in the thin film regime and provides analytical formulas for compact skyrmion radius, rotation angle and energy[2,3]. We demonstrate that the DMI threshold at which a compact skyrmion loses its Néel character is a factor of 3 higher than that for a single domain wall. A reorientation of the skyrmion rotation angle from Néel to intermediate Néel-Bloch angles is predicted as the layer thickness is increased in the low DMI regime, which is confirmed by micromagnetic simulations. We will also present an extension of our model which includes applied magnetic field.