Analytic theory for Néel skyrmion size, accounting for finite film thickness

Abstract

The energy of a Néel skyrmion in a magnetic film with interfacial DMI is developed analytically, with the finite thickness of the film taken into account. Many theories for the energy of magnetic skyrmions use models that are only accurate in the limit that the film thickness vanishes. One theory [Büttner et al. Sci. Rep. 8, 4464 (2018)] provides exceptionally accurate expressions for the skyrmion energy for finite thickness films, but does not yield an analytic solution for the skyrmion size, plus it contains contributions which are found using fitting methods. Here, we approximate the demagnetizing energy of a skyrmion by examining the form of magnetostatic Green’s function expressions for the demagnetizing fields. Then, analytic expressions for the skyrmion radius and the skyrmion domain wall width are derived in the limit of zero applied field. Our methods are rigorously compared to other calculation methods and compare well to numerical calculations and experimental results, both with and without an applied magnetic field. The approximations that allow for an analytic result lead to a loss of accuracy for the skyrmion radius of up to 20% but this loss in accuracy may be offset by the simplicity of the developed physical expressions. Finite thickness is important to consider for skyrmions existing in multilayer stacks with total thickness of a few nanometers.