A theory on skyrmion size and profile

A magnetic skyrmion is a topological object consisting of an inner domain, an outer domain, and a wall that separates the two domains. The skyrmion size and wall width are two fundamental quantities of a skyrmion that depend sensitively on material parameters such as exchange energy, magnetic anisotropy, Dzyaloshinskii-Moriya interaction, and magnetic field. However, there is no quantitative understanding of the two quantities so far. Here, we present general expressions for the skyrmion size and wall width obtained from energy considerations. The two formulas agree almost perfectly with simulations and experiments for a wide range of parameters, including all existing materials that support skyrmions. Furthermore, it is found that skyrmion profiles agree very well with the Walker-like 360\textdegree{} domain wall formula.