Vortex energy and 360° Néel walls in thin‐film micromagnetics

Source abstract

Abstract We study the vortex pattern in ultrathin ferromagnetic films of circular crosssection. The model is based on the following energy functional: for in‐plane magnetizations m : B 2 → S 1 in the unit disc $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}B^2 \subset \R^2$ . The avoidance of volume charges ▿ · m ≠ 0 in B 2 and surface charges m · ν ≠ 0 on δ B 2 leads to the formation of a vortex in the limit ε → 0. At the level ε > 0 the vortex is regularized by the formation of a 360° Néel wall (a one‐dimensional transition layer with core of scale ε) concentrated along a radius of B 2 . We derive the limiting energy of the vortex by matching upper and lower bounds. Our analysis on the lower bound is based on a dynamical system argument and an interpolation inequality with sharp leading‐order constant, while the upper bound uses the leading‐order energy for 360° Néel walls. © 2010 Wiley Periodicals, Inc.