Theory of Polar Skyrmions in Layered Structure of Ferroelectric Perovskites

Discovery of polar skyrmions on the (PbTiO$_3$)$_n$/(SrTiO$_3$)$_n$ superlattice in the absence of any known interaction, which can orient electric polarizations at a point with respect to its neighboring points, is enigmatic. We here show that the coupling of the electric polarization and strain at each layer is responsible for orienting polarizations. We formulate coupled Euler equations for electric polarization and elastic displacement vectors, and solve to find skyrmion solution for a range of electric fields with unique topological number in each layer. The types of skyrmions vary from Neel to Bloch depending on the position of the layers in the superlattice, as observed in the experiments.