Magnetic vortex dynamics in the non-circular potential of a thin elliptic ferromagnetic nanodisk with applied fields

Spontaneous vortex motion in thin ferromagnetic nanodisks of elliptical shape is dominated by a natural gyrotropic orbital part, whose resonance frequency $\omega_G=\overline{k}/G$ depends on a force constant and gyrovector charge, both of which change with the disk size and shape and applied in-plane or out-of-plane fields. The system is analyzed via a dynamic Thiele equation and also using numerical simulations of the Landau-Lifshitz-Gilbert (LLG) equations for thin systems, including temperature via stochastic fields in a Langevin equation for the spin dynamics. A vortex is found to move in an elliptical potential with two principal axis force constants $k_x$ and $k_y$, whose ratio determines the eccentricity of the vortex motion, and whose geometric mean $\overline{k}=\sqrt{k_x k_y}$ determines the frequency. The force constants can be estimated from the energy of quasi-static vortex configurations or from an analysis of the gyrotropic orbits. $k_x$ and $k_y$ get modified either by an applied field perpendicular to the plane or by an in-plane applied field that changes the vortex equilibrium location. Notably, an out-of-plane field also changes the vortex gyrovector $G$, which directly influences $\omega_G$. The vortex position and velocity distributions in thermal equilibrium are found to be Boltzmann distributions in appropriate coordinates, characterized by the force constants.