Abstract
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.
Highlights
Magnetic vortices in thin ferromagnetic disks of sub-micron size offer an interesting system for the study of collective dynamics of fundamental excitations [1]
Spontaneous vortex motion in thin ferromagnetic nanodisks of elliptical shape is dominated by a natural gyrotropic orbital part, whose resonance frequency ωG = 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
A vortex is found to move in an elliptical potential with two principal axis force constants kx and ky, whose ratio determines the eccentricity of the vortex motion, and whose geometric mean k = kxky determines the frequency
Summary
Magnetic vortices in thin ferromagnetic disks of sub-micron size offer an interesting system for the study of collective dynamics of fundamental excitations [1]. With a magnetic field applied in the plane of the disk, the vortex equilibrium position will be displaced away from the disk center, perpendicular to the field in a direction depending on chirality c. A field Hxext along the long axis, shifting the vortex minimum position along the short axis, does not significantly change the frequency We confirm these results, showing how the vortex effective potential and force constants are modified by the displaced vortex equilibrium location. If a field is applied instead perpendicular to the disk plane (z-axis), there will be two non-degenerate vortex gyrotropic modes, as has been seen in resonance experiments [8] and micromagnetics [9] for circular disks. This analysis is considered first for the zero temperature motion as obtained from Landau-Lifshitz-Gilbert (LLG)
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