Coupled oscillations of vortex cores confined in a ferromagnetic elliptical disk

Abstract

By solving the Thiele equation with simultaneous application of a radio-frequency (rf) magnetic field $({\mathbit{h}}_{\mathrm{rf}})$ and an rf spin current $({\mathbit{j}}_{\mathrm{sp}})$, the dynamic susceptibility of exchange-coupled vortices in response to ${\mathbit{h}}_{\mathrm{rf}}$ and ${\mathbit{j}}_{\mathrm{sp}}$ was obtained. It was found that the four eigenmodes expected for two vortices trapped in a magnetic elliptical disk were coupled to different components of ${\mathbit{h}}_{\mathrm{rf}}$ and ${\mathbit{j}}_{\mathrm{sp}}$. As a consequence, orthogonal ${\mathbit{h}}_{\mathrm{rf}}$ and ${\mathbit{j}}_{\mathrm{sp}}$ (which are simultaneously generated by the application of an rf current to an elliptical disk) can excite two modes with different eigenfrequencies. This result suggests that a fieldlike nonadiabatic torque caused by an rf spin current can be spectroscopically distinguished from the one caused by the rf magnetic field.