Exchange-mediated, non-linear, out-of-plane magnetic field dependence of   the ferromagnetic vortex gyrotropic mode frequency driven by core deformation
  We have performed micromagnetic simulations of the vortex gyrotropic mode resonance in a range of disk geometries subject to spatially uniform out-of-plane magnetic fields. For disks of small lateral dimensions, we observe a drop-off in the mode's frequency for field amplitudes approaching the disk saturation field. This non-linear frequency response is shown to be associated with an increased vortex core deformation, which results from the demagnetizing field created when the core is shifted laterally. Such deformation results in an increase in the average out-of-plane magnetization of the displaced vortex state, which through an exchange contribution, leads to a sharp decrease in the vortex stiffness coefficient. It is this decrease in the vortex stiffness coefficient which leads to the non-linear field dependence of the gyrotropic mode frequency. 
Jasper P. Fried, Hans Fangohr, Mikhail Kostylev, Peter J. Metaxas
cond-mat.mes-hall