Vortex state oscillations in soft magnetic cylindrical dots
  We have studied magnetic vortex oscillations in soft sub-micron cylindrical dots with variable thickness and diameter by an analytical approach and micromagnetic simulations. We have considered two kinds of modes of the vortex magnetization oscillations: 1) low-frequency translation mode, corresponding to the movement of the vortex as a whole near its equilibrium position; 2) high-frequency vortex modes, which correspond to radially symmetric oscillations of the vortex magnetization, mainly outside the vortex core. The vortex translational eigenmode was calculated numerically in frequency and time domains for different dot aspect ratios. To describe the discrete set of vortex high-frequency modes we applied the linearized equation of motion of dynamic magnetization over the vortex ground state. We considered only radially symmetric magnetization oscillations modes. The eigenfrequencies of both kinds of excitation modes are determined by magnetostatic interactions. They are proportional to the thickness/diameter ratio and lie in the GHz range for typical dot sizes. 
K. Yu. Guslienko, W. Scholz, R. W. Chantrell, V. Novosad