The curling-type or vortex magnetization state is a ground state of ferromagnetic disk-shaped sub-micron elements. This essentially non-linear state can be interpreted as a part of a two-dimensional magnetic topological soliton adapted to the particle shape to minimize the total magnetic energy. The purpose of the present work is to investigate spin excitation spectra of soft-magnetic cylindrical elements in the vicinity of the equilibrium vortex state. Spin excitations in the vortex state are substantially different from those in the uniformly magnetized state. An example is the appearance of a low-frequency mode associated with movement of the vortex as a whole [1]. We demonstrate that the vortex core in thin submicron dots undergoes a spiral motion with sub-GHz frequency. In general, both the short-range exchange and the long-range magnetostatic interactions contribute to eigenfrequencies of the collective spin excitations. The adequate description of high-frequency magnetization dynamics in such systems is a challenge for modern magnetism theory. The low-lying part of excitation spectrum in sub-micron magnetic particles is mainly determined by the long-range dipolar forces arising from oscillations of surface and volume magnetic charges. Therefore the eigenfrequencies are determined by the demagnetizing fields of the magnetic element. These fields can be strongly non-uniform in non-ellipsoidal samples or if the static magnetization distribution essentially differs from uniform state. The spectrum of vortex spin excitations is quantized due to particle's restricted geometry. Our approach is based on the consideration of small dynamic oscillations over the centered vortex ground state. These dipole dominated spin excitation modes have eigenfrequencies well above the vortex core translation frequency for the given dot sizes. Analytical calculations of the vortex eigenfrequencies are supported by numerical micromagnetic simulations. The high-frequency radial vortex excitation modes were recently detected experimentally [2]. (Fig.2 )

The "translational mode" of the vortex excitations corresponds to the spiral vortex core rotation around the dot center. Its direction (counter-clockwise or clockwise) is defined by the combination of the vortex polarization (p) and chirality. The trajectory depends on magnetization damping (a).

[1] K.Y. Guslienko, B.A. Ivanov, V. Novosad et al., J. Appl. Phys. 91, 8037 (2002).

[2] V. Novosad, M. Grimsditch, K.Y. Guslienko et al., Phys. Rev. B 66, 052407 (2002).