  The spin dynamics are calculated for a model system consisting of
magnetically soft, layered nanomagnets, in which two ferromagnetic (F)
cylindrical dots, each with a magnetic vortex ground state, are separated by a
non-magnetic spacer (N). This permits a study of the effects of interlayer
magnetostatic interactions on the vortex dynamics. The system was explored by
applying the equations of motion for the vortex core positions. The restoring
force was calculated taking into account the magnetostatic interactions
assuming a realistic surface charge free spin distribution. For tri-layer F/N/F
dots with opposite chiralities and the same core polarizations (lowest energy
state), two eigenmodes are predicted analytically and confirmed via
micromagnetic simulations. One mode is in the sub-GHz range for submicron dot
diameters and corresponds to quasi-circular rotation of the cores about the dot
center. A second mode is in the MHz range corresponding to a small amplitude
rotation of the mean core position. The eigenfrequencies depend strongly on the
geometrical parameters of the system, suggesting that magnetostatic effects
play a dominant role in determining the vortex dynamics.
