9.5 Conclusions

A detailed comparison of the rigid vortex model for magnetic vortex states in soft magnetic nanodots with the finite element simulations has revealed some special features of the magnetic vortex state:

• The magnetization distribution near the vortex core radius ($r=a$) deviates essentially from Usov's analytical model. Especially a non-vanishing radial component $M_\rho$ has been found (Fig. 9.39).

• In addition to the magnetic surface charges in the core of the vortex, the finite element simulations have revealed a ring of weak surface charges with opposite sign around the core of the nanodot.

• The shape of the vortex core and its exchange energy have been found to be very stable (``rigid'') even for large vortex shifts in an external field.

• However, the surface charges on the circumference of the nanodot are overestimated by the rigid vortex model, because the magnetization distribution is distorted from the perfectly circular shape by the magnetostatic stray field. As a result, the surface charges and the magnetostatic energy are reduced as compared to the rigid vortex model.

• The phase diagram of magnetic ground states shows sharp transitions from the ``in-plane'' state to the perpendicular magnetization distribution and the magnetic vortex state, whereas the transition from the perpendicular magnetization to the magnetic vortex state is not well defined.

• Vortex precession in an in-plane external field has been observed and compared with finite difference simulation. Very good agreement for the precession frequency has been found for suitable finite element meshes.

• The radial mode has been studied in detail. The results show, that the core remains almost undisturbed and this (radially symmetric) magnetostatic eigenfunction can be approximately described as a uniform mode without side surface charges.