Vortex-in-nanodot potentials in thin circular magnetic dots.

Vortex states in thin circular magnetic nanodots are studied using auxiliary constraining fields as a way to map out the potential energy space of a vortex, while avoiding a rigid vortex approximation. In the model, isotropic Heisenberg exchange competes with the demagnetization field caused by both surface and volume magnetization charge densities. The system energy is minimized while applying a constraint on the vortex core position, using Lagrange's method of undetermined multipliers. The undetermined multiplier is seen to be the external field needed to hold the vortex core in place at a desired radial distance r from the dot center. This auxiliary field is applied only in the core region of the vortex. For a uniform nanodot, the potential energy is found to be very close to parabolic with r, as in the rigid vortex approximation, while the constraining field increases linearly with r. Effects of nonmagnetic impurities and holes in the medium can also be estimated. An impurity or hole in the dot can lead to bistable operation between the two minima that result under the application of a transverse applied magnetic field.