Energetics of magnetic ring and disk elements: Uniform versus vortex state

Magnetic energy expressions for the uniform and vortex states of ring elements are derived and compared with the results of micromagnetic simulations. In particular, the effect of roughness on the energy of the vortex state is considered and an expression for the magnetometric demagnetizing factor of rings is found. These energy expressions allow us to calculate the phase diagram separating the uniform from the vortex state. Our results suggest that the roughness contribution to the magnetic energy is sizable in the vortex state and is magnetostatic in origin and its effect is to increase the energy of the vortex state. For the case of rings, the vortex state is favoured with respect to the uniform state on account of the extra magnetostatic energy arising from the inner surface of the ring.