Vortex states in superconducting rings

The superconducting state of a thin superconducting disk with a hole is studied within the non-linear Ginzburg-Landau theory in which the demagnetization effect is accurately taken into account. We find that the flux through the hole is not quantized, the superconducting state is stabilized with increasing size of the hole for fixed radius of the disk, and a transition to a multi-vortex state is found if the disk is sufficiently large. Breaking the circular summetry through a non central location of the hole in the disk enhances the multi-vortex state.