Thermodynamically stable vortex structures in thin superconducting disks

We investigate the formation of thermodynamically stable vortex structures in a small, thin superconducting disk. We formulate the Gibbs free energy of the disk with an arbitrary number of vortices as a function of applied magnetic field and valid at finite temperatures. We minimize the energy to obtain the optimal position of vortices for different applied fields and temperatures. We also analyze the stability of the different vortex states inside the disk and compare our results with available experimental observations. Our results are in very good agreement with experiments.