Variational approach to vortex penetration and vortex interaction

Abstract

A variational calculation for vortex penetration is presented. Variational trial functions for the Meissner state are combined with variational functions for a vortex near the surface. The latter is based on Clem's trial solutions for a vortex in bulk, which were adapted to include surface effects through consideration of an image vortex. Three variational parameters are considered, corresponding to the effective coherence length of the vortex, the effective penetration length for the Meissner currents, and the value of the order parameter at the surface. The results show that the last two variational parameters are independent of vortex position. Explicit calculations are presented for several $\ensuremath{\kappa}$ values. The energy barrier for vortex penetration is shown to be in good agreement with full numerical calculations of the Ginzburg--Landau equations. We consider the variation of the magnetic flux carried by a vortex as it gets inside the superconductor, and agreement with known experimental and theoretical results is obtained. The model was extended to calculate the force between two vortices, and the results show that the force goes to zero as the pair comes close to the surface. This result can be of interest for the study of the melting of the vortex lattice and for vortices confined in mesoscopic superconductors. The variational approach can be very helpful for intermediate $\ensuremath{\kappa}$ values when numerical calculations become computationally demanding because it provides manageable expressions for all physically relevant quantities.