Vortices in small superconducting disks

We study the Ginzburg-Landau equations in order to describe a two-dimensional superconductor in a bounded domain. Using the properties of a particular integrability point (kappa=1/root 2) of these nonlinear equations which allows vortex solutions, we obtain a closed expression for the energy of the superconductor. The presence of the boundary provides a selection mechanism for the number of vortices. A perturbation analysis around kappa = 1/root 2 enables us to include the effects of the vortex interactions and to describe quantitatively the magnetization curves recently measured on small superconducting disks [A.K, Geim, I.V. Grigorieva, S.V. Dubonos, J.G.S. Lok, J.C. Maan, A.E. Filippov. F.M. Peeters, Nature (London) 390 (1997) 259]. We also calculate the optimal vortex configuration and obtain an expression fur the confining potential away from the London limit. (C) 2000 Elsevier Science B.V. All rights reserved.