J Low Temp Phys (2007) 148: 869–873 DOI 10.1007/s10909-007-9442-6 Spontaneous Generation of Single Vortices and Anti-Vortices in Superconductors with Restricted Geometries M.B. Sobnack · F.V. Kusmartsev · J.C.H. Fung Published online: 1 June 2007 © Springer Science+Business Media, LLC 2007 Abstract We study the nucleation of vortices in a thin (thickness ≪ penetration depth) mesoscopic superconducting disc in an applied magnetic field perpendicu- lar to the disc (i.e., parallel to the axis of the disc). We write down an expression for the free energy of the system with an arbitrary number of vortices and anti-vortices at finite (non-zero) temperatures. For a given applied field, we minimize the free energy to find the optimal position of the vortices and anti-vortices (the configuration which minimizes the energy). We show that, whereas at zero temperature anti-vortices do not nucleate, anti-vortices do penetrate the disc at finite temperatures. We also cal- culate the magnetization of the disc as a function of the applied field and hence de- termine the different configurations possible in which a fixed number of fluxoids can penetrate the disc. PACS 67.40.Vs · 73.23.-b · 67.40.w · 74.78.w 1 Introduction Over the years, there have been numerous studies (see, for example, [1–11]) of vortex nucleation in both thin superconducting films of infinite extent and in thin bounded films with transverse dimensions R. Fetter [6] and Buzdin [8] both studied the prob- lem of flux penetration in a thin superconducting disc. More recently, Chibotaru et al. [11] looked at flux penetration in thin mesoscopic superconductors of various M.B. Sobnack ( ) · F.V. Kusmartsev Department of Physics, Loughborough University, Loughborough LE11 3TU, UK e-mail: M.B.Sobnack@lboro.ac.uk J.C.H. Fung Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong, Hong Kong