Boundary conditions, dimensionality, topology and size dependence of the superconducting transition temperature

Abstract

For various sample geometries (slabs, cylinders, spheres, hypercubes), de Gennes' boundary condition parameter b is used to study its effect upon the transition temperature T _c of a superconductor. For b > 0 the order parameter at the surface is decreased, and as a consequence T _c is reduced, while for b < 0 the order parameter at the surface is increased, thereby enhancing T _c of a specimen in zero magnetic field. Exact solutions, derived by Fink and Haley (Int. J. mod. Phys. B, 17, 2171 (2003)), of the order parameter of a slab of finite thickness as a function of temperature are presented, both for reduced and enhanced transition (nucleation) temperatures. At the nucleation temperature the order parameter approaches zero. This concise review closes with a link established between de Gennes' microscopic boundary condition and the Ginzburg–Landau phenomenological approach, and a discussion of some relevant experiments. For example, applying the boundary condition with b < 0 to tin whiskers elucidates the increase of T _c with strain.

Acknowledgements

J.O.I. wishes to thank Ben Widom, to whom this paper is dedicated, for his guidance, over the years, during numerous expeditions in the wondrous field of critical phenomena. Discussions with D.S. McLachlan are acknowledged. This research is supported, in part, by K.U. Leuven Fellowship F/03/066 for V.F.K. and by FWO project G.0237.05.