Theory of Boundary Effects of Superconductors

Abstract

An extension of the phenomenological London equations to take into account a space variation of the concentration of superconducting electrons is presented. The theory differs from that of Ginsburg and Landau in that it makes use of the Gorter-Casimir two-fluid model rather than an order parameter to derive an expression for the free energy. An effective wave function is used for the superconducting electrons. The theory is applied to calculate the boundary energy between normal and superconducting phases and the relative change $\frac{\ensuremath{\Delta}\ensuremath{\lambda}}{\ensuremath{\lambda}}$ of penetration depth with magnetic field. Calculated values of boundary energies are somewhat larger, and of $\frac{\ensuremath{\Delta}\ensuremath{\lambda}}{\ensuremath{\lambda}}$ somewhat smaller, than observed. It is suggested that additional nonlinear terms are required to account for the observed $\frac{\ensuremath{\Delta}\ensuremath{\lambda}}{\ensuremath{\lambda}}$ at low temperatures. The connection of the theory with Pippard's ideas on range of order is discussed briefly.