Susceptibilities and Critical Fields of Superconducting Films

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Abstract
The present report concerns the calculation of field distributions, susceptibilities, and critical fields of a superconducting film using the nonlocal and nonlinear (Ginzburg-Landau) theories with diffuse scattering boundary conditions. Both the Pippard and the BCS kernels are considered. The main tool in obtaining these results is a numerical calculation of the vector potential, but an analytical treatment is possible in the very thin film and bulk limits. A comparison between the results obtained with the two different kernels is made for field distributions and susceptibilities. The present susceptibilities are compared with those for diffuse scattering calculated by Rogers and Schrieffer and with Toxen's results for specular reflection. Maximum fields are obtained from a nonlinear-nonlocal generalization of the Ginzburg-Landau equations due to Bardeen. These equations are solved by a mixture of perturbation and numerical methods using the Pippard kernel. The dependence of these maximum fields on the coherence length is studied, and the present results are compared with Toxen's critical fields for specular reflection. In the thin film limit, the present calculation establishes on a rigorous basis the proportionality of the critical field to the negative three-halves power of thickness. It is shown that there exist two types of transition and a critical thickness in the nonlocal-nonlinear case, just as in the Ginzburg-Landau theory. The type of transition changes, for fixed thickness, from first to second order when the coherence length is raised beyond a certain value.