Superheating fields of superconductors: Asymptotic analysis and numerical results

Abstract

The superheated Meissner state in type-I superconductors is studied both analytically and numerically within the framework of Ginzburg-Landau theory. Using the method of matched asymptotic expansions we have developed a systematic expansion for the solutions of the Ginzburg-Landau equations in the limit of small $κ$, and have determined the maximum superheating field $H_{\rm sh}$ for the existence of the metastable, superheated Meissner state as an expansion in powers of $κ^{1/2}$. Our numerical solutions of these equations agree quite well with the asymptotic solutions for $κ<0.5$. The same asymptotic methods are also used to study the stability of the solutions, as well as a modified version of the Ginzburg-Landau equations which incorporates nonlocal electrodynamics. Finally, we compare our numerical results for the superheating field for large-$κ$ against recent asymptotic results for large-$κ$, and again find a close agreement. Our results demonstrate the efficacy of the method of matched asymptotic expansions for dealing with problems in inhomogeneous superconductivity involving boundary layers.