Energy minimization and flux domain structure in the intermediate state of a type-I superconductor
This paper examines the scaling law of the minimum energy and the qualitative properties of domain patterns, achieving this law, in the intermediate state of a type-I superconductor, demonstrating a microscopic mixture of normal and superconducting domains. This analysis is restricted to the case of a plate under a transverse applied field. Author's motivation is not that the system actually minimizes its energy. Basing on the viewpoint, due to Landau, that the realizable domain patterns are local minima of a nonconvex variational problem, the methods developed include explicit geometric constructions leading to upper bounds and ansatz-free inequalities leading to lower bounds. Two different approaches are developed. One uses a sort of ``dual problem'' to estimate the magnetic energy between two cross-sections. The other takes advantage of an analogy to micromagnets. The analysis provides insight concerning several aspects of the intermediate state and includes issues of the hysteresis of flux patterns, in-plane complexity of the domain patterns for intermediate applied fields, and a change of the length scale of the microstructure with depth. First, it is provided physical and mathematical background based on the variational sharp-interface viewpoint. Then, a mathematically precise formulation of the underlying singularly perturbed variational problem is presented with corresponding variational characterization for the correction term in the minimum energy, due to positive surface energy. Upper bounds for the scaling law are obtained by considering specific constructions, in which flux tubes branch but remain clustered. Several constructions of branched domain patterns, achieving the better scaling law, than an unbranched, laminar structure, are stated. It is shown, that these constructions are more or less optimal, given their topology. The energy of a domain pattern is mainly determined by its local length scale. Then, it is clarified the constructions developed by discussing 1-D variational problems for optimizing the local length scale. Final section addresses the issue of geometric complexity by proving ansatz-independent lower bounds on the base of a well-known analogy between the intermediate state and micromagnets.
scientific article; zbMATH DE number 2133627