Branched Microstructures in the Ginzburg-Landau Model of Type-I Superconductors

TL;DR

This paper analyzes the minimal energy scaling laws of type-I superconductors modeled by Ginzburg-Landau energy, revealing two regimes of magnetic field inhomogeneity near the surface.

Contribution

It establishes the optimal energy scaling laws and constructs explicit branching patterns without ansatz assumptions, identifying two distinct regimes.

Findings

Optimal energy scaling laws are derived for small magnetic fields and thick samples.

Two regimes are identified: one with uniform boundary magnetic field, another with persistent inhomogeneity.

Explicit branching constructions are provided to match the lower bounds.

Abstract

We consider the Ginzburg-Landau energy for a type-I superconductor in the shape of an infinite three-dimensional slab, with two-dimensional periodicity, with an applied magnetic field which is uniform and perpendicular to the slab. We determine the optimal scaling law of the minimal energy in terms of the parameters of the problem, when the applied magnetic field is sufficiently small and the sample sufficiently thick. This optimal scaling law is proven via ansatz-free lower bounds and an explicit branching construction which refines further and further as one approaches the surface of the sample. Two different regimes appear, with different scaling exponents. In the first regime, the branching leads to an almost uniform magnetic field pattern on the boundary; in the second one the inhomogeneity survives up to the boundary.