On the Ginzburg--Landau Functional in the Surface Superconductivity Regime

TL;DR

This paper analyzes the Ginzburg-Landau energy in surface superconductivity, establishing new bounds and showing that the energy is dominated by a 1D boundary problem, with refined estimates for disc-shaped samples.

Contribution

It provides new energy lower bounds and a refined analysis for disc samples, confirming a conjecture about the uniformity of surface superconductivity layers.

Findings

Energy is determined by a 1D boundary functional

Established pointwise estimates for the order parameter in discs

Proved uniformity of the surface superconductivity layer

Abstract

We present new estimates on the two-dimensional Ginzburg-Landau energy of a type-II superconductor in an applied magnetic field varying between the second and third critical fields. In this regime, superconductivity is restricted to a thin layer along the boundary of the sample. We provide new energy lower bounds, proving that the Ginzburg-Landau energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. Estimates relating the density of the Ginzburg-Landau order parameter to that of the 1D problem follow. In the particular case of a disc sample, a refinement of our method leads to a pointwise estimate on the Ginzburg-Landau order parameter, thereby proving a strong form of uniformity of the surface superconductivity layer, related to a conjecture by Xing-Bin Pan.