Boundary Behavior of the Ginzburg-Landau Order Parameter in the Surface Superconductivity Regime

TL;DR

This paper investigates the boundary behavior of the Ginzburg-Landau order parameter in surface superconductivity, proving uniformity of the boundary approximation by analyzing second order energy contributions and curvature effects.

Contribution

It confirms that the boundary approximation holds uniformly in the surface superconductivity regime, incorporating curvature effects into the energy analysis.

Findings

Boundary approximation is uniform along the boundary.

Second order energy expansion accounts for curvature effects.

Surface superconductivity layer is proven to be uniform.

Abstract

We study the 2D Ginzburg-Landau theory for a type-II superconductor in an applied magnetic field varying between the second and third critical value. In this regime the order parameter minimizing the GL energy is concentrated along the boundary of the sample and is well approximated to leading order by a simplified 1D profile in the direction perpendicular to the boundary. Motivated by a conjecture of Xing-Bin Pan, we address the question of whether this approximation can hold uniformly in the boundary region. We prove that this is indeed the case as a corollary of a refined, second order energy expansion including contributions due to the curvature of the sample. Local variations of the GL order parameter are controlled by the second order term of this energy expansion, which allows us to prove the desired uniformity of the surface superconductivity layer.