Persistence of superconductivity in thin shells beyond $H_{c1}$

TL;DR

This paper investigates the persistence of superconductivity in thin shell superconductors under strong magnetic fields, revealing that superconductivity can survive near the zero locus of the normal magnetic component and exhibits a phenomenon called freezing of the boundary.

Contribution

It introduces a novel analysis of superconductivity persistence beyond the first critical field using a mean field reduction and obstacle problem approach, highlighting the freezing phenomenon.

Findings

Superconductivity persists in a neighborhood of size (H_{c1}/h)^{1/3} near the zero locus of H.

A freezing boundary phenomenon occurs, making part of the superconductivity region insensitive to small field variations.

The study describes intermediate regimes and extends results from symmetric to general models.

Abstract

In Ginzburg-Landau theory, a strong magnetic field is responsible for the breakdown of superconductivity. This work is concerned with the identification of the region where superconductivity persists, in a thin shell superconductor modeled by a compact surface $\mathcal M\subset\mathbb R^3$, as the intensity $h$ of the external magnetic field is raised above $H_{c1}$. Using a mean field reduction approach devised by Sandier and Serfaty as the Ginzburg-Landau parameter $\kappa$ goes to infinity, we are led to studying a two-sided obstacle problem. We show that superconductivity survives in a neighborhood of size $(H_{c1}/h)^{1/3}$ of the zero locus of the normal component $H$ of the field. We also describe intermediate regimes, focusing first on a symmetric model problem. In the general case, we prove that a striking phenomenon we call freezing of the boundary takes place: one component of the superconductivity region is insensitive to small changes in the field.