Type-I Superconductors in the Limit as the London Penetration Depth Goes to 0

TL;DR

This paper derives an explicit approximate solution formula for the static London equations in Type-I superconductors as the penetration depth approaches zero, providing insights into magnetic fields and currents near the boundary.

Contribution

It introduces an explicit solution formula with optimal error estimates for the London equations in the small penetration depth limit, utilizing Hodge decomposition techniques.

Findings

Explicit approximate solution formula for small penetration depth

Error estimates that are essentially optimal

Applicable to scattering problems in Type-I superconductors

Abstract

This paper provides an explicit formula for the approximate solution of the static London equations. These equations describe the currents and magnetic fields in a Type-I superconductor. We represent the magnetic field as a 2-form and the current as a 1-form, and assume that the superconducting material is contained in a bounded, connected set, $\Omega,$ with smooth boundary. The London penetration depth gives an estimate for the thickness of the layer near $\partial\Omega$ where the current is largely carried. In an earlier paper, we introduced a system of Fredholm integral equations of second kind, on $\partial\Omega,$ for solving the physically relevant scattering problems in this context. In real Type-I superconductors the penetration depth is very small, typically about $100$nm, which often renders the integral equation approach computationally intractable. In this paper we provide an explicit formula for approximate solutions, with essentially optimal error estimates, as the penetration depth tends to zero. Our work makes extensive use of the Hodge decomposition of differential forms on manifolds with boundary, and thus evokes Kohn's work on the tangential Cauchy-Riemann equations.