On the Meissner state for type-II inhomogeneous superconductors

TL;DR

This paper analyzes the behavior of vortexless states in inhomogeneous type-II superconductors modeled by Ginzburg--Landau energy, estimating critical fields and characterizing Meissner solutions below and near the superheating field.

Contribution

It provides new estimates for the first critical field and characterizes the Meissner solution behavior in inhomogeneous superconductors near critical magnetic fields.

Findings

Estimated the first critical field $H_{c_1}$ for inhomogeneous superconductors.

Characterized the unique vortexless Meissner solution below $H_{c_1}.

Demonstrated the existence of a Meissner-type solution near the superheating field $H_{sh}$.

Abstract

We consider extreme type-II superconductors modeled by the Ginzburg--Landau energy with a pinning term $a_\varepsilon(x)$, which we assume to be a bounded measurable function such that $b\leq a_\varepsilon(x)\leq 1$ for some constant $b>0$. A crucial feature of this type of superconductors is the occurrence of vortices, which appear above the so-called first critical field $H_{c_1}$. In this paper we estimate this value and characterize the behavior of the Meissner solution, the unique vortexless configuration that globally minimizes the energy below $H_{c_1}$. In addition, we show that beyond this value, for applied fields whose strength is slightly below the so-called superheating field $H_{sh}$, there exists a unique Meissner-type solution that locally minimizes the energy.