Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Meissner state for type-II inhomogeneous superconductors

Matías Díaz-Vera, Carlos Román

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.

On the Meissner state for type-II inhomogeneous superconductors

Abstract

We consider extreme type-II superconductors modeled by the Ginzburg--Landau energy with a pinning term , which we assume to be a bounded measurable function such that for some constant . A crucial feature of this type of superconductors is the occurrence of vortices, which appear above the so-called first critical field . 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 . In addition, we show that beyond this value, for applied fields whose strength is slightly below the so-called superheating field , there exists a unique Meissner-type solution that locally minimizes the energy.

Paper Structure

This paper contains 18 sections, 20 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. The problem and brief state of the art
  3. Main results
  4. Preliminaries and Energy Splitting
  5. The (weighted) Ginzburg--Landau equations.
  6. The function ρ
  7. Estimates for critical points in the Coulomb gauge
  8. Vorticity estimate
  9. Approximation of the Meissner state
  10. Proof of Proposition \ref{['prop-splitting']}
  11. First critical field
  12. Heuristic derivation of the first critical field
  13. Proof of Theorem \ref{['crit field lower bound']}
  14. Proof of Theorem \ref{['crit field upper bound']}
  15. Existence and uniqueness of a Meissner configuration above the first critical field
  16. ...and 3 more sections

Key Result

Proposition 1.1

Given any configuration $(\mathbf{u,A}) \in H^1(\Omega,\mathbb{C}) \times H^1(\Omega,\mathbb{R}^2)$, letting $(u,A)$ be defined through the relation $(\mathbf{u,A}) = \left(\rho_\varepsilon u, A + h_{\mathrm{ex}}A_\varepsilon^0\right)$, we have where

Theorems & Definitions (49)

  • Proposition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • ...and 39 more