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Boundary Superconductivity in the BCS Model

Christian Hainzl, Barbara Roos, Robert Seiringer

Abstract

We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one-dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary, an effect that cannot be modeled by effective Neumann boundary conditions on the order parameter as often imposed in Ginzburg-Landau theory. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.

Boundary Superconductivity in the BCS Model

Abstract

We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one-dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary, an effect that cannot be modeled by effective Neumann boundary conditions on the order parameter as often imposed in Ginzburg-Landau theory. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.
Paper Structure (12 sections, 20 theorems, 126 equations, 1 figure, 1 table)

This paper contains 12 sections, 20 theorems, 126 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and Main Result
  2. Preliminaries
  3. Existence of Boundary Superconductivity
  4. Weak Coupling Limit
  5. Proofs of Intermediate Results
  6. Strong Coupling Limit
  7. Proof of Lemma \ref{['lea:limits']}
  8. Proofs of Auxiliary Results
  9. From Section \ref{['sec:pre']}
  10. From Section \ref{['sec:ex']}
  11. From Section \ref{['sec:weak']}
  12. From Section \ref{['sec:strong']}

Key Result

Theorem 1.1

Let $\mu>0$.

Figures (1)

  • Figure 1: The nine regions of the domain of $p,q$ in the proof of Lemma \ref{['lea:supB']}.

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: Momentum representation of $A_{T,\mu}^{\mathbb{R}}$
  • Lemma 2.4: Momentum representation of $A_{T,\mu}^{{\mathbb{R}}_+}$
  • proof : Proof or Lemma \ref{['A_T_momentum']}
  • Remark 2.5
  • Lemma 2.6
  • ...and 34 more