Pair correlations and the survival of superconductivity in and around a super-conducting impurity

TL;DR

This paper investigates how superconductivity persists in small grains near negative-U impurities, revealing size-dependent pairing behavior and the influence of electron gas screening.

Contribution

It provides a detailed analysis of pair correlations and superconducting gap behavior around negative-U impurities in one and two dimensions, highlighting size effects and sensitivity to chemical potential.

Findings

Pairing occurs when the system has degeneracy between electron pair numbers.

Pair correlations can decay exponentially or be long-ranged depending on interaction strength.

A minimal island size exists below which pairing vanishes, scaling as a power-law.

Abstract

The problem of the survival of superconductivity in a small super-conducting grain placed in a metal substrate is addressed. For this aim the pair correlations and super-conducting gap around and inside a negative-U impurity in one and two dimensions is calculated, in a discrete tight-binding model and a continuous model. Using the Hartree-Fock-Gorkov mean-field decomposition, it is found that pairing inside the gap occurs when the system has a degeneracy between successive number of electron pairs, and is highly sensitive to the chemical potential. For finite pairing in the island, pair correlations in the normal part may either decay exponentially or be long-ranged, depending on the strength of interaction. In addition, it is shown analytically that there is a minimal island size under-which pairing vanishes, and that it scales as a power-law, rather then exponentially as in isolated grains. These results are interpreted in terms of screening of the negative-U impurity by the electron gas.