Applicability of Bardeen-Cooper-Schrieffer theory to small-sized superconductors: role of Cooper-pair binding energy

TL;DR

This paper examines the limits of BCS theory in small superconductors, showing it fails when size quantization effects make energy level spacing comparable to the Cooper-pair binding energy.

Contribution

It identifies the specific conditions under which BCS theory is not applicable to small-sized superconductors, emphasizing the role of energy level spacing.

Findings

BCS theory fails when level spacing approaches the Cooper-pair binding energy.

In conventional superconductors, level spacing is much smaller than the superconducting gap.

Size quantization impacts the validity of mean-field superconductivity models.

Abstract

We analyze conditions of applicability of grand-canonical mean-field Bardeen-Cooper-Schrieffer theory to the evaluation of an interaction energy in the ground state of small-sized superconductors. We argue that this theory fails to describe correctly an interaction energy, when an average distance between energy levels near the Fermi energy due to the size quantization becomes of the order of the single-pair binding energy. In conventional superconductors, this quantity is much smaller than the superconducting gap.