Effect of Level Statistics on Superconductivity in Ultrasmall Metallic Grains

TL;DR

This paper investigates how the reduction in size of metallic grains affects superconductivity, focusing on the statistical behavior of energy levels and pairing destruction as the level spacing approaches the BCS order parameter.

Contribution

It introduces a statistical analysis of superconductivity suppression in ultrasmall metallic grains considering random matrix theory for energy levels.

Findings

Average critical level spacing is larger than in equally spaced level models.

Derived probabilities for pairing presence in grains based on level spacing.

Quantified the impact of size reduction on superconducting pairing.

Abstract

We examine the destruction of superconducting pairing in metallic grains as their size is decreased for both even and odd numbers of electrons. This occurs when the average level spacing d is of the same order as the BCS order parameter. The energy levels of these grains are randomly distributed according to random matrix theory, and we must work statistically. We find that the average value of the critical level spacing is larger than for the model of equally spaced levels for both parities, and derive numerically the probabilities $P_{o,e}(d)$ that a grain of mean level spacing d shows pairing.