Fixed-N Superconductivity: The Crossover from the Bulk to the Few-Electron Limit

TL;DR

This paper develops a canonical theory of superconductivity in ultrasmall metallic grains, describing the crossover from bulk BCS behavior to the few-electron regime, and confirms the persistence of a parity effect in spectral gaps.

Contribution

It provides the first full description of the superconducting crossover in small grains using fixed-N projected BCS wave-functions.

Findings

The BCS limit is recovered when level spacing is much smaller than the bulk gap.

Pairing correlations become delocalized in energy space in the few-electron regime.

Parity effects in spectral gaps persist after fixed-N projection.

Abstract

We present a truly canonical theory of superconductivity in ultrasmall metallic grains by variationally optimizing fixed-N projected BCS wave-functions, which yields the first full description of the entire crossover from the bulk BCS regime (mean level spacing $d \ll$ bulk gap $\tilde\Delta$) to the ``fluctuation-dominated'' few-electron regime ($d\gg\tilde\Delta$). A wave-function analysis shows in detail how the BCS limit is recovered for $d\ll \tilde \Delta$, and how for $d \gg \tilde \Delta$ pairing correlations become delocalized in energy space. An earlier grand-canonical prediction for an observable parity effect in the spectral gaps is found to survive the fixed-N projection.