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

Abstract

We use two truly canonical approaches to describe superconductivity in ultrasmall metallic grains: (a) a variational fixed-N projected BCS-like theory and (b) an exact solution of the model Hamiltonian developed by Richardson in context with Nuclear Physics. Thereby we obtain a 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 and how for $d\gg \tilde{\Delta }$ pairing correlations become delocalized in energy space.

Comments: Short overview (6 pages, 2 figures), comparison to exact results obtained by Richardson in Nuclear Physics