The BCS-Bose Crossover Theory

TL;DR

This paper compares four versions of the BCS-Bose crossover theory, highlighting the importance of including both two-electron and two-hole Cooper pairs for accurate superconducting predictions across temperatures and couplings.

Contribution

It introduces a generalized Bose-Einstein condensation theory that incorporates both 2e- and 2h- Cooper pairs, extending previous models and clarifying their impact on superconducting properties.

Findings

Only models including both 2e- and 2h-CPs recover full BCS condensation energy.

Ignoring either type of CPs results in only half the BCS condensation energy.

Critical temperatures in 2D require unphysically large couplings in simplified models.

Abstract

We contrast {\it four} distinct versions of the BCS-Bose statistical crossover theory according to the form assumed for the electron-number equation that accompanies the BCS gap equation. The four versions correspond to explicitly accounting for two-hole-(2h) as well as two-electron-(2e) Cooper pairs (CPs), or both in equal proportions, or only either kind. This follows from a recent generalization of the Bose-Einstein condensation (GBEC) statistical theory that includes not boson-boson interactions but rather 2e- and also (without loss of generality) 2h-CPs interacting with unpaired electrons and holes in a single-band model that is easily converted into a two-band model. The GBEC theory is essentially an extension of the Friedberg-T.D. Lee 1989 BEC theory of superconductors that excludes 2h-CPs. It can thus recover, when the numbers of 2h- and 2e-CPs in both BE-condensed and noncondensed states are separately equal, the BCS gap equation for all temperatures and couplings as well as the zero-temperature BCS (rigorous-upper-bound) condensation energy for all couplings. But ignoring either 2h- {\it or} 2e-CPs it can do neither. In particular, only {\it half} the BCS condensation energy is obtained in the two crossover versions ignoring either kind of CPs. We show how critical temperatures $T_{c}$ from the original BCS-Bose crossover theory in 2D require unphysically large couplings for the Cooper/BCS model interaction to differ significantly from the $T_{c}$s of ordinary BCS theory (where the number equation is substituted by the assumption that the chemical potential equals the Fermi energy).