O. Rojo

Cooper pair dispersion relation for weak to strong coupling

Cooper pairing in two dimensions is analyzed with a set of renormalized equations to determine its binding energy for any fermion number density and all coupling assuming a generic pairwise residual interfermion interaction. Also considered are Cooper pairs (CPs) with nonzero center-of-mass momentum (CMM) and their binding energy is expanded analytically in powers of the CMM up to quadratic terms. A Fermi-sea-dependent linear term in the CMM dominates the pair excitation energy in weak coupling (also called the BCS regime) while the more familiar quadratic term prevails in strong coupling (the Bose regime). The crossover, though strictly unrelated to BCS theory per se, is studied numerically as it is expected to play a central role in a model of superconductivity as a Bose-Einstein condensation of CPs where the transition temperature vanishes for all dimensionality

Bose-Einstein condensation with a BCS model interaction

dl~2

Linear to quadratic crossover of Cooper-pair dispersion relation

for quadratic dispersion, but is nonzero for all

Low-dimensional BEC

dg~1

INTRIGUING ROLE OF HOLE-COOPER-PAIRS IN SUPERCONDUCTORS AND SUPERFLUIDS

for linear dispersion.

Superconductivity as a Bose-Einstein condensation?

Abstract A simple model of a boson-fermion mixture of unpaired fermions plus linear-dispersion-relation Cooper pairs that includes pair-breaking effects leads to Bose-Einstein condensation for dimensions greater than unity, at critical temperatures substantially greater than those of the BCS theory of superconductivity, for the same BCS model interaction between the fermions.

Bose–Einstein condensation of nonzero-center-of-mass-momentum Cooper pairs

Abstract Cooper pairing is studied in three dimensions to determine its binding energy for all coupling using a general separable interfermion interaction. Also considered are Cooper pairs (CPs) with nonzero center-of-mass momentum (CMM). A coupling-independent linear term in the CMM dominates the pair excitation energy in weak coupling and/or high fermion density, while the more familiar quadratic term prevails only in the extreme low-density (i.e., vacuum) limit for any nonzero coupling. The linear-to-quadratic crossover of the CP dispersion relation is analyzed numerically, and is expected to play a central role in a model of superconductivity (and superfluidity) simultaneously accommodating a Bardeen–Cooper–Schrieffer condensate as well as a Bose–Einstein condensate of CP bosons.

Cooper pair dispersion relation in two dimensions

The Bose-Einstein condensation (BEC) temperature Tc of Cooper pairs (CPs) created from a general interfermion interaction is determined for a linear, as well as the usually assumed quadratic, energy vs center-of-mass momentum dispersion relation. This explicit Tc is then compared with a widely applied implicit one of Wen & Kan (1988) in d=2+∈ dimensions, for small ∈, for a geometry of an infinite stack of parallel (e.g., copperoxygen) planes as in, say, a cuprate superconductor, and with a new result for linear-dispersion CPs. The implicit formula gives Tc values only slightly lower than those of the explicit formula for typical cuprate parameters.

BEC-DRIVEN SUPERCONDUCTIVITY IN THE CUPRATES

The role in superconductors of hole-Cooper-pairs (CPs) are examined and contrasted with the more familiar electron-CPs, with special emphasis on their “background” effect in enhancing superconducting transition temperatures Tc — even when electron-CPs drive the transition. Both kinds of CPs are, of course, present at all temperatures. An analogy is drawn between the hole CPs in any many-fermion system with the antibosons in a relativistic ideal Bose gas that appear in substantial numbers only at higher and higher temperatures. Their indispensable role in yielding a lower Helmholtz free energy equilibrium state is established. For superconductors, the problem is viewed in terms of a generalized Bose-Einstein condensation (GBEC) theory that is an extension of the Friedberg-T.D. Lee 1989 boson-fermion BEC theory of high-Tc superconductors in that the GBEC theory includes hole CPs as well as electron-CPs — thereby containing as well as further extending BCS theory to higher temperatures with the same weak-coupling electron-phonon interaction parameters. We show that the Helmholtz free energy of both 2e- and 2h-CP pure condensates has a positive second derivative, and are thus stable equilibrium states. Finally, it is conjectured that the role of hole pairs in ultra-cold fermionic atom gases will likely be negligible because the very low densities involved imply a “shallow” Fermi sea.

BCS and BEC Finally Unified: A Brief Review

Abstract Bose-Eisntein condensation (BEC) in two dimensions (2D) (e.g., to describe the quasi-2D cuprates) is suggested as the possible mechanism widely believed to underlie superconductivity in general. A crucial role is played by nonzero center-of-mass momentum Cooper pairs (CPs) usually neglected in BCS theory. Also vital is the unique linear dispersion relation appropriate to weakly-coupled “bosonic” CPs moving in the Fermi sea—rather than in vacuum where the dispersion would be quadratic but only for very strong coupling, and for which BEC is known to be impossible in 2D.