M. A. Solís

The BCS–Bose crossover theory

Abstract We contrast 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–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 non-condensed 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- or 2e-CPs it can do neither. In particular, only 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).

Cooper pairs as resonances

Abstract Using the Bethe–Salpeter (BS) equation, Cooper pairing can be generalized to include contributions from holes as well as particles from the ground state of either an ideal Fermi gas (IFG) or of a BCS many-fermion state. The BCS model interfermion interaction is employed throughout. In contrast to the better-known original Cooper pair (CP) problem for either two particles or two holes, the generalized Cooper equation in the IFG case has no real-energy solutions. Rather, it possesses two complex-conjugate solutions with purely imaginary energies. This implies that the IFG ground state is unstable when an attractive interaction is switched on. However, solving the BS equation for the BCS ground state reveals two types of real solutions: one describing moving (i.e., having nonzero total, or center-of-mass, momenta) CPs as resonances (or bound composite particles with a finite lifetime), and another exhibiting superconducting collective excitations analogous to Anderson–Bogoliubov–Higgs RPA modes. A Bose–Einstein-condensation-based picture of superconductivity is addressed.

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

The Cooper pair dispersion relation

for quadratic dispersion, but is nonzero for all

Two-dimensional Bose-Einstein condensation in cuprate superconductors

dg~1

Linear to quadratic crossover of Cooper-pair dispersion relation

for linear dispersion.

Statistical model of superconductivity in a 2D binary boson–fermion mixture

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.

Low-dimensional BEC

The binding energy of a Cooper pair formed with the BCS model interaction potential is obtained numerically for all coupling in two and three dimensions for all non-zero center-of-mass momentum (CMM) of the pair. The pair breaks up for very small CMM, at most about four orders of magnitude smaller than the maximum CMM allowed by the BCS model interaction, and its binding energy is remarkably linear over the entire range of the CMM up to breakup.

Dimensional crossover of a boson gas in multilayers

Abstract A binary gas of noninteracting, temperature-dependent Cooper pairs in chemical/thermal equilibrium with unpaired fermions is studied in a two-dimensional (2D) boson-fermion statistical model analogous to an atom plus diatomic-molecule system. The model naturally suggests a more convenient definition for the bosonic chemical potential whereby access into the degenerate Fermi region of positive fermion chemical potential is now possible. The linear (as opposed to quadratic) dispersion relation of the pairs yields substantially higher T c s than with BCS or pure-boson Bose-Einstein condensation theories, and fall within the range of empirical T c s for quasi-2D copper oxide superconductors.