The new generalized Bose–Einstein condensation (GBEC) quantum-statistical theory starts from a noninteracting ternary boson-fermion (BF) gas of two-hole Cooper pairs (2hCPs) along with the usual two-electron Cooper pairs (2eCPs) plus unpaired electrons. Here we obtain the entropy and heat capacity and confirm once again that GBEC contains as a special case the Bardeen–Cooper–Schrieffer (BCS) theory. The energy gap is first calculated and compared with that of BCS theory for different values of a new dimensionless coupling parameter n/n[Formula: see text] where n is the total electron number density and n[Formula: see text] that of unpaired electrons at zero absolute temperature. Then, from the entropy, the heat capacity is calculated. Results compare well with elemental-superconductor data suggesting that 2hCPs are indispensable to describe superconductors (SCs).
The new generalized Bose–Einstein condensation (GBEC) quantum-statistical theory starts from a noninteracting ternary boson-fermion (BF) gas of two-hole Cooper pairs (2hCPs) along with the usual two-electron Cooper pairs (2eCPs) plus unpaired electrons. Here we obtain the entropy and heat capacity and confirm once again that GBEC contains as a special case the Bardeen–Cooper–Schrieffer (BCS) theory. The energy gap is first calculated and compared with that of BCS theory for different values of a new dimensionless coupling parameter n/n[Formula: see text] where n is the total electron number density and n[Formula: see text] that of unpaired electrons at zero absolute temperature. Then, from the entropy, the heat capacity is calculated. Results compare well with elemental-superconductor data suggesting that 2hCPs are indispensable to describe superconductors (SCs).
The generalized Bose–Einstein condensation (GBEC) theory of superconductivity (SC) is briefly surveyed. It hinges on three distinct new ingredients: (i) Treatment of Cooper pairs (CPs) as actual bosons since they obey Bose statistics, in contrast to BCS pairs which do not obey Bose commutation relations; (ii) inclusion of two-hole Cooper pairs (2hCPs) on an equal footing with two-electron Cooper pairs (2eCPs), thus making this a complete boson–fermion (BF) model; and (iii) inclusion in the resulting ternary ideal BF gas with particular BF vertex interactions that drive boson formation/disintegration processes. GBEC subsumes as special cases both BCS (having its 50–50 symmetry of both kinds of CPs) and ordinary BEC theories (having no 2hCPs), as well as the now familiar BCS-Bose crossover theory. We extended the crossover theory with the explicit inclusion of 2hCPs and construct a phase diagram of [Formula: see text] versus [Formula: see text], where [Formula: see text] and [Formula: see text] are the critical and Fermi temperatures, [Formula: see text] is the total number density and [Formula: see text] that of unbound electrons at [Formula: see text]. Also, with this extended crossover one can construct the energy gap [Formula: see text] versus [Formula: see text] for some elemental SCs by solving at least two equations numerically: a gap-like and a number equation. In 50–50 symmetry, the energy gap curve agrees quite well with experimental data. But ignoring 2hCPs altogether leads to the gap curve falling substantially below that with 50–50 symmetry which already fits the data quite well, showing that 2hCPs are indispensable to describe SCs.
Abstract The superfluid density ns(T) n s ( T ) of a superconductor is calculated based on the generalized Bose–Einstein condensation (GBEC) theory that addresses a fully-interacting ternary boson-fermion gas mixture of free electrons as fermions, plus two-electron Cooper pairs (2eCPs) and also, explicitly, two-hole Cooper pairs (2hCPs), both as bosons. Here we consider two special cases (i) 100%–0% (i.e., with no condensed 2hCPs) and (ii) 0%–100% (i.e., with no condensed 2eCPs). Subsumed in GBEC are the Bardeen–Cooper–Schrieffer (BCS) and Bose–Einstein condensation (BEC) theories along with the BCS-BEC crossover theory extended with 2hCPs. We find that in the weak-coupling regime ns(0) n s ( 0 ) agrees with data from the Uemura et al. (2004) graph for several elemental SCs by taking in 3D with a quadratic energy-dispersion relation while in 2D with a linear relation are much too far below the data. In the strong-coupling regime the linear behavior of critical temperature Tc T c vs ns(0) n s ( 0 ) obtained here is just as Božović et al. (2016) found. However, in 2D with a linear relation accounting for 0%–100%, ns(T)/ns(0) n s ( T ) / n s ( 0 ) compares well with some high- Tc T c -cuprate SC data between the two coupling regimes.
Unification of the Bardeen, Cooper and Schrieffer (BCS) and the Bose‐Einstein condensation (BEC) theories is surveyed in terms of a generalized BEC (GBEC) finite‐temperature statistical formalism. A vital distinction is that Cooper pairs (CPs) are true bosons that may suffer a BEC since they obey BE statistics, in contrast with BCS pairs that are “hard‐core bosons” at best. A second crucial ingredient is the explicit presence of hole‐pairs (2h) alongside the usual electron‐pairs (2e). A third critical element (particularly in 2D where ordinary BEC does not occur) is the linear dispersion relation of CPs in leading order in the center‐of‐mass momentum (CMM) power‐series expansion of the CP energy. The GBEC theory reduces in limiting cases to all five continuum (as opposed to “spin”) statistical theories of superconductivity, from BCS on one extreme to the BEC theory on the other, as well as to the BCS‐Bose “crossover” picture and the 1989 Friedberg‐Lee BEC theory. It accounts for 2e‐ and 2h‐CPs in arbitrary proportions while BCS theory can be deduced from the GBEC theory but allows only equal (50%‐50%) BE condensed‐mixtures of both kinds of CPs. As it yields the precise BCS gap equation for all temperatures as well as the precise BCS zero‐temperature condensation energy for all couplings, it suggests that the BCS condensate is a BE condensate of a ternary mixture of kinematically independent unpaired electrons coexisting with equally proportioned weakly‐bound zero‐CMM 2e‐ and 2h‐CPs. Without abandoning the electron‐phonon mechanism in moderately weak coupling, and fortuituously insensitive to the BF interactions, the GBEC theory suffices to reproduce the unusually high values of Tc (in units of the Fermi temperature TF) of 0.01–0.05 empirically found in the so‐called “exotic” superconductors of the Uemura plot, including cuprates, in contrast to the low values of Tc/TF⩽10−3 roughly reproduced by BCS theory for conventional (mostly elemental) superconductors.
Applying the generalized Bose–Einstein condensation (GBEC) formalism, we extend the BCS-Bose crossover theory by explicitly including hole Cooper pairs (2hCPs). From this follows a phase diagram with two pure phases, one with 2hCPs and the other with electron Cooper pairs (2eCPs), plus a mixed phase with arbitrary proportions of 2eCPs and 2hCPs. One has a special-case phase when there is perfect symmetry (i.e., with ideal 50–50 proportions between 2eCPs and 2hCPs). Explicitly including 2hCPs leads to an extended BCS-Bose crossover which predicts [Formula: see text] values for some well-known conventional superconductors (SCs) (i.e., assuming electron–phonon dynamics). These compare reasonably well with experimental data. We compare with experimental [Formula: see text] values for some conventional SCs associated with the new dimensionless number density [Formula: see text] with theoretical curves associated with the extended crossover for the special case of perfect symmetry. They all obey the Bogoliubov et al. upper limit, thus vindicating it.
Unification of the BCS and the Bose-Einstein condensation (BEC) theories is surveyed in detail via a generalized BEC (GBEC) finite-temperature statistical formalism. Its major difference with BCS theory is that it can be diagonalized exactly. Under specified conditions it yields the precise BCS gap equation for all temperatures as well as the precise BCS zero-temperature condensation energy for all couplings, thereby suggesting that a BCS condensate is a BE condensate in a ternary mixture of kinematically independent unpaired electrons coexisting with equally proportioned weakly-bound two-electron and two-hole Cooper pairs. Without abandoning the electron-phonon mechanism in moderately weak coupling it suffices, in principle, to reproduce the unusually high values of Tc (in units of the Fermi temperature TF) of 0.01-0.05 empirically reported in the so-called "exotic" superconductors of the Uemura plot, including cuprates, in contrast to the low values of Tc/TF ≤ 10-3 roughly reproduced by BCS theory for conventional (mostly elemental) superconductors. Replacing the characteristic phonon-exchange Debye temperature by a characteristic magnon-exchange one more than twice in size can lead to a simple interaction model associated with spin-fluctuation-mediated pairing.