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.
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.
By recognizing the vital importance of two‐hole Cooper pairs (CPs) in addition to the usual two‐electron ones in a strongly‐interacting many‐electron system, the concept of CPs was re‐examined with striking conclusions: namely, they are gapped and linearly‐dispersive resonances with a finite lifetime—but provided the ideal‐gas Fermi sea is replaced by a BCS‐correlated unperturbed ground‐state “sea.” Based on this, Bose‐Einstein condensation (BEC) theory has been generalized to include not boson‐boson interactions (also neglected in BCS theory) but rather boson‐fermion (BF) interaction vertices reminiscent of the Fröhlich electron‐phonon interaction in metals. Instead of phonons, the bosons in the generalized BEC (GBEC) theory are now both particle and hole CPs. Unlike BCS theory, the GBEC model is not a mean‐field theory restricted to weak‐coupling as it can be diagonalized exactly. In weak coupling it reproduces the BCS condensation energy, and the next‐order‐in‐coupling term increases its magnitude with respect to BCS. Each kind of CP is responsible for only half the condensation energy. The GBEC theory reduces to all the old known statistical theories as special cases—including the so‐called “BCS‐Bose crossover” picture which in turn generalizes BCS theory by not assuming that the interelectronic chemical potential equals the Fermi energy. Indeed, a BCS condensate is precisely the weak‐coupling limit of a GBE condensate with equal numbers of both types of CPs. With feasible Cooper/BCS model interelectonic interaction parameter values, and even without BF interactions, the GBEC theory yields transition temperatures [including room‐temperature superconductivity (RTSC)] substantially higher than the BCS ceiling of around 45K, without relying on non‐phonon dynamics involving excitons, plasmons, magnons or otherwise purely‐electronic mechanisms. The results are expected to shed light on the experimental search for RTSC.
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.
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.
We review Cooper pairing starting from its simplest, original 1956 version of two electrons interacting above the Fermi sea of an ideal Fermi gas (IFG). The two-electron interaction assumed extensively (if not exclusively), is the attractive two-parameter Cooper, and then BCS, model interactions. Hole Cooper pairs (CPs) and electron-hole CPs are then included along with the initial electron-CPs in terms of the single-fermion Green functions implied by the Bethe-Salpeter (BS) integral equation in the ladder approximation. A purely-imaginary CP energy "instability" is recovered that is well-documented in the literature at least since the late 1950's. A novel interpretation of this instability is that an unperturbed Hamiltonian different from the IFG one first used by Cooper suffices to obtain meaningful CPs. Instead of the IFG sea, a BCS-correlated Fermi "sea" used in the BS equation interpreted as the associated unperturbed Hamiltonian leads to real CP energies (with small imaginary terms implying damping). We survey how this has been achieved in 1D, 2D and 3D, and give a more detailed treatment in 2D. A vital distinction is that the original and generalized CPs are true bosons in contrast with BCS pairs that are not ordinary bosons but rather "hard-core bosons" as they do not obey strict Bose commutation rules. Another important common element of the original or generalized CPs (particularly in 2D where ordinary Bose-Einstein condensation (BEC) does not occur) is their linear dispersion relation in leading order in the total (or, center-of-mass) momentum power-series expansion of the CP energy. This theory encompasses, in principle, all empirically known superconductors including quasi-2D superconductors such as cuprates and the ET organic compounds, as well as quasi-1D ones such as the organometallic Bechgaard salts and nanotubes.
Though commonly unrecognized, a superconducting BCS condensate consists of equal numbers of two-electron (2e) and two-hole (2h) Cooper pairs (CPs). A complete (in the sense that 2h-CPs are not ignored) boson-fermion statistical model, however, is able to depart from this perfect 2e-/2h-CP symmetry and yields robustly higher T_c's without abandoning electron-phonon dynamics. The complete model reduces to all the known statistical theories of superconductors, including the BCS-Bose "crossover" picture but going considerably beyond it.