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International Journal of Modern Physics B

Generalized BCS-Bose Crossover Picture of Superconductivity

Josep Amengual i Batle, M. Casas, MANUEL DE LLANO, Mauricio Fortes, Francisco J. Sevilla

Published Aug 10, 2003• 0 citations
Publisher

Abstract

A recent unification of the BCS theory with that of the Bose–Einstein condensation (BEC) through a "complete" boson-fermion model is discussed as a generalization of the "BCS-Bose crossover" picture of superconductivity. Good first-principles Tc predictions in 2D are calculated with no adjustable parameters for the so-called "exotic" cuprate superconductors of the "Uemura plot", without abandoning the phonon interaction mechanism. The only condition is that one depart moderately from the perfect electron-/hole-Cooper-pair symmetry to which BCS (as well as the "crossover") theory are restricted by construction.

DOI
10.1142/s0217979203020843

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A recent unification of the BCS theory with that of the Bose–Einstein condensation (BEC) through a "complete" boson-fermion model is discussed as a generalization of the "BCS-Bose crossover" picture of superconductivity. Good first-principles Tc predictions in 2D are calculated with no adjustable parameters for the so-called "exotic" cuprate superconductors of the "Uemura plot", without abandoning the phonon interaction mechanism. The only condition is that one depart moderately from the perfect electron-/hole-Cooper-pair symmetry to which BCS (as well as the "crossover") theory are restricted by construction.
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.
We review efforts to unify both the Bardeen, Cooper and Schrieffer (BCS) and Bose-Einstein condensation (BEC) pictures of superconductivity. We have finally achieved this in terms of a "\textit{complete} boson-fermion (BF) model" (CBFM) that reduces in special cases to all the main continuum (as opposed to "spin") statistical theories of superconductivity. Our BF model is "complete" in the sense that not only two-electron (2e) but also two-hole (2h) Cooper pairs (CPs) are allowed in arbitrary proportions. In contrast, BCS-Bogoliubov theory--which can also be considered as the theory of a mixture of kinematically independent electrons, 2e- and 2h-CPs--allows only equal, 50%-50%, mixtures of the two kinds of CPs. This is obvious from the perfect symmetry about μμμ, the electron chemical potential, of the well-known Bogoliubov v2(ε)v^{2}(ε)v2(ε) and u2(ε)u^{2}(ε)u2(ε) coefficients, where εεε is the electron energy. The CBFM is then applied to see: a) whether the BCS model interaction for the electron-phonon dynamical mechanism is sufficient to predict the unusually high values of TcT_{c}Tc​ (in units of the Fermi temperature) of ≃0.01−0.1\simeq 0.01-0.1≃0.01−0.1 exhibited by the so-called ``exotic'' superconductors \cite{Brandow} in both 2D and 3D--relative to the low values of ≲10−3\lesssim 10^{-3}≲10−3 more or less correctly predicted by BCS theory for conventional, elemental superconductors; and b) whether it can at least suggest, if not explain, why "hole superconductors" have higher TcT_{c}Tc​'s.
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.
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.
For the Cooper/BCS model interaction in superconductors (SCs) it is shown: a) how BCS-Bose crossover picture transition temperatures Tc, defined self-consistently by both the gap and fermion-number equations, require unphysically large couplings to differ significantly from the Tc of ordinary BCS theory defined without the number equation since here the chemical potential is assumed equal to the Fermi energy; how although ignoring either hole- or electron-Cooper-pairs in the recent "complete boson-fermion model": b) one obtains the precise BCS gap equation for all temperatures T, but c) only half the zero-temperature BCS condensation energy emerges. Results (b) and (c) are also expected to hold for neutral-fermion superfluids (SFs)--such as liquid 3^33He, neutron matter and trapped ultra-cold fermion atomic gases--where the pair-forming two-fermion interaction of course differs from the Cooper/BCS one for SCs.
The (mean field based) BCS theory is considered one of the most successful theories in condensed matter physics. It is justified in ordinary metal superconductors the coherence length ξξξ is large, with two important features: the order parameter (OP) and excitation gap (EG) are identical, and the pair formation and their Bose condensation take place at the same temperature Tc. It fails to explain the underdoped cuprate superconductivity: EG is finite at Tc and thus distinct from OP. Since these superconductors belong to a large class of small ξξξ materials, this failure has the potential for widespread impact. Here we have extended BCS theory in a natural way to short ξξξ superconductors, based on a BCS--BEC crossover scenario, and arrived at a simple physical picture in which incoherent, finite momentum pairs become progressively more important as the pairing interaction becomes stronger, leading to the distinction between EG and OP. The superconductivity from the fermionic perspective and BEC from the bosonic perspective are just two sides of the same coin. Our theory is capable of making verifiable quantitative predictions. We obtain a cuprate phase diagram (with one free parameter) , in (semi-)quantitative agreement with experiment. The mutually compensating contributions from fermionic quasiparticles and bosonic pair excitations provides a natural explanation for the quasi-universal behavior of the in-plane superfluid density versus T. Our bosonic pair excitations also provide an intrinsic mechanism for the long mysterious linear T terms in the specific heat. Incoherent pair contributions lead to new low T power laws, consistent with existing experiments. Finally, we demonstrated that the onset of superconducting long range order leads to sharp features in the specific heat at Tc, consistent with experiment.