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
ξ 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
ξ materials, this failure has the potential for widespread impact. Here we have extended BCS theory in a natural way to short
ξ 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.
The BCS coherence length in low-dimensional superconductors is dramatically modified by quantum-size effects. In particular, for nanowires made of conventional superconducting materials, we show that the longitudinal zero-temperature coherence length exhibits width-dependent drops by 2--3 orders of magnitude each time when the bottom of one of single-electron subbands formed due to the transverse quantization of electron motion is situated in a close vicinity to the Fermi level. This phenomenon has strong similarities to the well-known BCS-BEC (Bose-Einstein condensation) crossover in ultracold fermionic condensates but with an important exception: it is driven by the transverse quantization of the electron motion rather than by the externally controlled strength of the fermion-fermion interaction.
The BCS superconductivity to Bose condensation crossover problem is studied in two dimensions in S, P, and D waves, for a simple anisotropic pairing, with a finite-range separable potential at zero temperature. The gap parameter and the chemical potential as a function of Cooper-pair binding
Bc exhibit universal scaling. In the BCS limit the results for coherence length \ensuremath{\xi} and the critical temperature
Tc are appropriate for high-
Tc cuprate superconductors and also exhibit universal scaling as a function of
Bc.