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Physical review. B, Condensed matter

Scaling in the BCS to Bose crossover problem in different partial waves

Sadhan K. Adhikari, Angsula Ghosh

Published Jan 1, 1997• 20 citations
Publisher

Abstract

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 ${\mathrm{B}}_{\mathrm{c}}$ exhibit universal scaling. In the BCS limit the results for coherence length \ensuremath{\xi} and the critical temperature ${\mathrm{T}}_{\mathrm{c}}$ are appropriate for high-${\mathrm{T}}_{\mathrm{c}}$ cuprate superconductors and also exhibit universal scaling as a function of ${\mathrm{B}}_{\mathrm{c}}$.

DOI
10.1103/physrevb.55.1110

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arXiv:cond-mat/0412418•Dec 16, 2004•0 citations
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{\mathrm{B}}_{\mathrm{c}}Bc​ exhibit universal scaling. In the BCS limit the results for coherence length \ensuremath{\xi} and the critical temperature Tc{\mathrm{T}}_{\mathrm{c}}Tc​ are appropriate for high-Tc{\mathrm{T}}_{\mathrm{c}}Tc​ cuprate superconductors and also exhibit universal scaling as a function of Bc{\mathrm{B}}_{\mathrm{c}}Bc​.
The solutions of a renormalized BCS model are studied in two space dimensions for s, p and d waves for finite-range separable potentials. The gap parameter, the critical temperature , the coherence length and the jump in specific heat at as a function of the zero-temperature condensation energy exhibit universal scalings. In the weak-coupling limit, the present model yields a small and large , appropriate for high- cuprates. The specific heat, penetration depth and thermal conductivity as functions of temperature show universal scaling for p and d waves.
We argue that the natural variable to establish the crossover from Cooper-pair-based superconductivity to Bose-Einstein condensation of bound-electron pairs is the product kF{\mathit{k}}_{\mathit{F}}kF​\ensuremath{\xi} of Fermi electron wave vector times the coherence length for two-electron correlation, and that Cooper-pair-based superconductivity is stable against bosonization down to kF{\mathit{k}}_{\mathit{F}}kF​\ensuremath{\xi}\ensuremath{\simeq}2\ensuremath{\pi}. We also propose that the experimental plot by Uemura et al., relating the superconducting temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ to the Fermi temperature TF{\mathit{T}}_{\mathit{F}}TF​ for a variety of ``exotic'' as well as conventional superconductors, can be rationalized by correlating the ratio Tc{\mathit{T}}_{\mathit{c}}Tc​/TF{\mathit{T}}_{\mathit{F}}TF​ to kF{\mathit{k}}_{\mathit{F}}kF​\ensuremath{\xi}. It appears in this way that the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors lie in the plot by Uemura et al. near the ``instability'' line kF{\mathit{k}}_{\mathit{F}}kF​\ensuremath{\xi}=2\ensuremath{\pi}.
In a two-dimensional $t\text{\ensuremath{-}}J$ model that includes anisotropy in the hopping frequency of the charge carriers, the crossover scenario from a BCS superconductor to a condensate of tightly bound bosonic pairs is studied. The important length scales, such as the mean pair radius and penetration depth, are obtained as a function of the anisotropy parameter that distinguishes hopping in one direction with respect to the other. The results indicate a smooth evolution from a picture of highly overlapping Cooper pairs to a condensate of short-range bosonic pairs with increasing anisotropy. The claim for the crossover phenomena is supported by calculating the renormalized chemical potential which slips below the band minimum in the extreme anisotropy limit, implying the onset of a Bose phase.
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.
We study a two-dimensional continuum model of a dilute gas of fermions at T=0 interacting via a given two-body potential, with an aim to investigate superconductors with coherence length of the order of the interparticle spacing [kF{k}_{F}kF​${\ensuremath{\xi}}_{0}$\ensuremath{\sim}O(1)], a striking feature of the high-Tc{T}_{c}Tc​ materials. We find that a two-body bound state in vacuum is a necessary and sufficient condition for an s-wave pairing instability. We also find that the existence of such a bound state in a higher-angular-momentum (l>0) channel is not a necessary condition for an l-wave pairing instability. We further investigate using a variational ansatz the evolution from a state with large overlapping Cooper pairs (kF{k}_{F}kF​${\ensuremath{\xi}}_{0}$\ensuremath{\gg}1) to one with Bose condensation of composite bosons (kF{k}_{F}kF​${\ensuremath{\xi}}_{0}$\ensuremath{\ll}1). For the s-wave case an exact solution of the variational equations shows a smooth crossover from one regime to the other at T=0. The p-wave solution has a weak singularity when the chemical potential goes through zero, which is the bottom of the band. We show, quite generally, independent of the dimensionality and of model details, that the gap--to--single-particle excitations is nodeless, even if the anisotropic pair wave function has nodes, when the coupling is strong enough that the chemical potential is negative.
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.