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Physical review. B, Condensed matter

Evolution from BCS superconductivity to Bose condensation: Role of the parameter kF ξ

F. Pistolesi, Giancarlo Calvanese Strinati

Published Mar 1, 1994• 140 citations
Publisher

Abstract

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 ${\mathit{k}}_{\mathit{F}}$\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 ${\mathit{k}}_{\mathit{F}}$\ensuremath{\xi}\ensuremath{\simeq}2\ensuremath{\pi}. We also propose that the experimental plot by Uemura et al., relating the superconducting temperature ${\mathit{T}}_{\mathit{c}}$ to the Fermi temperature ${\mathit{T}}_{\mathit{F}}$ for a variety of ``exotic'' as well as conventional superconductors, can be rationalized by correlating the ratio ${\mathit{T}}_{\mathit{c}}$/${\mathit{T}}_{\mathit{F}}$ to ${\mathit{k}}_{\mathit{F}}$\ensuremath{\xi}. It appears in this way that the high-${\mathit{T}}_{\mathit{c}}$ superconductors lie in the plot by Uemura et al. near the ``instability'' line ${\mathit{k}}_{\mathit{F}}$\ensuremath{\xi}=2\ensuremath{\pi}.

DOI
10.1103/physrevb.49.6356

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Comment on "Evolution from BCS Superconductivity to Bose-Einstein Condensation: Role of the Parameter $k_{\rm F} \xi$ in Interpreting the Experimental Plot by Uemura et al.''}

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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 ${\mathit{T}}_{\mathit{c}}$/${\mathit{T}}_{\mathit{F}}$ to ${\mathit{k}}_{\mathit{F}}$\ensuremath{\xi}. It appears in this way that the high-${\mathit{T}}_{\mathit{c}}$ superconductors lie in the plot by Uemura et al. near the instability'' line kF{\mathit{k}}_{\mathit{F}}kF​\ensuremath{\xi}=2\ensuremath{\pi}.
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.
We show that the natural variable to follow the crossover from Cooper-pair-based superconductivity to Bose–Einstein condensation within the model of Noziéres and Schmitt-Rink is the product k F ξ, where k F is the Fermi wave vector and ξ, is the coherence length for two-electron correlation. In terms of this product, the results of the model do not depend on the detailed form of the (separable) pairing potential, and the crossover turns out to be restricted to the universal region π −1 ≲k F ξ ≲ 2π. Experimental estimates indicate that k F ξ ≈10 (> 2π) for high-T c superconductors. Evolution from weak to strong coupling superconductivity has been considered by Nozières and Schmitt-Rink [1] (hereafter referred to as NSR) following the pioneering work by Leggett [2]. After the discovery of high- T c superconductivity, the interest in this problem has grown, and many papers on this subject have appeared [3]. In the present work, we show that working within the simplified treatment by NSR, it is already possible to isolate the essential features of the crossover. Central to the work of NSR and Leggett is the argument [4] that the BCS wave function has the Bose–Einstein condensation (BEC) built in as a limiting case. (See the review by Randeria in this volume.) NSR study the evolution from BCS to BEC through the increase of the coupling strength associated to an effective fermionic attractive potential, and conclude that the evolution is “smooth”.
We consider a fermionic system at zero temperature interacting through an effective nonretarded potential of the type introduced by Nozi\`eres and Schmitt-Rink, and calculate the phase coherence length ${\ensuremath{\xi}}_{\mathrm{phase}}$ (associated with the spatial fluctuations of the superconducting order parameter) by exploiting a functional-integral formulation for the correlation functions and the associated loop expansion. This formulation is especially suited to follow the evolution of the fermionic system from a BCS-type superconductor for weak coupling to a Bose-condensed system for strong coupling, since in the latter limit a direct mapping of the original fermionic system onto an effective system of bosons with a residual boson-boson interaction can be established. Explicit calculations are performed at the one-loop order. The phase coherence length ${\ensuremath{\xi}}_{\mathrm{phase}}$ is compared with the coherence length ${\ensuremath{\xi}}_{\mathrm{pair}}$ for two-electron correlation, which is relevant to distinguish the weak- (kF{\mathit{k}}_{\mathit{F}}kF​${\ensuremath{\xi}}_{\mathrm{pair}}$\ensuremath{\gg}1) from the strong- (kF{\mathit{k}}_{\mathit{F}}kF​${\ensuremath{\xi}}_{\mathrm{pair}}$\ensuremath{\ll}1) coupling limits (kF{\mathit{k}}_{\mathit{F}}kF​ being the Fermi wave vector) as well as to follow the crossover in between. It is shown that ${\ensuremath{\xi}}_{\mathrm{phase}}$ coincides with ${\ensuremath{\xi}}_{\mathrm{pair}}$ down to kF{\mathit{k}}_{\mathit{F}}kF​${\ensuremath{\xi}}_{\mathrm{pair}}$\ensuremath{\simeq}10, ${\ensuremath{\xi}}_{\mathrm{pair}}$ in turn coinciding with the Pippard coherence length. In the strong-coupling limit we find instead that ${\ensuremath{\xi}}_{\mathrm{phase}}$\ensuremath{\gg}${\ensuremath{\xi}}_{\mathrm{pair}}$, with ${\ensuremath{\xi}}_{\mathrm{pair}}$ coinciding with the radius of the bound-electron pair. From the mapping onto an effective system of bosons in the strong-coupling limit we further relate ${\ensuremath{\xi}}_{\mathrm{pair}}$ with the ``range'' of the residual boson-boson interaction, which is physically the only significant length associated with the dynamics of the bosonic system. \textcopyright{} 1996 The American Physical Society.
Comment on "Evolution from BCS Superconductivity to Bose-Einstein Condensation: Role of the Parameter kFξk_{\rm F} \xikF​ξ in Interpreting the Experimental Plot by Uemura et al.''}
The problem of crossover from BCS to Bose-Einstein condensation has recently attracted considerable interest owing to the discovery of high-TcT_{\rm c}Tc​ cuprates. Their short coherence length, ξ\xiξ, places these materials in the interesting region between BCS and Bose-Einstein condensation. In the paper of F. Pistolesi and G.C. Strinati (Phys. Rev. B {\bf 49}, 6356 (1994)), the Nozi\`eres and Schmitt-Rink approach (NSR) is taken, which is valid for the weak coupling regime. They derive a relation between TcT_{\rm c}Tc​ and TFT_{\rm F}TF​, which they insist is valid for any coupling strength. We present arguments that their assumptions are incorrect by using our fully self-consistent T-matrix formalism in two dimensions, and show that the NSR approach produces unphysical results in this case.
This article gives a detailed review of our current theoretical understanding of the crossover from cooperative Cooper pairing to independent bound-state formation and Bose–Einstein condensation in Fermi systems with increasing attractive interactions. Introduction There are two well-known paradigms for understanding the phenomena of superconductivity and superfluidity: (I) BCS theory [1], in which the normal state is a degenerate Fermi liquid that undergoes a pairing instability at a temperature T c ≪ ε f , the degeneracy scale. The formation of Cooper pairs and their condensation (macroscopic occupation of a single quantum state) both occur simultaneously at the transition temperature T c . (II) Bose–Einstein condensation (BEC) of bosons at a T c of the order of their degeneracy temperature. At a fundamental level these bosons (for example, 4 He or excitons) are invariably composite objects made up of an even number of fermions. The composite particles form at some very high temperature scale of the order of their dissociation temperature T dissoc , and these “pre-formed” bosons then condense at the BEC T c ≪ T dissoc . In most cases of experimental interest the system under consideration clearly falls into one category or the other. For instance, 3 He is a Fermi superfluid described by (I) whereas 4 He is a Bose superfluid (II). Essentially all cases of superconductivity in metals that we understand reasonably well are much closer to (I) than to (II), as evidenced by the existence of fermionic quasiparticles above T c .
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.