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arXiv:cond-mat/0109486 · Physical review. B, Condensed matter

Coherence length in superconductors from weak to strong coupling

L. Benfatto, A. Toschi, S. Caprara, Claudio Castellani

Published Aug 12, 2002• 32 citations
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Abstract

We investigate the evolution of the superconducting coherence length ${\ensuremath{\xi}}_{0}$ from weak to strong coupling, within both an s-wave and a d-wave lattice model. We show that the identification of ${\ensuremath{\xi}}_{0}$ with the Cooper-pair size ${\ensuremath{\xi}}_{\mathrm{pair}}$ in the weak-coupling regime is meaningful only for a fully gapped (e.g., s-wave) superconductor. Instead in a d-wave superconductor, where ${\ensuremath{\xi}}_{\mathrm{pair}}$ diverges, we show that ${\ensuremath{\xi}}_{0}$ is finite (and of the expected order of magnitude) when properly defined as the characteristic length scale for the correlation function of the amplitude of the superconducting order parameter. The strong-coupling regime is quite intriguing, since the interplay between the particle-particle and the particle-hole channel is no more negligible. In the case of s-wave pairing, which allows for an analytical treatment, we show that ${\ensuremath{\xi}}_{0}$ is of the order of the lattice spacing at finite densities. In the diluted regime ${\ensuremath{\xi}}_{0}$ diverges, recovering the behavior of the coherence length of an effective bosonic system. Similar results are expected to hold for d-wave superconductors. We also comment on the consequences of our results with respect to the physics of high-${T}_{c}$ superconducting cuprates.

DOI
10.1103/physrevb.66.054515

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We investigate the evolution of the superconducting coherence length \ensuremathξ0{\ensuremath{\xi}}_{0}\ensuremathξ0​ from weak to strong coupling, within both an s-wave and a d-wave lattice model. We show that the identification of \ensuremathξ0{\ensuremath{\xi}}_{0}\ensuremathξ0​ with the Cooper-pair size \ensuremathξpair{\ensuremath{\xi}}_{\mathrm{pair}}\ensuremathξpair​ in the weak-coupling regime is meaningful only for a fully gapped (e.g., s-wave) superconductor. Instead in a d-wave superconductor, where \ensuremathξpair{\ensuremath{\xi}}_{\mathrm{pair}}\ensuremathξpair​ diverges, we show that \ensuremathξ0{\ensuremath{\xi}}_{0}\ensuremathξ0​ is finite (and of the expected order of magnitude) when properly defined as the characteristic length scale for the correlation function of the amplitude of the superconducting order parameter. The strong-coupling regime is quite intriguing, since the interplay between the particle-particle and the particle-hole channel is no more negligible. In the case of s-wave pairing, which allows for an analytical treatment, we show that \ensuremathξ0{\ensuremath{\xi}}_{0}\ensuremathξ0​ is of the order of the lattice spacing at finite densities. In the diluted regime \ensuremathξ0{\ensuremath{\xi}}_{0}\ensuremathξ0​ diverges, recovering the behavior of the coherence length of an effective bosonic system. Similar results are expected to hold for d-wave superconductors. We also comment on the consequences of our results with respect to the physics of high-Tc{T}_{c}Tc​ superconducting cuprates.
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.
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.
It has been argued that inhomogeneity generally can enhance superconductivity (SC) in the cuprate high-Tc{T}_{c}Tc​ materials. To check the validity of this claim, we study ddd-wave SC on the checkerboard Hubbard model on a square lattice using the cellular dynamical mean-field theory method with an exact diagonalization solver at zero temperature. The ddd-wave order parameter is computed for various inhomogeneity levels over the entire doping range of interest in both strong- and weak-coupling regimes. At a given doping, the size of the ddd-wave order parameter manifests itself directly in the height of the coherence peaks and, hence, is an appropriate measure of the strength of SC. The weak-coupling results reveal a suppression of the order parameter in the presence of inhomogeneity for small-to-intermediate hole dopings, while it is enhanced for large dopings. In contrast, for strong coupling, there is a monotonic decrease in the maximum amplitude of the SC order parameter with inhomogeneity over the entire doping range of interest. Furthermore, at moderately high inhomogeneity, the system undergoes a first-order transition from the SC to the normal state in the underdoped regime. In the overdoped regime, the change in the value of the SC order parameter correlates with the height of the lowest-energy peak in the spectral weight of antiferromagnetic spin fluctuations, confirming the connection between antiferromagnetic fluctuations and ddd-wave SC found in earlier papers on the homogeneous case. Our results are benchmarked by comparisons with numerically exact results on the checkerboard Hubbard ladder.
The pseudogap state of cuprate high-temperature superconductors has been often viewed as either a yet unknown competing order or a precursor state to superconductivity. While awaiting the resolution of the pseudogap problem in cuprates, we demonstrate that local pairing fluctuations, vortex liquid dynamics, and other precursor phenomena can emerge quite generally whenever fermionic excitations remain gapped across the superconducting transition, regardless of the gap origin. Our choice of a tractable model is a lattice band insulator with short-range attractive interactions between fermions in the sss-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes can be identified in any band insulator above two dimensions, while in two dimensions only the BEC regime exists. The superconducting transition is ``unconventional'' (non-pair-breaking) in the BEC regime, identified by either the bosonic mean field or XY\mathit{XY}XY universality class. The insulator adjacent to the superconductor in the BEC regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to charge density wave ordering. We construct a function of the many-body excitation spectrum whose nonanalytic changes define a sharp distinction between band and Mott insulators. The corresponding ``second-order transition'' can be observed out of equilibrium by driving a Cooper pair laser in the Mott insulator. We explicitly show that the gap for charged bosonic excitations lies below the threshold for Cooper pair breakup in any BEC regime, despite quantum fluctuations. Our discussion ends with a view of possible consequences for cuprates, where antinodal pair dynamics has certain features in common with our simple sss-wave picture.
The Cooper-pair size (i.e., 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 coherence length exhibits size-dependent drops by two-three orders of magnitude and reaches values found in high-TcT_cTc​ superconductors. This phenomenon is surprisingly similar to the well-known BCS-BEC crossover but with one important exception: it is driven by the transverse quantization of the electron spectrum rather than by the strength of the fermion-fermion interaction. Similar results can be expected for other systems with the same structure of the single-particle spectrum, e.g, for superfluid Fermi gases confined in a quantum-wire or quantum-well geometry.
We use a variational approach to gain insight into the strongly correlated ddd-wave superconducting state of the high Tc{T}_{c}Tc​ cuprates at T=0T=0T=0. We show that strong correlations lead to qualitatively different trends in pairing and phase coherence: the pairing scale decreases monotonically with hole doping while the superconducting order parameter shows a nonmonotonic dome. We obtain detailed results for the doping dependence of a large number of experimentally observable quantities, including the chemical potential, coherence length, momentum distribution, nodal quasiparticle weight and dispersion, incoherent features in photoemission spectra, optical spectral weight, and superfluid density. Most of our results are in remarkable quantitative agreement with existing data and some of our predictions, first reported in Phys. Rev. Lett. 87, 217002 (2001), have been recently verified.