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Physical review. B, Condensed matter

Superconductivity in a two-dimensional Fermi gas: Evolution from Cooper pairing to Bose condensation

Mohit Randeria, Ji-Min Duan, Lih-Yir Shieh

Published Jan 1, 1990• 367 citations
Publisher

Abstract

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 [${k}_{F}$${\ensuremath{\xi}}_{0}$\ensuremath{\sim}O(1)], a striking feature of the high-${T}_{c}$ materials. We find that a two-body bound state in vacuum is a necessary and sufficient condition for an s-wave pairing instability. We also find that the existence of such a bound state in a higher-angular-momentum (l>0) channel is not a necessary condition for an l-wave pairing instability. We further investigate using a variational ansatz the evolution from a state with large overlapping Cooper pairs (${k}_{F}$${\ensuremath{\xi}}_{0}$\ensuremath{\gg}1) to one with Bose condensation of composite bosons (${k}_{F}$${\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.

DOI
10.1103/physrevb.41.327

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Bound states, Cooper pairing, and Bose condensation in two dimensions

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For a dilute gas of fermions interacting via an arbitrary pair potential in d=2 dimensions, we show that the many-body ground state is unstable to s-wave pairing at T=0 if and only if a two-body bound state exists. We further obtain, within a variational pairing Ansatz, a smooth crossover from a Cooper-paired state (${\ensuremath{\xi}}_{0}$${\mathrm{k}}_{\mathrm{f}}$\ensuremath{\gg}1) to a Bose condensed state of tightly bound pairs (${\ensuremath{\xi}}_{0}$${\mathrm{k}}_{\mathrm{f}}$\ensuremath{\ll}1). We briefly discuss non--s-wave superconductors. Insofar as this model is appli- cable to the high-${\mathrm{T}}_{\mathrm{c}}$ materials, they are in the interesting regime with the coherence length ${\ensuremath{\xi}}_{0}$ comparable to the interparticle spacing ${\mathrm{K}}_{\mathrm{f}}^{\mathrm{\ensuremath{-}}1}$.

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BCS-to-BEC Quantum Phase Transition in High-Tc Superconductors and Fermionic Atomic Gases: A Functional Integral Approach

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The problem of the evolution from BCS theory with cooperative Cooper pairing to the formation and condensation of composite bosons has attracted considerable attention for the past several decades. It has gained renewed impetus in the mid-eighties with the discovery of the high-Tc superconductors, which have a coherence length comparable to the interparticle spacing. More recently, this subject has spurred a great deal of research activity in connection with experiments involving dilute atomic gases of fermionic atoms. The initial objective of this work will be to use functional integral techniques to analyze the low-temperature BCS-to-BEC evolution of d-wave superconductors within the saddle point (mean field) approximation for a continuum model. Then, the same mathematical formalism will be applied to the problem of the BCS-to-BEC evolution of fully spin-polarized p-wave Fermi gases in two dimensions. We find that a quantum phase transition occurs for both systems as they are driven from the BCS-like regime of weakly interacting fermionic pairs to the opposite BEC-like regime of strongly interacting bosonic molecules. This is in contrast to the smooth crossover predicted and observed in systems that exhibit s-wave pairing symmetry. We calculate several spectroscopic and thermodynamic properties that signal the occurrence of this phase transition, and suggest some possible experimental realizations. Finally, fluctuations about the saddle point solution are included in the calculations, and the effects of such correction are analyzed in the low (T~0) and high (T~Tc) temperature limits. We conclude that, at high temperatures, the bosonic degrees of freedom that arise from two-particle bound states become essential to describe the strong coupling limit,\nas the saddle point approximation alone becomes unreliable.

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Collective excitations in the ground state of a two-dimensional attractive Fermi gas

Sergei V. Traven

The spectrum of collective (pair) excitations in the ground state of a dilute twodimensional (2D) attractive Fermi-gas is studied within the functional integral formalism. The linearized equations for the fluctuations about the non-trivial saddle point are analyzed for all coupling regimes, which are characterized by the ratio ε 0 /ε F , where ε 0 is the two-fermion binding energy and ε F is the Fermi energy. The approximation takes into account propagation of the fluctuations and their interaction with the condensate. In the strong-coupling, or ‘;Bosegas’, regime (ε 0 /ε F > 1) the spectrum is continuous and has the Bogolubov form, but in the weak-coupling limit (ε 0 /ε F «1) there are two types of excitations (different from the two-fermion scattering states): (i) long-wavelength sound-like excitations with the cut-off at momentum q c ≃ l/ξ 0 (where ξ 0 is the Cooper pair size), and (ii) pair excitations with q ≃ (8 m μ) ½ , where μ is the chemical potential and m is the fermion mass. The crossover between weakand strong-coupling behavior of the excitation spectrum is found to occur at the value of the coupling parameter ε 0 /ε F ≃¼. Introduction Since the discovery of high- T c superconductivity there has been growing interest in studying 2D Fermi gases with attractive interaction, especially in the crossover regime, when the pair size ξ 0 is of the order of the interparticle distance. The importance of such a model, which can be regarded as a semiphenomenological model of a 2D superconductor, is highlighted by experimental evidence that the high- T c superconductors, most of which have a layered structure, have a short coherence length, so that k F ξ 0 ≃1, where k F is the Fermi momentum.

Cambridge University Press eBooks•Sep 7, 1995•0 citations

Coherence lengths for three-dimensional superconductors in the BCS-Bose picture

R. M. Carter, M. Casas, J. M. Getino +4 more

Following an approach similar to that of Miyake or Randeria, Duan, and Shieh in two dimensions, we study a three-dimensional many-fermion gas at zero temperature interacting via some short-ranged two-body potential. To accommodate a possible singularity (e.g., the Coulomb repulsion) in the interaction, the potential is eliminated in favor of the two-body scattering t-matrix, the low-energy form of which is expressible in terms of the s-wave scattering length ${\mathit{a}}_{\mathit{s}}$. The BCS gap equation for s-wave pairing is then solved simultaneously with the number equation in order to self-consistently obtain the zero-temperature BCS gap \ensuremath{\Delta} as well as the chemical potential \ensuremath{\mu} as functions of the dimensionless coupling variable \ensuremath{\lambda}\ensuremath{\equiv}${\mathit{k}}_{\mathit{Fa}\mathit{s}}$, where ${\mathit{k}}_{\mathit{F}}$ is the Fermi momentum. Results are valid for arbitrary coupling strength, and in the weak coupling limit reproduce the standard BCS results. Finally, root-mean-square pair sizes are obtained as a function of \ensuremath{\lambda} and compared with experimental values.

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Cooper-pair and Bose-Einstein condensations in two dimensions: A critical analysis based on the Nozières and Schmitt-Rink formalism

Akio Tokumitu, Kazumasa Miyake, Kazuo Yamada

The crossover between the Cooper-pair condensation and the Bose-Einstein condensation of ``di-electronic'' molecules in two-dimensional superconductors is investigated in detail on the basis of the Nozi\`eres and Schmitt-Rink formalism. It is shown that temperature dependence of the chemical potential \ensuremath{\mu} so calculated is classified into two classes as decreasing temperatures; i.e., class (a) where \ensuremath{\mu} approaches the point of Bose-Einstein condensation of two-dimensional ideal Bose gas of ``di-electronic'' molecules, and class (b) where \ensuremath{\mu} diverges positively along the line of BCS-type mean-field pair condensation. This feature is rather universal irrespective of strength V of the attractive interaction of the s-wave type. While the former class (a) has been found by Schmitt-Rink, Varma, and Ruckenstein, existence of the latter class (b) is recognized here. In the case where V is fixed, class (a) is realized for electron number density N smaller than ${\mathit{N}}_{\mathrm{cr}}$, which is an increasing function of V, and class (b) is realized for N larger than ${\mathit{N}}_{\mathrm{cr}}$. If N\ensuremath{\gg}${\mathit{N}}_{\mathrm{cr}}$ in particular, there exists a regime, where the Fermi-liquid-like description is valid, between the BCS-type mean-field transition temperature and the Fermi temperature. In the situation where V is changed with N being fixed, low-temperature states for the strong-coupling case belong to class (a) while those for the weak-coupling case belong to class (b). Therefore, with decreasing V, the chemical potential \ensuremath{\mu}(T), at temperatures far below the Fermi temperature, shows a discontinuous jump at V=${\mathit{V}}_{\mathrm{cr}}$(N) corresponding to the transition from class (a) to (b). However, this is in contradiction to a physical picture that the chemical potential should smoothly cross over between the above two limits unless the liquid-gas transition occurs. This shows in turn a necessity of improving the Nozi\`eres and Schmitt-Rink formalism itself especially in two dimensions. A preliminary approach beyond their formalism is briefly discussed.

Physical review. B, Condensed matter•May 1, 1993•43 citations

Hopping anisotropies: A candidate for BCS-BEC crossover

Poulumi Dey, Saurabh Basu

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.

Physical Review B•May 15, 2007•1 citation
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 [{k}_{F}$${\ensuremath{\xi}}_{0}\ensuremath{\sim}O(1)], a striking feature of the high-Tc{T}_{c}Tc​ materials. We find that a two-body bound state in vacuum is a necessary and sufficient condition for an s-wave pairing instability. We also find that the existence of such a bound state in a higher-angular-momentum (l>0) channel is not a necessary condition for an l-wave pairing instability. We further investigate using a variational ansatz the evolution from a state with large overlapping Cooper pairs ({k}_{F}$${\ensuremath{\xi}}_{0}\ensuremath{\gg}1) to one with Bose condensation of composite bosons ({k}_{F}$${\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 a dilute gas of fermions interacting via an arbitrary pair potential in d=2 dimensions, we show that the many-body ground state is unstable to s-wave pairing at T=0 if and only if a two-body bound state exists. We further obtain, within a variational pairing Ansatz, a smooth crossover from a Cooper-paired state (${\ensuremath{\xi}}_{0}$kf{\mathrm{k}}_{\mathrm{f}}kf​\ensuremath{\gg}1) to a Bose condensed state of tightly bound pairs (${\ensuremath{\xi}}_{0}$kf{\mathrm{k}}_{\mathrm{f}}kf​\ensuremath{\ll}1). We briefly discuss non--s-wave superconductors. Insofar as this model is appli- cable to the high-Tc{\mathrm{T}}_{\mathrm{c}}Tc​ materials, they are in the interesting regime with the coherence length ${\ensuremath{\xi}}_{0}$ comparable to the interparticle spacing ${\mathrm{K}}_{\mathrm{f}}^{\mathrm{\ensuremath{-}}1}$.
The problem of the evolution from BCS theory with cooperative Cooper pairing to the formation and condensation of composite bosons has attracted considerable attention for the past several decades. It has gained renewed impetus in the mid-eighties with the discovery of the high-Tc superconductors, which have a coherence length comparable to the interparticle spacing. More recently, this subject has spurred a great deal of research activity in connection with experiments involving dilute atomic gases of fermionic atoms. The initial objective of this work will be to use functional integral techniques to analyze the low-temperature BCS-to-BEC evolution of d-wave superconductors within the saddle point (mean field) approximation for a continuum model. Then, the same mathematical formalism will be applied to the problem of the BCS-to-BEC evolution of fully spin-polarized p-wave Fermi gases in two dimensions. We find that a quantum phase transition occurs for both systems as they are driven from the BCS-like regime of weakly interacting fermionic pairs to the opposite BEC-like regime of strongly interacting bosonic molecules. This is in contrast to the smooth crossover predicted and observed in systems that exhibit s-wave pairing symmetry. We calculate several spectroscopic and thermodynamic properties that signal the occurrence of this phase transition, and suggest some possible experimental realizations. Finally, fluctuations about the saddle point solution are included in the calculations, and the effects of such correction are analyzed in the low (T~0) and high (T~Tc) temperature limits. We conclude that, at high temperatures, the bosonic degrees of freedom that arise from two-particle bound states become essential to describe the strong coupling limit,\nas the saddle point approximation alone becomes unreliable.
The spectrum of collective (pair) excitations in the ground state of a dilute twodimensional (2D) attractive Fermi-gas is studied within the functional integral formalism. The linearized equations for the fluctuations about the non-trivial saddle point are analyzed for all coupling regimes, which are characterized by the ratio ε 0 /ε F , where ε 0 is the two-fermion binding energy and ε F is the Fermi energy. The approximation takes into account propagation of the fluctuations and their interaction with the condensate. In the strong-coupling, or ‘;Bosegas’, regime (ε 0 /ε F > 1) the spectrum is continuous and has the Bogolubov form, but in the weak-coupling limit (ε 0 /ε F «1) there are two types of excitations (different from the two-fermion scattering states): (i) long-wavelength sound-like excitations with the cut-off at momentum q c ≃ l/ξ 0 (where ξ 0 is the Cooper pair size), and (ii) pair excitations with q ≃ (8 m μ) ½ , where μ is the chemical potential and m is the fermion mass. The crossover between weakand strong-coupling behavior of the excitation spectrum is found to occur at the value of the coupling parameter ε 0 /ε F ≃¼. Introduction Since the discovery of high- T c superconductivity there has been growing interest in studying 2D Fermi gases with attractive interaction, especially in the crossover regime, when the pair size ξ 0 is of the order of the interparticle distance. The importance of such a model, which can be regarded as a semiphenomenological model of a 2D superconductor, is highlighted by experimental evidence that the high- T c superconductors, most of which have a layered structure, have a short coherence length, so that k F ξ 0 ≃1, where k F is the Fermi momentum.
Following an approach similar to that of Miyake or Randeria, Duan, and Shieh in two dimensions, we study a three-dimensional many-fermion gas at zero temperature interacting via some short-ranged two-body potential. To accommodate a possible singularity (e.g., the Coulomb repulsion) in the interaction, the potential is eliminated in favor of the two-body scattering t-matrix, the low-energy form of which is expressible in terms of the s-wave scattering length as{\mathit{a}}_{\mathit{s}}as​. The BCS gap equation for s-wave pairing is then solved simultaneously with the number equation in order to self-consistently obtain the zero-temperature BCS gap \ensuremath{\Delta} as well as the chemical potential \ensuremath{\mu} as functions of the dimensionless coupling variable \ensuremath{\lambda}\ensuremath{\equiv}kFas{\mathit{k}}_{\mathit{Fa}\mathit{s}}kFas​, where kF{\mathit{k}}_{\mathit{F}}kF​ is the Fermi momentum. Results are valid for arbitrary coupling strength, and in the weak coupling limit reproduce the standard BCS results. Finally, root-mean-square pair sizes are obtained as a function of \ensuremath{\lambda} and compared with experimental values.
The crossover between the Cooper-pair condensation and the Bose-Einstein condensation of ``di-electronic'' molecules in two-dimensional superconductors is investigated in detail on the basis of the Nozi\`eres and Schmitt-Rink formalism. It is shown that temperature dependence of the chemical potential \ensuremath{\mu} so calculated is classified into two classes as decreasing temperatures; i.e., class (a) where \ensuremath{\mu} approaches the point of Bose-Einstein condensation of two-dimensional ideal Bose gas of ``di-electronic'' molecules, and class (b) where \ensuremath{\mu} diverges positively along the line of BCS-type mean-field pair condensation. This feature is rather universal irrespective of strength V of the attractive interaction of the s-wave type. While the former class (a) has been found by Schmitt-Rink, Varma, and Ruckenstein, existence of the latter class (b) is recognized here. In the case where V is fixed, class (a) is realized for electron number density N smaller than Ncr{\mathit{N}}_{\mathrm{cr}}Ncr​, which is an increasing function of V, and class (b) is realized for N larger than Ncr{\mathit{N}}_{\mathrm{cr}}Ncr​. If N\ensuremath{\gg}Ncr{\mathit{N}}_{\mathrm{cr}}Ncr​ in particular, there exists a regime, where the Fermi-liquid-like description is valid, between the BCS-type mean-field transition temperature and the Fermi temperature. In the situation where V is changed with N being fixed, low-temperature states for the strong-coupling case belong to class (a) while those for the weak-coupling case belong to class (b). Therefore, with decreasing V, the chemical potential \ensuremath{\mu}(T), at temperatures far below the Fermi temperature, shows a discontinuous jump at V=Vcr{\mathit{V}}_{\mathrm{cr}}Vcr​(N) corresponding to the transition from class (a) to (b). However, this is in contradiction to a physical picture that the chemical potential should smoothly cross over between the above two limits unless the liquid-gas transition occurs. This shows in turn a necessity of improving the Nozi\`eres and Schmitt-Rink formalism itself especially in two dimensions. A preliminary approach beyond their formalism is briefly discussed.
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.