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Journal of Mathematical Physics

Cooper pairing in one, two, and three dimensions

Carlos Esebbag, J. M. Getino, MANUEL DE LLANO, Steven A. Moszkowski, U. Oseguera, Angel Ricardo Plastino, H. Rubio

Published Mar 1, 1992• 10 citations
Publisher

Abstract

The Cooper pair eigenvalue equation is solved exactly for the free-electron density of states in one, two, and three dimensions. The two-dimensional (2-D) case is essentially the original derivation of Cooper, as is the three-dimensional case if the interaction is restricted to the vicinity of the Fermi surface. The essential singularity in coupling, familiar in 2-D quantum binding in a well, and sometimes erroneously attributed to the two-dimensional character of the BCS model interaction which is nonzero only very near the Fermi surface, appears in all three cases.

DOI
10.1063/1.529733

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Linear to quadratic crossover of Cooper pair dispersion relation ∗

Sadhan K. Adhikari, M. Casas, Antonio Puente +7 more

Cooper pairing is studied in three dimensions to determine its binding energy for all coupling using a general separable interfermion interaction. Also considered are Cooper pairs (CPs) with nonzero center-of-mass momentum (CMM). A coupling-independent linear term in the CMM dominates the pair excitation energy in weak coupling and/or high fermion density, while the more familiar quadratic term prevails only in the extreme low-density (i.e., vacuum) limit for any nonzero coupling. The linear-to-quadratic crossover of the CP dispersion relation is analyzed numerically, and is expected to play a central role in a model of superconductivity (and superfluidity) simultaneously accommodating a BCS condensate as well as a Bose-Einstein condensate of CP bosons. PACS #: 74.20.Fg; 64.90+b; 05.30.Fk; 05.30.Jp KEYWORDS: Cooper pairs; Bose-Einstein condensation; exotic superconductivity; superfluidity

•Jan 1, 2000•17 citations
The Cooper pair eigenvalue equation is solved exactly for the free-electron density of states in one, two, and three dimensions. The two-dimensional (2-D) case is essentially the original derivation of Cooper, as is the three-dimensional case if the interaction is restricted to the vicinity of the Fermi surface. The essential singularity in coupling, familiar in 2-D quantum binding in a well, and sometimes erroneously attributed to the two-dimensional character of the BCS model interaction which is nonzero only very near the Fermi surface, appears in all three cases.
The dynamical similarities between the three-dimensional (3D) electron fluid ``jellium'' model and the exactly soluble 1D fermion fluid with attractive \ensuremath{\delta}-function pairwise potentials motivates solving the Cooper-pair problem in this model for all couplings and/or densities. This is accomplished exactly in graphical form. For weak coupling, the essential singularity characteristic of standard 3D low-temperature superconductivity reemerges, and for infinite coupling, BCS theory reproduces the exact ground-state system energy.
Analytic expressions for the Bardeen–Cooper–Schrieffer (BCS) gap of a many-electron system with the BCS model interaction are obtained in one, two, and three dimensions in the weak coupling limit, but for arbitrary interaction width ν≡ℏωD/EF, 0<ν<∞. Here ℏωD is the maximum energy of a force-mediating boson, and EF is the Fermi energy (which is fixed by the electronic density). Our results address both phononic (ν≪1) as well as nonphononic (e.g., exciton, magnon, plasmon, etc.) pairing mechanisms where the mediating boson energies are not small compared with EF, provided weak electron–boson coupling prevails. The essential singularity in coupling, sometimes erroneously attributed to the two-dimensional character of the BCS model interaction with ν≪1, is shown to appear in one, two, and three dimensions before the limit ν→0 is taken.
Cooper pairing in two dimensions is analyzed with a set of renormalized equations to determine its binding energy for any fermion number density and all coupling assuming a generic pairwise residual interfermion interaction. Also considered are Cooper pairs (CP's) with nonzero center-of-mass momentum (CMM) and their binding energy is expanded analytically in powers of the CMM up to quadratic terms. A Fermi-sea-dependent linear term in the CMM dominates the pair excitation energy in weak coupling (also called the BCS regime) while the more familiar quadratic term prevails in strong coupling (the Bose regime). The crossover, though strictly unrelated to BCS theory per se, is studied numerically as it is expected to play a central role in a model of superconductivity as a Bose-Einstein condensation of CPs where the transition temperature vanishes for all dimensionality d< 2d<~2d< 2 for quadratic dispersion, but is nonzero for all d> 1d>~1d> 1 for linear dispersion.
Relation between the binding energy of a pair of electrons in Cooper's configuration and the magnitude of the attractive interaction between them is investigated theoretically for a two-dimensional electron gas when the electron pair has (a) zero and (b) finite center of mass momentum (CMM) first (i) in the background of a passive Fermi sea and (ii) with inclusion of the effect of Pauli exclusion principle in the presence of an active Fermi sea, at zero temperature. In the Cooper's approach (Cooper 1956 Phys. Rev. 104 1189) two pairing electrons are considered above the Fermi sea. Due to Pauli exclusion principle, these two electrons are forbidden to go into the states whose energies are less than the Fermi energy. Consequently, the role of the electrons which form the Fermi sea is passive in formation of electron pair (Fujita and Godoy 2001 Theory of High Temperature Superconductivity (Kluwer Academic Publishers)). However, in the case of an active Fermi sea, implying possible presence of multiple pairs of electrons coming out of the Fermi sea, the two electrons will have less available region in the momentum space for pair formation due to the Pauli blocking effect. In both the situations, threshold values of attractive interaction are found to exist to bind the electrons having finite CMM even with respect to the Fermi energy while with arbitrarily small value of attractive interaction they can be bound only in the case of zero CMM and that too in the presence of a passive Fermi sea. However, the presence of an active Fermi sea strongly opposes the pair formation having lower CMM. Furthermore, raising the range of the attractive interaction below and above the Fermi surface makes the pair formation more difficult due to enhanced Pauli blocking in this case. In the case of a passive Fermi sea however, with the increase in the range of the attractive interaction it becomes easier for the electron pairs to be in both Cooper pair state (CP) and true bound state (TB). These results are totally distinct from those obtained by Randeria et al (1989 Phys. Rev. Lett. 62 981, 1990 Phys. Rev. B 41 327) that a fermionic bound state in true vacuum is a necessary and sufficient condition for the Cooper pairing instability in the presence of a passive Fermi sea, independent of attractive interaction strength implying that for a potential which is attractive everywhere in two dimension, a two-body bound state exists for an arbitrarily weak attraction. Our calculations show, in the mean field approximation, that co-existence of TB state and CP state with various CMMs can occur only beyond a critical attractive interaction strength. This could lead to the formation of incoherent pairs even under strong coupling situation, which may hint onto the pseudo gap formation leading to some of the exotic features in the anomalous normal state seen in some of the exotic superconductors.
Cooper pairing is studied in three dimensions to determine its binding energy for all coupling using a general separable interfermion interaction. Also considered are Cooper pairs (CPs) with nonzero center-of-mass momentum (CMM). A coupling-independent linear term in the CMM dominates the pair excitation energy in weak coupling and/or high fermion density, while the more familiar quadratic term prevails only in the extreme low-density (i.e., vacuum) limit for any nonzero coupling. The linear-to-quadratic crossover of the CP dispersion relation is analyzed numerically, and is expected to play a central role in a model of superconductivity (and superfluidity) simultaneously accommodating a BCS condensate as well as a Bose-Einstein condensate of CP bosons. PACS #: 74.20.Fg; 64.90+b; 05.30.Fk; 05.30.Jp KEYWORDS: Cooper pairs; Bose-Einstein condensation; exotic superconductivity; superfluidity