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Progress of Theoretical Physics

Structure of Cooper Pairs in Superconductors

Yasushi Wada

Published Jul 1, 1972• 0 citations
Publisher

Abstract

In the generalized random-phase approximation, introduced by Anderson and applied to the theory of superconductivity, existence of a collective excitation with vanishing excitation energy and momentum was known. It is shown to correspond to the creation of a Cooper pair. The creation operator satisfies a nonlinear operator equation which does not depend on the magnitude of order parameter. It is, therefore, independent of dynamics, that is, of the strength of the pairing interactions. Nevertheless it has the assumed matrix elements between a given state in an N-particle system and the corresponding state in the N+2 particle system. Discussions are confined to the BCS superconductor at absolute zero temperature.

DOI
10.1143/ptp.48.19

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In the generalized random-phase approximation, introduced by Anderson and applied to the theory of superconductivity, existence of a collective excitation with vanishing excitation energy and momentum was known. It is shown to correspond to the creation of a Cooper pair. The creation operator satisfies a nonlinear operator equation which does not depend on the magnitude of order parameter. It is, therefore, independent of dynamics, that is, of the strength of the pairing interactions. Nevertheless it has the assumed matrix elements between a given state in an N-particle system and the corresponding state in the N+2 particle system. Discussions are confined to the BCS superconductor at absolute zero temperature.
We demonstrate the existence of a collective excitation branch in the pair-breaking continuum of superfluid Fermi gases and BCS superconductors. At zero temperature, we analytically continue the equation on the collective mode energy in Anderson's Random Phase Approximation or Gaussian fluctuations through its branch cut associated with the continuum, and obtain the full complex dispersion relation, including in the strong coupling regime. The branch exists as long as the chemical potential μ is positive and the wave number below sqrt[2mμ]/ℏ (with m the fermion mass). In the long wavelength limit, the branch varies quadratically with the wave number, with a complex effective mass that we compute analytically for an arbitrary interaction strength.
A generalization of the random-phase approximation of the theory of Coulomb correlation energy is applied to the theory of superconductivity. With no further approximations it is shown that most of the elementary excitations have the Bardeen-Cooper-Schrieffer energy gap spectrum, but that there are collective excitations also. The most important of these are the longitudinal waves which have a velocity ${v}_{F}{{\frac{1}{3}[1\ensuremath{-}4N(0)|V|]}}^{\frac{1}{2}}$ in the neutral Fermi gas, and are essentially unperturbed plasma oscillations in the charged case. Other collective excitations resembling higher bound pair states may or may not exist but do not seriously affect the energy gap. The theory obeys the sum rules and is gauge invariant to an adequate degree throughout.
We study the collective excitations of a system with local attractive interactions as a function of the coupling strength. The two-particle excitations are obtained in the random-phase approximation. For the strong-coupling limit we reproduce the excitation spectrum of a hard-core Bose gas, while in the weak-coupling limit we obtained a spectrum consistent with the BCS picture. The evolution of the excitation spectrum from the weak- to the strong-coupling limits is continuous. We briefly discuss the effect of these excitations in the thermodynamic properties of the system.
In the BCS theory of superconductivity, one assumes that all Cooper pairs have the same center-of-mass momentum. This is indeed enforced by self-consistency if the pairing interaction is momentum independent. Here, we show that for an attractive nearest-neighbor interaction, this is different. In this case, stable solutions with pairs with momenta q\mathbf{q}q and $\ensuremath{-}\mathbf{q}$ coexist and, for a sufficiently strong interaction, one of these states becomes the ground state of the superconductor. The possibility for a finite-momentum pairing state emerges only for nodal superconductors and is accompanied by a charge order with wave vector 2q2\mathbf{q}2q. For a weak pairing interaction, the ground state is a ddd-wave superconductor.
The dynamics of BCS (Bardeen-Cooper-Schrieffer) superconductors is fairly well understood due to the availability of a mean-field solution for the pairing Hamiltonian, a solution which gives the quantum state of a superconductor as a state of almost-free fermions interacting only with a condensate. As a result, transition probabilities may be computed, and expressed in terms of matrix elements of electron creation and annihilation operators between approximate eigenstates. These matrix elements are also called ``coherence factors.'' Mean-field theory is however not sufficient to describe all eigenstates of a superconductor, a deficiency which is hardly important in (or very close to) equilibrium, but one that becomes relevant in certain out-of-equilibrium situations. We report here on a computation of matrix elements (coherence factors) for the pairing Hamiltonian between any ``two-arc'' eigenstates in the thermodynamic limit.
We study the thermodynamics of a superconductor with strongly bound Cooper pairs by solving the self-consistent T-matrix equations for the attractive Hubbard model within a local approximation. We determine the transition temperature as a function of the interaction strength at quarter filling, and compute the temperature dependence of the superconducting order parameter, the specific heat, the spin susceptibility, and the density of local pairs in the normal and superconducting phase. Pronounced deviations from BCS behavior are found already for moderate interactions, where superconductivity sets in at a critical temperature above the coherence temperature for Fermi liquid behavior.