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CRC Press eBooks

Applications of the Pairing Theory

J. R. Schrieffer

Published Mar 5, 2018• 0 citations
Publisher

Abstract

This chapter reviews the calculation of a number of system properties within the pairing approximation. Since the BCS theory was originally proposed, attempts to justify the pairing correlations basic to the theory have proceeded along two lines. The first approach has been to apply the BCS theory to a wide variety of phenomena in superconductors and check the theoretical predictions of the pairing approximation against experiment. The second approach has been to treat by various approximate methods the residual interactions neglected within the pairing scheme, hoping to show that these residual interactions introduce no major change in the predicted properties of the system. The interaction Hamiltonian, being a one-body operator, can at most change the state of one electron. While the majority of the electron–phonon interaction has been accounted for in forming the wave functions for the normal and superconducting states, there remains the part corresponding to resonant phonon absorption and emission processes.

DOI
10.1201/9780429495700-3

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This chapter reviews the calculation of a number of system properties within the pairing approximation. Since the BCS theory was originally proposed, attempts to justify the pairing correlations basic to the theory have proceeded along two lines. The first approach has been to apply the BCS theory to a wide variety of phenomena in superconductors and check the theoretical predictions of the pairing approximation against experiment. The second approach has been to treat by various approximate methods the residual interactions neglected within the pairing scheme, hoping to show that these residual interactions introduce no major change in the predicted properties of the system. The interaction Hamiltonian, being a one-body operator, can at most change the state of one electron. While the majority of the electron–phonon interaction has been accounted for in forming the wave functions for the normal and superconducting states, there remains the part corresponding to resonant phonon absorption and emission processes.
Motivation With few exceptions [131] it is widely accepted that the conventional Bardeen–Cooper–Schrieffer (BCS) theory [12] and its intermediate-coupling Eliashberg extension [14] do not suffice to explain high-temperature superconductivity. As shown in the preceding chapters the true origin of high-temperature superconductivity could be found in a proper combination of the bare Coulomb repulsion with the bare electron–phonon interaction in the framework of the many-body polaron theory [51] beyond the BCS–Eliashberg approximation. Nevertheless some researchers have maintained that the repulsive electron–electron interaction alone should provide pairing at high temperatures without phonons despite substantial experimental evidence that the EPI is an important player and needs to be included in the successful theory of high- T c superconductivity. Following the original proposal by P. W. Anderson, many authors [22, 74] argued that the electron–electron interaction in high-temperature superconductors was strong but repulsive providing high T c without phonons via superexchange and/or spin fluctuations in the d -wave pairing channel ( l = 2). A motivation for this concept can be found in the earlier works by Kohn and Luttinger (KL) [132, 133] who showed that the Cooper pairing of fermions with any weak repulsion was possible since the two-particle interaction induced by many-body effects is attractive for pairs with large orbital momenta, l ≫ 1.
Abstract One of the important steps leading to the successful formulation of the microscopic theory of superconductivity was the model calculation performed by Cooper (1956) which showed that two electrons, interacting above a filled Fermi sea, had a bound state in the presence of an arbitrarily weak attractive interaction. This chapter first examines Cooper’s calculation. The success of the Cooper model calculation in producing a bound state, with its implication that a proper many-body theory would contain a gap in the excitation spectrum, strongly suggests that the ground state wave function should be constructed from pairs of electrons. However, it is difficult to perform calculations using the many-body pair wave function from Cooper’s model. The chapter then introduces the wave function proposed by Bardeen, Cooper, and Schrieffer (BCS) as an alternative.
This chapter introduces the methods of quantum field theory as it appies to many-particle systems, particularly to superconductors. It focuses the ideas underlying the Green's function method, that is useful in the study of superconductivity. In dealing with an electron-phonon system it is reasonably realistic to approximate the effects of the phonons by an effective interaction between electrons that is of short range in space and retarded in time. It is especially appropriate to study the Green's function method in connection with the theory of superconductivity, because there are two reasons for this. The superconductor is the first system in solid-state physics in which the interaction between electrons qualitatively changes the spectrum of excitations. The other reason is the interaction that causes superconductivity is not describable by an ordinary potential: An electronic charge fluctuation with a frequency just below a characteristic frequency of the ionic lattice excites in the lattice a resonant sympathetic vibration that overcompensates for the electronic charge.
BCS superconductivity is explained by a simple Hamiltonian describing an attractive pairing interaction between pairs of electrons. The Hamiltonian may be treated using a mean-field method, which is adequate to study equilibrium properties and a variety of nonequilibrium effects. Nevertheless, in certain nonequilibrium situations, even in a macroscopic rather than a microscopic superconductor, the application of mean-field theory may not be valid. In such cases, one may resort to the full solution of the Hamiltonian, as given by Richardson in the 1960s. The relevance of Richardson's solution to macroscopic nonequilibrium superconductors was pointed out recently based on the existence of quantum instabilities out of equilibrium. It is then of interest to obtain analytical expressions for expectation values between exact eigenvalues of the pairing Hamiltonian within the Richardson approach for macroscopic systems. We undertake this task in the current paper. It should be noted that Richardson's approach yields the full set of eigenvalues of the Hamiltonian, while BCS theory yields only a subset. The results obtained here, then, generalize the familiar BCS expressions (e.g., for the electron occupation or pairing correlations) to cases where the spectrum of excitations diverges from BCS theory (e.g., in cases where the spectrum exhibits multiple gaps).
A strongly coupled many-body system (a system of fermions confined in a shell surrounding the Fermi surface for which the variation in the single-particle kinetic energy is neglected) is defined. Some general properties of this system are obtained, in particular a criterion for permissible approximations. It is shown that a set of diagrams in which an electron of momentum $\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ and of spin \ensuremath{\downarrow} interacts with many electrons of momentum $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}+\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ and spin \ensuremath{\uparrow}, where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ does not grow too large, can be consistently extracted from the totality of graphs generated by this strongly coupled Hamiltonian. When analyzed, these lead to a ground-state and single-particle spectrum displaying the important qualitative features of a BCS superconductor.
A nonperturbative argument is used to derive a one-electron Hamiltonian whose eigenfunctions are in one-to-one correspondence with electron or hole quasiparticles of an N-electron system. Nonlocal exchange is included exactly, and self-energy terms due to electronic correlation are included at the level of approximation of a given N-electron wave function. When parametrized by quasiparticle occupation numbers, this one-electron Hamiltonian is the functional derivative of an exact N-electron energy functional, as in the Landau theory of interacting fermion systems. The present nonperturbative derivation extends the Landau theory to ordered or localized systems for which many-body perturbation theory is not directly applicable. Analysis of the Bardeen-Cooper-Schrieffer (BCS) reduced Hamiltonian for superconductors leads to a temperature-dependent energy gap and a transition temperature in good agreement with the BCS theory, in a formalism that conserves the number of electrons.