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Oxford University Press eBooks

The Cooper Pair Problem

Monique Combescot, Shiue-Yuan Shiau

Published Dec 1, 2015• 0 citations
Publisher

Abstract

Abstract Chapter 6 introduces the BCS theory of superconductivity, proposed in 1957 by Bardeen, Cooper, and Schrieffer and which is based on the idea that electrons form a coherent state of electron pairs with opposite spins and opposite momenta. It then briefly explains how, along the Fröhlich idea, the electron-phonon interaction can lead to the ion-mediated attractive potential for Cooper pairing. Next, it introduces the reduced BCS potential, which acts between electrons having opposite spins and opposite momenta. Finally, this chapter discusses the drastic consequence of the reduced BCS potential on Cooper pair interaction and explains why Cooper pairs can strongly overlap without dissociating.

DOI
10.1093/acprof:oso/9780198753735.003.0006

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Science•Apr 8, 2021•0 citations
Abstract Chapter 6 introduces the BCS theory of superconductivity, proposed in 1957 by Bardeen, Cooper, and Schrieffer and which is based on the idea that electrons form a coherent state of electron pairs with opposite spins and opposite momenta. It then briefly explains how, along the Fröhlich idea, the electron-phonon interaction can lead to the ion-mediated attractive potential for Cooper pairing. Next, it introduces the reduced BCS potential, which acts between electrons having opposite spins and opposite momenta. Finally, this chapter discusses the drastic consequence of the reduced BCS potential on Cooper pair interaction and explains why Cooper pairs can strongly overlap without dissociating.
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.
The crucial demonstration that superfluidity was linked to Bose particles and Bose-Einstein condensation came after the discovery of He3 in 1949, which failed to show the characteristic A transition within a reasonable wide temperature interval around the critical temperature for the onset of superfluidity in He4. The idea of the bosonization of electrons due to their pairing, introduced by Ogg (1946) became very attractive as an explanation for superconductivity after Schafroth (1955) showed that a gas of charged bosons with charge 2e would indeed show a Meissner effect, which is one of the most crucial features of superconductors (see Chapter 1). However this phenomenological Bose-gas picture was condemned and later on practically forgotten because it was unable to account quantitatively for the critical parameters of the classical (low-temperature) superconductors. Using for Tc the Bose-Einstein condensation temperature Eq.(l.S) one obtains the utterly meaningless result Tc = lOOOO.R&s; with the atomic density of electron pairs ~ 1022cm~3 and with the effective mass m** = 2me. The failure of the Schafroth picture stems from a very large size £ of pairs in classical superconductors, which can be estimated from the uncertainty principle: The uncertainty of the momentum 8p is estimated from the uncertainty of the kinetic energy 8E ~ vpSp, which should be of the order of Tc, the only characteristic energy of the superconducting state. Therefore and turns out to be more than one micron for simple metals with Tc ~ IK and the Fermi velocity VF — I0scm/s. That means that pairs in classical superconductors overlap strongly and the Shafroth model of real space pairs is wrong. The finally successful theory for classical superconductors was formulated by Bardeen, Cooper and Schrieffer (1957). In this chapter we present the main points of the BCS theory, using the Bogoliubov transformation (1947,1958) for the weak-coupling regime and the Green&s;s function technique for the intermediate coupling in Chapter 3. More details will be found in the excellent books by Schrieffer(1965), De Gennes (1989) and Abrikosov, Gor&s;kov and Dzyaloshinskii (1963).
We review efforts to unify both the Bardeen, Cooper and Schrieffer (BCS) and Bose-Einstein condensation (BEC) pictures of superconductivity. We have finally achieved this in terms of a "\textit{complete} boson-fermion (BF) model" (CBFM) that reduces in special cases to all the main continuum (as opposed to "spin") statistical theories of superconductivity. Our BF model is "complete" in the sense that not only two-electron (2e) but also two-hole (2h) Cooper pairs (CPs) are allowed in arbitrary proportions. In contrast, BCS-Bogoliubov theory--which can also be considered as the theory of a mixture of kinematically independent electrons, 2e- and 2h-CPs--allows only equal, 50%-50%, mixtures of the two kinds of CPs. This is obvious from the perfect symmetry about μμμ, the electron chemical potential, of the well-known Bogoliubov v2(ε)v^{2}(ε)v2(ε) and u2(ε)u^{2}(ε)u2(ε) coefficients, where εεε is the electron energy. The CBFM is then applied to see: a) whether the BCS model interaction for the electron-phonon dynamical mechanism is sufficient to predict the unusually high values of TcT_{c}Tc​ (in units of the Fermi temperature) of ≃0.01−0.1\simeq 0.01-0.1≃0.01−0.1 exhibited by the so-called ``exotic'' superconductors \cite{Brandow} in both 2D and 3D--relative to the low values of ≲10−3\lesssim 10^{-3}≲10−3 more or less correctly predicted by BCS theory for conventional, elemental superconductors; and b) whether it can at least suggest, if not explain, why "hole superconductors" have higher TcT_{c}Tc​'s.
Superfluidity is one of the most striking manifestations of quantum mechanics at macroscopic scales, characterized by frictionless flow and the emergence of collective quantum phenomena. The theoretical foundation of superfluidity in fermionic systems was revolutionized by the Bardeen-Cooper-Schrieffer (BCS) theory, developed in 1957 to explain superconductivity in metals. This groundbreaking work provided a microscopic description of how attractive interactions between electrons near the Fermi surface can lead to the formation of Cooper pairs-correlated pairs of fermions that condense into a coherent quantum state. The BCS theory not only explained superconductivity but also laid the groundwork for understanding superfluidity in systems like liquid helium-3 and ultracold fermionic gases, becoming a cornerstone of condensed matter physics.
Superconductivity In conventional superconductors, the electron pairs responsible for superconductivity are large and overlapping. Starting from this so-called Bardeen-Cooper-Schrieffer (BCS) limit, increasing interactions can set the system on a path of crossover to the opposite limit of small, tightly bound electron pairs that undergo Bose-Einstein condensation (BEC). Nakagawa et al. intercalated lithium ions into the insulating material zirconium nitride chloride, varying the carrier density across a large range (see the Perspective by Randeria). This induced superconductivity and enabled the system to enter the crossover regime between the BCS and BEC limits. Science , this issue p. [190][1]; see also p. [132][2] [1]: /lookup/doi/10.1126/science.abb9860 [2]: /lookup/doi/10.1126/science.abg1752