Section 4: Gateways to a Theory of Superconductivity
Superconductivity—the ability of some metals at very low temperatures to carry electrical current without any appreciable resistance and to screen out external magnetic fields—is in many ways the poster child for the emergence of new states of quantum matter in the laboratory at very low temperatures. Gilles Holst, an assistant in the Leiden laboratory of the premier low-temperature physicist of his time, Kamerlingh Onnes, made the initial discovery of superconductivity in 1911. Although he did not share the Nobel Prize for its discovery with Kamerlingh Onnes, he went on to become the first director of the Phillips Laboratories in Eindhoven. But physicists did not understand the extraordinary properties of superconductors until 1957, when Nobel Laureate John Bardeen, his postdoctoral research associate Leon Cooper, and his graduate student Robert Schrieffer published their historic paper (known as "BCS") describing a microscopic theory of superconductivity.
Figure 13: Superconductors carry electrical current without resistance and are almost perfect diamagnets (a more fundamental aspect of their behavior), in that they can screen out external magnetic fields within a short distance known as the "penetration depth."
We now recognize the two gateways to the emergence of the superconducting state: an effective attractive interaction between electrons (the quasiparticles of Landau's Fermi liquid theory), whose energies put them close to their Fermi surface; and the condensation of pairs of these quasiparticles of opposite spin and momentum into a macroscopically occupied single quantum state, the superfluid condensate.
BCS theory explains the superfluidity of quantum fermionic matter. It applies to conventional superconductors in which phonons, the quantized vibrations of the lattice, serve as the pairing glue that makes possible an attractive quasiparticle interaction and those discovered subsequently, such as superfluid pairing phenomena in atomic nuclei, superfluid 3He, the cosmic superfluids of nuclear matter in the solid outer crust, and liquid interiors of rotating neutron stars. It also applies to the unconventional superconductors such as the cuprate, heavy electron, organic, and iron-based materials that take center stage for current work on superconductivity.
As we shall see, a remarkable feature of BCS theory is that, although it was based on an idealized model for quasiparticle behavior, it could explain all existing experiments and predict the results of many new ones. This occurs because the superconducting state is protected; its emergent behavior is independent of the details. As a result, a quite simple model that incorporates the "right stuff"—the gateways to superconducting behavior we noted above—can lead to a remarkably accurate description of its emergent behavior. In this section, we will trace the steps from 1950 to 1956 that led to the theory. The next section will outline the theory itself. And later in this unit, we will show how a simple extension of the BCS framework from the Standard Model considered in their original paper offers the prospect of explaining the properties of the unconventional superconductors at the center of current research on correlated electron matter.
Four decades of failed theories
In 1950, nearly 40 years after its discovery, the prospects for developing a microscopic theory of superconductivity still looked grim. Failed attempts to solve this outstanding physics challenge by the giants in the field, from Einstein, Bohr, Heisenberg, Bloch, and Landau to the young John Bardeen, led most theorists to look elsewhere for promising problems on which to work.
Figure 14: A photograph such as this of a levitating magnet is arguably the iconic image for superconductivity. It provides a vivid demonstration the way in which the near perfect diamagnetism of a superconducting material (the Meissner effect) makes it possible to levitate magnets above it.
Source: © Wikimedia Commons, GNU Free Documentation License, Version 1.2. Author: Mai-Linh Doan, 13 October 2007. More info
Despite that, experimentalists had made considerable progress on the properties of the superconducting state. They realized that a strong enough applied external magnetic field could destroy superconductivity and that a superconductor's almost perfect diamagnetism—its ability to shield out an external magnetic field within a short distance, known as the "penetration depth"—was key to an explanation. Theorists found that a two-fluid model analogous to that considered for superfluid 4He in the previous section could connect many experimental results. Moreover, London had argued eloquently that the perfect diamagnetism could be explained by the rigidity of the superfluid wavefunction in the presence of an external magnetic field, while progress occurred on the "phase transition" front as Vitaly Ginzburg and Landau showed how to extend Landau's general theory of second-order phase transitions to superconductors to achieve an improved phenomenological understanding of emergent superconducting behavior.
Superconductivity was obviously an amazing emergent electronic phenomenon, in which the transition to the superconducting state must involve a fundamental change in the ground and excited states of electron matter. But efforts to understand how an electron interaction could bring this about had come to a standstill. A key reason was that the otherwise successful nearly free electron model offered no clues to how an electron interaction that seemed barely able to affect normal state properties could turn some metals into superconductors.
A promising new path
Matters began to change in 1950 with the discovery of the isotope effect on the superconducting transition temperature, Tc, by Bernard Serin at Rutgers University, Emanuel Maxwell at the National Bureau of Standards, and their colleagues. They found that Tc for lead varied inversely as the square root of its isotopic mass. That indicated that quantized lattice vibrations must be playing a role in bringing about that transition, for their average energy was the only physical quantity displaying such a variation.
That discovery gave theorists a new route to follow. Herbert Fröhlich and Bardeen independently proposed theories in which superconductivity would arise through a change in the self-energy of individual electrons produced by the co-moving cloud of phonons that modify their mass. But, it soon became clear that efforts along these lines would not yield a satisfactory theory.
Frohlich then suggested in 1952 that perhaps the phonons played a role through their influence on the effective electron interaction. The problem with his proposal was that it was difficult to see how such an apparently weak phonon-induced interaction could play a more important role than the much stronger repulsive electrostatic interaction he had neglected. Two years later, Bardeen and his first postdoctoral research associate at the University of Illinois at Urbana-Champaign—myself—resolved that problem.
We did so by generalizing the collective coordinate formalism that David Bohm and I had developed for electron-electron interactions alone to derive their effective interaction between electrons when both the effects of the electrostatic interaction and the electron-phonon coupling are taken into account (Figure 15). Surprisingly, we found that, within the random phase approximation, the phonon-induced interaction could turn the net interaction between electrons lying within a characteristic phonon frequency of the Fermi surface from a screened repulsive interaction to an attractive one. We can imagine the phonon-induced interaction as the electronic equivalent of two children playing on a waterbed. One (the polarizer) makes a dent (a density wave) in the bed; this attracts the second child (the analyzer), so that the two wind up closer together.
Figure 15: The net effective interaction between electrons in a metal, with (a) depicting the net phonon-induced interaction and (b) their intrinsic effective screened electrostatic interaction.
Source: © David Pines. More info
Leon Cooper, who replaced me in Bardeen's research group in 1955, then studied the behavior of two electrons of opposite spin and momentum near the Fermi surface using a simplified version of the Bardeen-Pines attractive interaction. In a 1956 calculation that allowed for the multiple scattering of the pair above the Fermi surface, he showed that net attraction produced an energy gap in the form of a bound state for the electron pair.
During this period, in work completed just before he went to Stockholm to accept his 1956 Nobel Prize, Bardeen had showed that many of the key experiments on superconductivity could be explained if the "normal" elementary excitations of the two-fluid model were separated from the ground state by an energy gap. So the gap that Cooper found was intriguing. But there was no obvious way to go from a single bound pair to London's coherent ground state wavefunction that would be rigid against magnetic fields. The field awaited a breakthrough.