Section 5: The BCS Theory

The breakthrough came in January 1957, when Bardeen's graduate student, Robert Schrieffer, while riding a New York City subway train following a conference in Hoboken, NJ on The Many-Body Problem, wrote down a candidate wavefunction for the ground state and began to calculate its low-lying excited states. He based his wavefunction on the idea that the superconducting condensate consists of pairs of quasiparticles of opposite spin and momenta. This gateway to emergent superconducting behavior is a quite remarkable coherent state of matter; because the pairs in the condensate are not physically located close to one another, their condensation is not the Bose condensation of pairs that preform above the superconducting transition temperature in the normal state. Instead, the pairs condense only below the superconducting transition temperature. The typical distance between them, called the "coherence length," is some hundreds of times larger than the typical spacing between particles.

To visualize this condensate and its motion, imagine a dance floor in which one part is filled with couples (the pairs of opposite spin and momentum) who, before the music starts (that is, in the absence of an external electric field), are physically far apart (Figure 16). Instead of being distributed at random, each member of the couple is connected, as if by an invisible string, to his or her partner faraway. When the music begins (an electric field is applied), each couple responds by gliding effortlessly across the dance floor, moving coherently with the same velocity and never colliding with one another: the superfluid motion without resistance.

Figure 16: Like the condensate, these coupled dancers came together when the music started and continued in a fluid motion next to each other without bumping into each other or stepping on each other's toes.

Sorting out the details of the theory

Back in Urbana, Schrieffer, Bardeen, and Cooper quickly worked out the details of the microscopic theory that became known as BCS. A key feature was the character of the elementary excitations that comprise the normal fluid. We can describe these as quasiparticles. But in creating them, we have to break the pair bonds in the condensate. This requires a finite amount of energy—the energy gap. Moreover, each BCS quasiparticle carries with it a memory of its origin in the pair condensate; it is a mixture of a Landau quasiparticle and a Landau quasihole (an absence of a quasiparticle) of opposite spin.

Figure 17: An illustration of the physical process by which a BCS quasiparticle becomes a mixture of a normal state quasiparticle and a quasihole and in so doing acquires an energy gap.

Figure 17 illustrates the concept. Because the key terms in the effective interaction of the BCS quasiparticle with its neighbors are those that couple it to the condensate, a quasiparticle that scatters against the condensate emerges as a quasihole of opposite spin. The admixture of Landau quasiholes with quasiparticles in a BCS quasiparticle gives rise to interference effects that lead the superconductor to respond differently to probes that measure its density response and probes that measure its spin response. Indeed, the fact that BCS could explain the major difference between the results of acoustic attenuation measurements that probe the density and nuclear spin relaxation measurements of the spin response of the superconducting state in this way provided definitive proof of the correctness of the theory.

The basic algebra that leads to the BCS results is easily worked out. It shows that the energy spectrum of the quasiparticles in the superconducting state takes an especially simple form:

Here, is the normal state quasiparticle energy and the temperature-dependent superconducting energy gap, which also serves as the order parameter that characterizes the presence of a superconducting condensate. When it vanishes at Tc, the quasiparticle and the metal revert to their normal state behavior.

The impact of BCS theory

The rapid acceptance by the experimental low-temperature community of the correctness of the BCS theory is perhaps best epitomized by a remark by David Shoenberg at the opening of a 1959 international conference on superconductivity in Cambridge: "Let us now see to what extent the experiments fit the theoretical facts." Acceptance by the theoretical community came less rapidly. Some of those who had failed to devise a theory were particularly reluctant to recognize that BCS had solved the problem. (Their number did not include Feynman, who famously recognized at once that BCS had solved the problem to which he had just devoted two years of sustained effort, and reacted by throwing into the nearest wastebasket the journal containing their epochal result.) The objections of the BCS deniers initially centered on the somewhat arcane issue of gauge invariance. With the rapid resolution of that issue, the objections became more diffuse. Some critics persisted until they died, a situation not unlike the reaction of the physics community to Planck's discovery of the quantum.

For most physicists, however, the impact of BCS was rapid and immense. It led to the 1957 proposal of nuclear superfluidity, a 15-year search for superfluid 3He, and to the exploration of the role played by pair condensation in particle physics, including the concept of the Higgs boson as a collective mode of a quark-gluon condensate by Philip W. Anderson and Peter Higgs. It led as well to the suggestion of cosmic hadron superfluidity, subsequently observed in the behavior of pulsars following a sudden jump in their rotational frequency, as we will discuss in Section 8.

In addition, BCS gave rise to the discovery of emergent behavior associated with condensate motion. That began with the proposal by a young Cambridge graduate student, Brian Josephson, of the existence of currents associated with the quantum mechanical tunneling of the condensate wavefunction through thin films, called "tunnel junctions," that separate two superconductors. In retrospect, Josephson's 1962 idea was a natural one to explore. If particles could tunnel through a thin insulating barrier separating two normal metals, why couldn't the condensate do the same thing when one had a tunnel junction made up of two superconductors separated by a thin insulating barrier? The answer soon came that it could, and such superconducting-insulating-superconductor junctions are now known as "Josephson junctions." The jump from a fundamental discovery to application also came rapidly. Superconducting quantum interference devices (SQUIDS) (Figure 18), now use such tunneling to detect minute electromagnetic fields, including an application in magnetoencephalography—using SQUIDS to detect the minute magnetic fields produced by neurocurrents in the human brain.

Figure 18: A Superconducting Quantum Interference Device (SQUID) is the most sensitive type of detector of magnetic fields known to science.

Still another fascinating property of a superconductor was discovered by a young Soviet theorist, Alexander Andreev, who realized that when an electron is reflected from the surface of a superconductor, because its wavefunction samples the condensate, it can break one of the pairs in the superconducting condensate, and so emerge as a hole. Measurements by point contact spectroscopy show this dramatic effect, known now as "Andreev reflection," which is the condensed matter equivalent of changing a particle into an antiparticle through a simple physical process.

Looking back at the steps that led to BCS as the Standard Model for what we now describe as conventional superconductors, a pattern emerges. Bardeen, who was key to the development of the theory at every stage from 1950 to 1957, consistently followed what we would now describe as the appropriate emergent strategy for dealing with any major unsolved problem in science:

• Focus first on the experimental results via reading and personal contact.
• Explore alternative physical pictures and mathematical descriptions without becoming wedded to any particular one.
• Thermodynamic and other macroscopic arguments have precedence over microscopic calculations.
• Aim for physical understanding, not mathematical elegance, and use the simplest possible mathematical description of system behavior.
• Keep up with new developments in theoretical techniques—for one of these may prove useful.
• Decide at a qualitative level on candidate organizing concepts that might be responsible for the most important aspect of the measured emergent behavior.
• Only then put on a "reductionist" hat, proposing and solving models that embody the candidate organizing principles.