ABSTRACT
In this chapter, we focus on the phenomenon of superconductivity and the Bardeen-Cooper-Schrieffer (BCS) theory behind it [BCS1957]. Superconductivity obtains when a finite fraction of the conduction electrons in a metal condense into a quantum state characterized by a unique quantum mechanical phase. The specific value of the quantum mechanical phase varies from one superconductor to another. The locking-in of the phase of a number of electrons on the order of Avogadro's number ensures the rigidity of the superconducting state. For example, electrons in the condensate find it impossible to move individually. Rather, the whole condensate moves from one end of the sample to the other as a single unit. Likewise, electron-scattering events that tend to disrupt the condensate must disrupt the phase of a macroscopic number of electrons for the superconducting state to be destroyed. Hence, phase rigidity implies collective motion as well as collective destruction of a superconducting condensate. The only other physical phenomenon that arises from a similar condensation of a macroscopic number of particles into a phase-locked state is that of Bose-Einstein condensation. There is a crucial difference between these effects, however. The particles that constitute the condensate in superconductivity are Cooper pairs, which do not obey Bose statistics. In fact, it is the Pauli principle acting on the electrons comprising a Cooper pair that prevents the complete mapping of the superconducting problem onto a simple one of Bose condensation. As we will see, it is the Pauli principle that makes BCS theory work so well. What do we mean by this? In BCS theory, it is assumed that electrons form Cooper pairs, and the pairs are strongly overlapping. Such strong overlap would imply a strong correlation between pairs. In fact, it is the correlations between pairs that accounts for most of the observed properties of superconductors, for example the energy gap and the Meissner effect. In BCS theory, however, there is no explicit dynamical interaction between Cooper pairs. The only interaction, if it can be thought of in these terms, is that arising from the Pauli exclusion principle that 206precludes two Cooper pairs from occupying the same momentum state. That BCS theory works so well speaks volumes for the real nature of pair-pair correlations in metals. It would suggest that real pair-pair interactions in a metal arise primarily from the Pauli exclusion principle rather than from some additional dynamical interaction. It is primarily for this reason that the simple pairing hypothesis of BCS has had such profound success.
