BCS theory is the first microscopic theory of superconductivity, proposed by Bardeen, Cooper, and Schrieffer in 1957 since the discovery of superconductivity in 1911. It describes superconductivity as a microscopic effect caused by a "condensation" of pairs of electrons into a bosonlike state.
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The mid 1950s saw rapid progress in the understanding of superconductivity. It began in the 1948 paper On the Problem of the Molecular Theory of Superconductivity where Fritz London proposed that the phenomonological London equations may be consequences of the coherence of a quantum state. In 1953 Brian Pippard, motivated by penetration experiments, proposed that this would modify the London equations via a new scale parameter called the coherence length. John Bardeen then argued in the 1955 paper Theory of the Meissner Effect in Superconductors that such a modification naturally occurs in a theory with an energy gap. The key ingredient was Leon Neil Cooper's calculation of the bound states of electrons subject to an attractive force in his 1956 paper Bound Electron Pairs in a Degenerate Fermi Gas.
In 1957 Bardeen and Cooper assembled these ingredients and constructed such a theory, the BCS theory, with Robert Schrieffer. The theory was first announced in February 1957 in the letter Microscopic theory of superconductivity. The demonstration that the phase transition is second order, that it reproduces the Meissner effect and the calculations of specific heats and penetration depths appeared in the July 1957 article Theory of superconductivity. They received the Nobel Prize in Physics in 1972 for this theory. The 1950 LandauGinzburg theory of superconductivity is not cited in either of the BCS papers.
In 1986, "hightemperature superconductivity" was discovered (i.e. superconductivity at temperatures considerably above the previous limit of about 30 K; up to about 130 K). It is believed that at these temperatures other effects are at play; these effects are not yet fully understood. (It is possible that these unknown effects also control superconductivity even at low temperatures for some materials).
At sufficiently low temperatures, electrons near the Fermi surface become unstable against the formation of Cooper pairs. Cooper showed such binding will occur in the presence of an attractive potential, no matter how weak. In conventional superconductors, an attraction is generally attributed to an electronlattice interaction. The BCS theory, however, requires only that the potential be attractive, regardless of its origin. In the BCS framework, superconductivity is a macroscopic effect which results from "condensation" of Cooper pairs. These have some bosonic properties, while bosons, at sufficiently low temperature, can form a large BoseEinstein condensate. Superconductivity was simultaneously explained by Nikolay Bogoliubov, by means of the socalled Bogoliubov transformations.
In many superconductors, the attractive interaction between electrons (necessary for pairing) is brought about indirectly by the interaction between the electrons and the vibrating crystal lattice (the phonons). Roughly speaking the picture is the following:
An electron moving through a conductor will attract nearby positive charges in the lattice. This deformation of the lattice causes another electron, with opposite "spin", to move into the region of higher positive charge density. The two electrons then become correlated. There are a lot of such electron pairs in a superconductor, so that they overlap very strongly, forming a highly collective "condensate". Breaking of one pair results in changing of energies of remained macroscopic number of pairs. If the required energy is higher than the energy provided by kicks from oscillating atoms in the conductor (which is true at low temperatures), then the electrons will stay paired and resist all kicks, thus not experiencing resistance. Thus, the collective behaviour of "condensate" is a crucial ingredient of superconductivity.
BCS theory starts from the assumption that there is some attraction between electrons, which can overcome the Coulomb repulsion. In most materials (in low temperature superconductors), this attraction is brought about indirectly by the coupling of electrons to the crystal lattice (as explained above). However, the results of BCS theory do not depend on the origin of the attractive interaction. The original results of BCS (discussed below) described an "swave" superconducting state, which is the rule among lowtemperature superconductors but is not realized in many "unconventional superconductors", such as the "dwave" hightemperature superconductors. Extensions of BCS theory exist to describe these other cases, although they are insufficient to completely describe the observed features of hightemperature superconductivity.
BCS is able to give an approximation for the quantummechanical manybody state of the system of (attractively interacting) electrons inside the metal. This state is now known as the "BCS state". In the normal state of a metal, electrons move independently, whereas in the BCS state, they are bound into "Cooper pairs" by the attractive interaction. The BCS formalism is based on the "reduced" potential for the electrons attraction. Within this potential, a variational ansatz for the wave function is proposed. This ansatz was later shown to be exact in the dense limit of pairs. Note that the continuous crossover between the dilute and dense regimes of attracting pairs of fermions is still an open problem, which now attracts a lot of attention within the field of ultracold gases.
BCS derived several important theoretical predictions that are independent of the details of the interaction, since the quantitative predictions mentioned below hold for any sufficiently weak attraction between the electrons and this last condition is fulfilled for many low temperature superconductors  the socalled "weakcoupling case". These have been confirmed in numerous experiments:
which is of the form suggested the previous year by M. J. Buckingham in Very High Frequency Absorption in Superconductors based on the fact that the superconducting phase transition is second order, that the superconducting phase has a mass gap and on Blevins, Gordy and Fairbank's experimental results the previous year on the absorption of millimeter waves by superconducting tin.
The BCS Papers:

