BCS theory successfully explains conventional superconductivity, the ability of certain metals at low temperatures to conduct electricity without resistance. BCS theory views superconductivity as a macroscopic quantum mechanical effect. It proposes that electrons with opposite spin can become paired, forming Cooper pairs.
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 cause a slight increase in concentration of positive charges in the lattice around it; this increase in turn can attract another electron. In effect, the two electrons are then held together with a certain binding energy. If this binding energy is higher than the energy provided by kicks from oscillating atoms in the conductor (which is true at low temperatures), then the electron pair will stick together and resist all kicks, thus not experiencing resistance.
BCS theory was developed in 1957 by John Bardeen, Leon Cooper, and Robert Schrieffer, who received the Nobel Prize for Physics in 1972 as a result.
In 1986, "high-temperature 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)
An excellent introduction to BCS theory and related areas of condensed matter physics at the graduate level is Schrieffer's book, Theory of Superconductivity, ISBN 0-7382-0120-0.
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. Note that
the original results of BCS (discussed below)
were describing an "s-wave" superconducting state,
which is the rule among low-temperature superconductors
but is not realized in many "unconventional superconductors",
such as the "d-wave" high-temperature superconductors.
Extensions of BCS theory exist to describe these other
cases, although they are insufficient to
completely describe the observed features of
high-temperature superconductivity.
BCS were able to give an
approximation for the quantum-mechanical state of the
system of (attractively interacting) electrons inside the metal. This state is
now known as the "BCS state". Whereas in the normal
metal electrons move independently, in the
BCS state they are bound into
"Cooper pairs" by the attractive interaction.
BCS have derived several
important theoretical predictions that are independent
of the details of the interaction (note that the quantitative
predictions mentioned below hold only for sufficiently
weak attraction between the electrons, which is however
fulfilled for many low temperature superconductors
- the so-called "weak-coupling case"). These have been
confirmed in numerous experiments:
Introduction
More details
Original reference:
J. Bardeen, L. N. Cooper, and J. R. Schrieffer, "Theory of Superconductivity",
Phys. Rev. 108 (5), 1175 (1957).