# BCS theory

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**BCS theory** (named for its creators, Bardeen, Cooper, and Schrieffer) 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.

## More details

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:

- Since the electrons are bound into Cooper pairs, a finite amount of energy is needed to break these apart into two independent electrons. This means there is an "energy gap" for "single-particle excitation", unlike in the normal metal (where the state of an electron can be changed by adding an arbitrarily small amount of energy). This energy gap is highest at low temperatures but vanishes at the transition temperature when superconductivity ceases to exist. BCS theory correctly predicts the variation of this gap with temperature. It also gives an expression that shows how the gap grows with the strength of the attractive interaction and the (normal phase) "density of states" at the Fermi energy. Furthermore, it describes how the "density of states" is changed on entering the superconducting state, where there are no electronic states any more at the Fermi energy. The energy gap is most directly observed in tunneling experiments and in reflection of microwaves from the superconductor.

- The ratio between the value of the energy gap at zero temperature and the value of the superconducting transition temperature (expressed in energy units) takes the universal value of 3.5, independent of material.

- Due to the energy gap, the specific heat of the superconductor is suppressed strongly (exponentially) at low temperatures, there being no thermal excitations left. However, before reaching the transition temperature, the specific heat of the superconductor becomes even higher than that of the normal conductor (measured immediately above the transition) and the ratio of these two values is found to be universally given by 2.5.

- BCS theory correctly predicts the Meissner effect, i.e. the expulsion of a magnetic field from the superconductor and the variation of the penetration depth (the extent of the screening currents flowing below the metal's surface) with temperature.

- It also describes the variation of the critical magnetic field (above which the superconductor can no longer expel the field but becomes normalconducting) with temperature. BCS theory relates the value of the critical field at zero temperature to the value of the transition temperature and the density of states at the Fermi energy.

*Original reference*:
J. Bardeen, L. N. Cooper, and J. R. Schrieffer, "Theory of Superconductivity",
*Phys. Rev.* **108** (5), 1175 (1957).

Four-fermion interactions | ||||
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Fermi's theory | Thirring model | Gross-Neveu model | Nambu-Jona-Lasinio model | BCS theory |

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