# BCS theory

**BCS theory**is a microscopic theory ofsuperconductivity , proposed by Bardeen, Cooper, and Schrieffer. It describessuperconductivity as a microscopic effect caused by Bose condensation of pairs of electrons.**History**BCS theory was developed in 1957 by

John Bardeen , Leon Cooper, and Robert Schrieffer and they received theNobel Prize in Physics in 1972 for this theory.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 somematerials).**Overview**In the BCS framework, superconductivity is a macroscopic effect which results from Bose condensation of electron (Cooper) pairs. These behave as bosons which, at sufficiently low temperature, form a large Bose-Einstein condensate. 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, such binding is generally attributed to an electron-lattice interaction. The BCS theory, however, requires only that the potential be attractive, regardless of its origin. Superconductivity was simultaneously explained by

Nikolay Bogoliubov , by means of the so-calledBogoliubov transformation s.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

phonon s). 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 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.

**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 thecrystal 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 "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 is able to give an approximation for the quantum-mechanical state of thesystem of (attractively interacting) electrons inside the metal. This state isnow 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.

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 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) single particle

density of states at theFermi 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 amagnetic 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 normal conducting) 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.

* In its simplest form, BCS gives the superconducting transition temperature in terms of the electron-phonon coupling potential and the

Debye cutoff energy::$k\_B,T\_c\; =\; 1.14E\_D,\{e^\{-1/N(0),V.,$**ee also***

Superconductivity **References**The BCS Papers:

*L. N. Cooper, "Bound Electron Pairs in a Degenerate Fermi Gas", [

*http://prola.aps.org/abstract/PR/v104/i4/p1189_1 "Phys. Rev"*] .**104**, 1189 - 1190 (1956)*J. Bardeen, L. N. Cooper, and J. R. Schrieffer, "Microscopic Theory of Superconductivity", [

*http://prola.aps.org/abstract/PR/v106/i1/p162_1 "Phys. Rev."*] .**106**, 162 - 164 (1957)*J. Bardeen, L. N. Cooper, and J. R. Schrieffer, "Theory of Superconductivity", [

*http://link.aps.org/abstract/PR/v108/p1175 "Phys. Rev."*] .**108**, 1175 (1957)**External links*** ScienceDaily: [

*http://www.sciencedaily.com/releases/2006/08/060817101658.htm Physicist Discovers Exotic Superconductivity*] (University of Arizona )August 17 ,2006

* [*http://hyperphysics.phy-astr.gsu.edu/hbase/solids/bcs.html Hyperphysics page on BCS*]

* [*http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/T.J_Barry/bcstheory.html BCS History*]**Further reading***

John Robert Schrieffer , "Theory of Superconductivity", (1964), ISBN 0-7382-0120-0

*Michael Tinkham , "Introduction to Superconductivity", ISBN 0-4864-3503-2

*Pierre-Gilles de Gennes , "Superconductivity of Metals and Alloys", ISBN 0-7382-0101-4.

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