BCS theory

BCS theory or Bardeen–Cooper–Schrieffer theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since its discovery in 1911. The theory describes superconductivity as a microscopic effect caused by a condensation of Cooper pairs into a boson-like state. The theory is also used in nuclear physics to describe the pairing interaction between nucleons in an atomic nucleus.

It was proposed by Bardeen, Cooper, and Schrieffer in 1957; they received the Nobel Prize in Physics for this theory in 1972.


Rapid progress in understanding superconductivity gained momentum in the mid-1950s. It began with the 1948 paper, "On the Problem of the Molecular Theory of Superconductivity",[1] where Fritz London proposed that the phenomenological 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",[2] 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".[3]

In 1957 Bardeen and Cooper assembled these ingredients and constructed such a theory, the BCS theory, with Robert Schrieffer. The theory was first published in April 1957 in the letter, "Microscopic theory of superconductivity".[4] 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 December 1957 article, "Theory of superconductivity".[5] They received the Nobel Prize in Physics in 1972 for this theory.

In 1986, high-temperature superconductivity was discovered in some materials at temperatures up to about 130 K, considerably above the previous limit of about 30 K. It is believed that BCS theory alone cannot explain this phenomenon and that other effects are in play.[6] These effects are still not yet fully understood; it is possible that they even control superconductivity 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 electron-lattice 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 the condensation of Cooper pairs. These have some bosonic properties, and bosons, at sufficiently low temperature, can form a large Bose–Einstein condensate. Superconductivity was simultaneously explained by Nikolay Bogolyubov, by means of the 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. Because there are a lot of such electron pairs in a superconductor, these pairs overlap very strongly and form a highly collective condensate. In this "condensed" state, the breaking of one pair will change the energy of the entire condensate - not just a single electron, or a single pair. Thus, the energy required to break any single pair is related to the energy required to break all of the pairs (or more than just two electrons). Because the pairing increases this energy barrier, kicks from oscillating atoms in the conductor (which are small at sufficiently low temperatures) are not enough to affect the condensate as a whole, or any individual "member pair" within the condensate. Thus the electrons stay paired together and resist all kicks, and the electron flow as a whole (the current through the superconductor) will not experience resistance. Thus, the collective behavior of the condensate is a crucial ingredient necessary for 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. For instance, Cooper pairs have been observed in ultracold gases of fermions where a homogeneous magnetic field has been tuned to their Feshbach resonance. 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 many-body 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.

Underlying evidence

The hyperphysics website pages at Georgia State University summarize some key background to BCS theory as follows:[7]

the existence of a critical temperature and critical magnetic field implied a band gap, and suggested a phase transition, but single electrons are forbidden from condensing to the same energy level by the Pauli exclusion principle. The site comments that "a drastic change in conductivity demanded a drastic change in electron behavior". Conceivably, pairs of electrons might perhaps act like bosons instead, which are bound by different condensate rules and do not have the same limitation.
The Debye frequency of phonons in a lattice is proportional to the inverse of the square root of the mass of lattice ions. It was shown that the superconducting transition temperature of mercury indeed showed the same dependence, by substituting natural mercury 202Hg with a different isotope 198Hg.[8]
An exponential increase in heat capacity near the critical temperature also suggests an energy bandgap for the superconducting material. As superconducting vanadium is warmed toward its critical temperature, its heat capacity increases massively in a very few degrees; this suggests an energy gap being bridged by thermal energy.
this suggests a type of situation where some kind of binding energy exists but it is gradually weakened as the critical temperature is approached. A binding energy suggests two or more particles or other entities that are bound together in the superconducting state. This helped to support the idea of bound particles - specifically electron pairs - and together with the above helped to paint a general picture of paired electrons and their lattice interactions.


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:

independent of material. Near the critical temperature the relation asymptotes to[10]
which is of the form suggested the previous year by M. J. Buckingham[11] 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.
where N(0) is the electronic density of states at the Fermi level. For more details, see Cooper pairs.

See also


  1. London, F. (September 1948). "On the Problem of the Molecular Theory of Superconductivity". Physical Review. 74 (5): 562–573. Bibcode:1948PhRv...74..562L. doi:10.1103/PhysRev.74.562. Retrieved March 3, 2012.
  2. Bardeen, J. (March 1955). "Theory of the Meissner Effect in Superconductors". Physical Review. 97 (6): 1724–1725. Bibcode:1955PhRv...97.1724B. doi:10.1103/PhysRev.97.1724.
  3. Cooper, Leon (November 1956). "Bound Electron Pairs in a Degenerate Fermi Gas". Physical Review. 104 (4): 1189–1190. Bibcode:1956PhRv..104.1189C. doi:10.1103/PhysRev.104.1189. ISSN 0031-899X.
  4. Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. (April 1957). "Microscopic Theory of Superconductivity". Physical Review. 106 (1): 162–164. Bibcode:1957PhRv..106..162B. doi:10.1103/PhysRev.106.162. Retrieved May 3, 2012.
  5. 1 2 Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. (December 1957). "Theory of Superconductivity". Physical Review. 108 (5): 1175–1204. Bibcode:1957PhRv..108.1175B. doi:10.1103/PhysRev.108.1175. Retrieved May 3, 2012.
  6. Mann, A. (July 2011). "High Temperature Superconductivity at 25: Still In Suspense". Nature. 475 (7356): 280–2. Bibcode:2011Natur.475..280M. doi:10.1038/475280a. PMID 21776057. Retrieved November 18, 2012.
  7. http://hyperphysics.phy-astr.gsu.edu/hbase/solids/bcs.html
  8. Maxwell, Emanuel. "Isotope Effect in the Superconductivity of Mercury". Physical Review. 78 (4): 477–477. Bibcode:1950PhRv...78..477M. doi:10.1103/PhysRev.78.477. Retrieved 27 July 2014.
  9. Ivar Giaever - Nobel Lecture. Nobelprize.org. Retrieved 16 Dec 2010. http://nobelprize.org/nobel_prizes/physics/laureates/1973/giaever-lecture.html
  10. 1 2 Tinkham, Michael (1996). Introduction to Superconductivity. Dover Publications. p. 63. ISBN 0-486-43503-2.
  11. Buckingham, M. J. (February 1956). "Very High Frequency Absorption in Superconductors". Physical Review. 101: 1431–1432. Bibcode:1956PhRv..101.1431B. doi:10.1103/PhysRev.101.1431.
  12. W. A. Little and R. D. Parks, “Observation of Quantum Periodicity in the Transition Temperature of a Superconducting Cylinder”, Physical Review Letters 9, 9 (1962), doi:10.1103/PhysRevLett.9.9

Primary sources

Further reading

External links

This article is issued from Wikipedia - version of the 11/11/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.