Cooper pair

A Cooper pair is the name given to electrons that are bound together in a certain manner first described by Leon Cooper. Cooper showed that an arbitrarily small attraction between electrons in a metal can cause a paired state of electrons to have a lower energy than the Fermi energy, which implies that the pair is bound. In normal superconductors, this attraction is due to the electron phonon interaction. The Cooper Pair state forms the basis of the BCS theory of superconductivity developed by John Bardeen, John Schrieffer and Leon Cooper for which they shared the 1972 Nobel Prize.

A simplified explanation: an electron in a metal normally behaves as basically a free particle. The electron is repelled from other electrons due to their similar charge, but it also attracts the positive ions that make up the rigid lattice of the metal. This attraction can distort the positively charged ions in such a way as to attract other electrons (the electron-phonon interaction). This attraction due to the displaced ions can overcome the electrons repulsion due to the electrons having the same charge and cause them to pair-up. Generally, the pairing only occurs at low temperatures and is quite weak, meaning the paired electrons may still be many hundreds of nanometers apart.

Cooper originally just considered the case of an isolated pair forming in a metal. When one considers the more realistic state consisting of many electrons forming pairs as is done in the full BCS Theory one finds that the pairing opens a gap in the continuous spectrum of allowed energy states of the electrons, meaning that all excitations of the system must possess some minimum amount of energy. This gap to excitations leads to superconductivity, since small excitations such as scattering of electrons are forbidden.

Herbert Fröhlich was first to suggest that the electrons might act as pairs coupled by lattice vibrations in the material. This was indicated by the isotope effect observed in superconductors. The isotope effect showed that materials with heavier ions (different nuclear isotopes) had lower superconducting transition temperatures. This can be explained nicely by the theory of Cooper pairing, since heavier ions are harder to move they would be less able to attract the electrons resulting in a smaller binding energy for Cooper pairs.

The pair are still Cooperic if k1 = k₂ and k1 − q = − (k1 − q) = − ( − k₂− q) = − (k₂ + q)

The theory of Cooper pairs is quite general and does not depend on the specific electron-phonon interaction. Condensed matter theorists have proposed pairing mechanisms based on other attractive interactions such as electron-exciton interactions or electron-plasmon interactions. Currently, none of these alternate pairing interactions has been observed in any material.

References

Original reference: L. N. Cooper, "Bound Electron Pairs in a Degenerate Fermi Gas", Phys. Rev. 104 (4), 1189 (1956).

See also

* BCS theory

* Superconductivity