In condensed matter physics, a Cooper pair is the name given to electrons that are bound together at low temperatures in a certain manner first described in 1956 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 is responsible for superconductivity, as described in the BCS theory developed by John Bardeen, John Schrieffer and Leon Cooper for which they shared the 1972 Nobel Prize.
The reason for the pairing can be seen from a simplified explanation. An electron in a metal normally behaves as 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 ion lattice in such a way as to attract other electrons (the electron-phonon interaction). At long distances this attraction between electrons due to the displaced ions can overcome the electrons' repulsion due to their negative charge, and cause them to pair-up.
The energy of the pairing interaction is quite weak, of the order of 10-3eV, and thermal energy can easily break the pairs up. So only at low temperatures are a significant number of the electrons in a metal in Cooper pairs. The electrons in a pair are not necessarily close together; because the interaction is long range, paired electrons may still be many hundreds of nanometers apart. This distance is usually greater than the average interelectron distance, so many Cooper pairs can occupy the same space. Since electrons are spin-1/2 fermions, a Cooper pair is a boson, to which the Pauli exclusion principle doesn't apply, so they are allowed to be in the same state. The tendency for all the Cooper pairs in a body to 'condense' into the same ground quantum state is responsible for the peculiar properties of superconductivity.
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.
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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.