In condensed matter physics, a Cooper pair or BCS pair is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist 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 conventional 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, Leon Cooper, and John Schrieffer for which they shared the 1972 Nobel Prize.
Although Cooper pairing is a quantum effect, the reason for the pairing can be seen from a simplified classical explanation. An electron in a metal normally behaves as a free particle. The electron is repelled from other electrons due to their negative charge, but it also attracts the positive ions that make up the rigid lattice of the metal. This attraction distorts the ion lattice, moving the ions slightly toward the electron, increasing the positive charge density of the lattice in the vicinity. This positive charge can attract other electrons. 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 rigorous quantum mechanical explanation shows that the effect is due to electron–phonon interactions, with the phonon being the collective motion of the positively-charged lattice.
The energy of the pairing interaction is quite weak, of the order of 10−3 eV, and thermal energy can easily break the pairs. So only at low temperatures, in metal and other substrates, are a significant number of the electrons 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 that many Cooper pairs can occupy the same space. Electrons have spin-1⁄2, so they are fermions, but the total spin of a Cooper pair is integer (0 or 1) so it is a composite boson. This means the wave functions are symmetric under particle interchange. Therefore, unlike electrons, multiple Cooper pairs are allowed to be in the same quantum state, which is responsible for the phenomena of superconductivity.
The BCS theory is also applicable to other fermion systems, such as helium-3. Indeed, Cooper pairing is responsible for the superfluidity of helium-3 at low temperatures. It has also been recently demonstrated that a Cooper pair can comprise two bosons. Here, the pairing is supported by entanglement in an optical lattice.
Relationship to superconductivity
Cooper originally considered only the case of an isolated pair's formation in a metal. When one considers the more realistic state of many electronic pair formations, as is elucidated 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. The gap appears due to many-body effects between electrons feeling the attraction.
R. A. Ogg, Jr., was first to suggest that 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 by the theory of Cooper pairing: heavier ions are harder for the electrons to attract and move (how Cooper pairs are formed), which results in smaller binding energy for the pairs.
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 other pairing interactions has been observed in any material.
It should be mentioned that Cooper pairing does not involve individual electrons pairing up to form "quasi-bosons". The paired states are energetically favored, and electrons go in and out of those states preferentially. This is a fine distinction that John Bardeen makes:
- "The idea of paired electrons, though not fully accurate, captures the sense of it." 
The mathematical description of the second-order coherence involved here is given by Yang.
- Cooper, Leon N. (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.
- Nave, Carl R. (2006). "Cooper Pairs". Hyperphysics. Dept. of Physics and Astronomy, Georgia State Univ. Retrieved 2008-07-24.
- Kadin, Alan M. (2005). "Spatial Structure of the Cooper Pair". Journal of Superconductivity and Novel Magnetism. 20 (4): 285. arXiv:cond-mat/0510279. doi:10.1007/s10948-006-0198-z.
- Fujita, Shigeji; Ito, Kei; Godoy, Salvador (2009). Quantum Theory of Conducting Matter. Springer Publishing. pp. 15–27. ISBN 978-0-387-88211-6.
- Feynman, Richard P.; Leighton, Robert; Sands, Matthew (1965). Lectures on Physics, Vol.3. Addison–Wesley. pp. 21–7, 8. ISBN 0-201-02118-8.
- Cooper Pairs of Bosons
- Nave, Carl R. (2006). "The BCS Theory of Superconductivity". Hyperphysics. Dept. of Physics and Astronomy, Georgia State Univ. Retrieved 2008-07-24.
- Ogg Jr, R.A., "Bose-Einstein Condensation of Trapped Electron Pairs. Phase Separation and Superconductivity of Metal-Ammonia Solutions," Phys. Rev. 69, 243-244, (1946).
- Poole Jr, Charles P, "Encyclopedic dictionary of condensed matter physics", (Academic Press, 2004), p. 576
- J. Bardeen, "Electron-Phonon Interactions and Superconductivity", in Cooperative Phenomena, eds. H. Haken and M. Wagner (Springer-Verlag, Berlin, Heidelberg, New York, 1973), p. 67.
- C. N. Yang, "Off-Diagonal Long-Range Order." Rev. Mod. Phys. 34, 694 (1962)