Boson 6: Cooper electron pairs explained by BCS theory

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.

BCS theory (named for its creators, Bardeen, Cooper, and Schrieffer) successfully explainsconventional superconductivity, the ability of certainmetals at low temperatures to conduct electricitywithout resistance. BCS theory viewssuperconductivity as a macroscopic quantum mechanical effect. It proposes that electrons with opposite spin can become paired, forming Cooper pairs. Independently and at the same time, superconductivity phenomenon was explained byNikolay Bogoliubov by means of the so-calledBogoliubov 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 (thephonons). 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.

BCS theory was developed in 1957 by John Bardeen, Leon Cooper, and Robert Schrieffer, who received theNobel Prize for Physics in 1972 as a result.

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 some materials).