Incondensed matter physics, aCooper pair orBCS pair (Bardeen–Cooper–Schrieffer pair) is a pair ofelectrons (or otherfermions) bound together atlow temperatures in a certain manner first described in 1956 by American physicistLeon Cooper.[1]
Cooper showed that an arbitrarily small attraction between electrons in ametal can cause a paired state of electrons to have a lower energy than theFermi energy, which implies that the pair is bound. In conventionalsuperconductors, this attraction is due to theelectron–phonon interaction. The Cooper pair state is responsible for superconductivity, as described in theBCS theory developed byJohn Bardeen,Leon Cooper, andJohn Schrieffer for which they shared the 1972Nobel Prize.[2]
Although Cooper pairing is a quantum effect, the reason for the pairing can be seen from a simplified classical explanation.[2][3] An electron in ametal normally behaves as afree particle. The electron is repelled from other electrons due to their negativecharge, but it also attracts the positiveions 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 toelectron–phonon interactions, with the phonon being the collective motion of the positively-charged lattice.[4]
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 bound 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 ofnanometers apart. This distance is usually greater than the average interelectron distance so that many Cooper pairs can occupy the same space.[5] Electrons havespin-1⁄2, so they arefermions, but thetotal spin of a Cooper pair is integer (0 or 1) so it is acomposite boson. This means thewave 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 phenomenon of superconductivity.
The BCS theory is also applicable to other fermion systems, such ashelium-3.[citation needed] Indeed, Cooper pairing is responsible for thesuperfluidity of helium-3 at low temperatures.[citation needed] In 2008 it was proposed that pairs ofbosons in anoptical lattice may be similar to Cooper pairs.[6]
The tendency for all the Cooper pairs in a body to "condense" into the sameground quantum state is responsible for the peculiar properties of 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. Thisgap to excitations leads to superconductivity, since small excitations such as scattering of electrons are forbidden.[7]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.[8][9] This was indicated by theisotope effect observed in superconductors. The isotope effect showed that materials with heavier ions (differentnuclear 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 mathematical description of the second-order coherence involved here is given by Yang.[11]
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