condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...

, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of

electrons The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...

(or other

fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...

s) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper.

Description

Cooper showed that an arbitrarily small attraction between electrons in a

metal A metal () is a material that, when polished or fractured, shows a lustrous appearance, and conducts electrical resistivity and conductivity, electricity and thermal conductivity, heat relatively well. These properties are all associated wit ...

can cause a paired state of electrons to have a lower energy than the

Fermi energy The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. In a Fermi ga ...

, which implies that the pair is bound. In conventional

superconductors Superconductivity is a set of physical properties observed in superconductors: materials where electrical resistance vanishes and magnetic fields are expelled from the material. Unlike an ordinary metallic conductor, whose resistance decreases ...

, this attraction is due to the

electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...

–

phonon A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. In the context of optically trapped objects, the quantized vibration mode can be defined a ...

interaction. The Cooper pair state is responsible for superconductivity, as described in the

BCS theory In physics, the Bardeen–Cooper–Schrieffer (BCS) theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory descr ...

developed by

John Bardeen John Bardeen (; May 23, 1908 – January 30, 1991) was an American solid-state physicist. He is the only person to be awarded the Nobel Prize in Physics twice: first in 1956 with William Shockley and Walter Houser Brattain for their inventio ...

, Leon Cooper, and John Schrieffer for which they shared the 1972

Nobel Prize in Physics The Nobel Prize in Physics () is an annual award given by the Royal Swedish Academy of Sciences for those who have made the most outstanding contributions to mankind in the field of physics. It is one of the five Nobel Prizes established by the ...

. Although Cooper pairing is a quantum effect, the reason for the pairing can be seen from a simplified classical explanation. An electron in a

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

–

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 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 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-, so they are

s, but the total spin of a Cooper pair is integer (0 or 1) so it is a composite boson. This means the

wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...

s 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. BEC-BCS

The BCS theory is also applicable to other fermion systems, such as

helium-3 Helium-3 (3He see also helion) is a light, stable isotope of helium with two protons and one neutron. (In contrast, the most common isotope, helium-4, has two protons and two neutrons.) Helium-3 and hydrogen-1 are the only stable nuclides with ...

. Indeed, Cooper pairing is responsible for the

superfluidity Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...

of helium-3 at low temperatures. In 2008 it was proposed that pairs of

boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...

s in an optical lattice may be similar to Cooper pairs.

Relationship to superconductivity

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. 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 Isotopes are distinct nuclear species (or ''nuclides'') of the same chemical element. They have the same atomic number (number of protons in their Atomic nucleus, nuclei) and position in the periodic table (and hence belong to the same chemica ...

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 An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb's law, Coulomb force resulting from their opposite charges. It is an electrically neutral quasiparticle regarded as ...

interactions or electron–

plasmon In physics, a plasmon is a quantum of plasma oscillation. Just as light (an optical oscillation) consists of photons, the plasma oscillation consists of plasmons. The plasmon can be considered as a quasiparticle since it arises from the quant ...

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.

Description

Relationship to superconductivity

See also

References

Further reading