Cooper Pair

A Cooper pair is a correlated pair of electrons, or more generally other fermions, formed through an effective attractive interaction in a many-particle system. Electron pairing is central to BCS theory, which explains conventional superconductivity. Unlike an ordinary molecule, a weakly bound Cooper pair can extend over many interatomic distances and overlap extensively with other pairs. Superconductivity arises from the collective quantum state of these pairs, rather than simply from the existence of isolated two-electron bound states. (journals.aps.org)

Origin and the Cooper instability

The concept originated in Leon N. Cooper’s 1956 paper, “Bound Electron Pairs in a Degenerate Fermi Gas.” Cooper considered two electrons interacting above an otherwise inert, filled sea of electron states. He showed that, within this idealized model, an arbitrarily weak attractive interaction can produce a state below the energy of the unpaired electrons. This result is known as the Cooper instability: the unpaired Fermi sea is unstable to pairing when an attractive pairing channel is available. (journals.aps.org)

The surrounding electrons are essential to this argument. The Pauli exclusion principle prevents the added electrons from occupying already filled states, restricting their motion to states above the Fermi surface, the boundary separating occupied and unoccupied states at zero temperature. The result therefore does not imply that any arbitrarily weak attraction binds two isolated electrons in empty three-dimensional space. (physics.umd.edu)

John Bardeen, Cooper, and John Robert Schrieffer developed the many-electron theory in 1957. Their work established how pairing produces a superconducting ground state and received the 1972 Nobel Prize in Physics. (journals.aps.org)

Effective attraction and pair structure

Electrons repel each other electrically, but their interaction inside a solid includes effects mediated by the surrounding material. In conventional superconductors, electron coupling to lattice vibrations—quantized as phonons—provides an effective attraction. A useful qualitative picture is that one electron distorts the positively charged lattice, and another responds to that distortion. Pairing becomes favorable when the attractive contribution dominates the screened repulsion in the relevant low-energy channel. The electrons’ electric charges do not change sign. (journals.aps.org)

In the simplest BCS state without a supercurrent, electrons pair in time-reversed states, conventionally written as $(\mathbf{k},\uparrow)$ and $(-\mathbf{k},\downarrow)$. Their total momentum is zero, and their spins form a singlet with total spin zero. The corresponding conventional s-wave pairing state has an isotropic gap. These properties describe the original model, not every possible paired state. (journals.aps.org)

Because a pair contains two fermions, it has bosonic character. Nevertheless, strongly overlapping electron pairs are not independent, elementary bosons. Their constituent electrons continue to obey fermionic statistics. The superconducting wave function describes a collective superposition of pair configurations, not a set of permanently identifiable electron partners. (nobelprize.org)

Collective coherence and the energy gap

BCS pairing creates a many-body state with macroscopic quantum coherence. Pair amplitudes possess a common phase relationship across the material; a spatial variation of that phase is associated with supercurrent. The pairing amplitude is related to the superconducting order parameter, which characterizes the ordered phase. Pair formation and coherent superconducting order are conceptually distinct, even though they appear together in the simplest BCS treatment. (nobelprize.org)

A fully gapped BCS superconductor has a superconducting energy gap: creating one quasiparticle requires at least an energy $\Delta$, while breaking a pair into two quasiparticles requires at least $2\Delta$. For a weak-coupling, isotropic s-wave superconductor,

where $k_{\mathrm B}$ is the Boltzmann constant and $T_{\mathrm c}$ is the superconducting transition temperature. This ratio is not universal for all superconductors. The gap and collective coherence help explain the stability of the superconducting state, but zero resistance should not be interpreted as two electrons mechanically shielding each other from collisions. (journals.aps.org)

The characteristic pair extent in clean, weak-coupling superconductors is related to the superconducting coherence length. Pair size and the length over which the order parameter varies are related but distinct quantities, especially outside that limit. (arxiv.org)

Electromagnetic evidence and devices

An electron pair carries electric charge $-2e$, where $e$ is the positive elementary charge. This charge appears in magnetic flux quantization through the flux quantum,

with $h$ the Planck constant. Quantized flux and the Josephson effect provide macroscopic evidence of coherent charge-$2e$ transport. (nvlpubs.nist.gov)

Across a Josephson junction, coherent pair transfer can support current without an applied voltage. The current depends on the phase difference between the superconductors. This behavior underlies SQUIDs and many superconducting qubits; the alternating-current Josephson effect also supports precision voltage standards. (nvlpubs.nist.gov)

Pairing beyond conventional superconductors

Cooper pairing is not restricted to phonon-mediated, spin-singlet s-wave superconductivity. Phase-sensitive experiments establish predominantly d-wave pairing in several cuprate materials associated with high-temperature superconductivity. Superfluid helium-3 instead exhibits spin-triplet, p-wave pairing of neutral fermionic atoms. Consequently, neither opposite-spin singlet pairing nor an isotropic gap belongs to the general definition. (journals.aps.org)

Ultracold atomic gases provide controllable realizations of fermion pairing and superfluidity. By tuning interactions near a Feshbach resonance, experiments explore the BCS–BEC crossover between extended, overlapping Cooper pairs and tightly bound molecules forming a Bose–Einstein condensate. This connects weak-coupling pairing to molecular condensation within a continuously tunable many-body system. (nature.com)