Condensed Matter

Cooper Pairs

At low T, phonon-mediated attraction binds two opposite-spin electrons into a boson — BCS state

A Cooper pair is a bound state of two electrons of opposite spin and momentum, held together by a weak attractive interaction mediated by lattice vibrations (phonons). Predicted by Leon Cooper (1956): even an arbitrarily weak attraction between two electrons at the Fermi surface binds them into a pair with total spin 0 (boson). Cooper pairs Bose-condense into a coherent ground state — the BCS superconducting state (Bardeen-Cooper-Schrieffer 1957, Nobel 1972). Pair size (coherence length): ~100 nm in Pb (much larger than atomic spacing — pairs strongly overlap). Energy gap Δ ~ 1.76 k_B T_c — minimum energy to break a pair, source of zero electrical resistance. High-T_c cuprates (1986+, Bednorz-Müller, Nobel 1987) have d-wave Cooper pairs with much shorter coherence length (~1-2 nm) and unclear pairing mechanism. Helium-3 superfluidity: Cooper pairs of fermionic He-3 atoms.

  • PredictedCooper 1956
  • BCS theory1957 (Nobel 1972)
  • Spin0 (boson)
  • Coherence length Pb~100 nm
  • Energy gap1.76 k_B T_c
  • Helium-3Fermionic Cooper pairs

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Why Cooper pairs matter

  • Superconductivity. The microscopic origin of zero resistance, perfect diamagnetism (Meissner effect), and flux quantization. Without pairing, conventional metals would not superconduct at any achievable temperature.
  • MRI magnets. Niobium-titanium and niobium-tin superconducting wires carry the kiloampere currents needed for medical MRI scanners. Operating well below 9 K, the wires sustain persistent fields without dissipation.
  • Qubit Josephson junctions. Superconducting qubits (used by Google, IBM, Rigetti) consist of Josephson junctions where Cooper pairs tunnel between two superconductors. Pair coherence and the gap protect quantum states from environmental noise.
  • Helium-3 superfluidity. Below 2.5 mK, fermionic He-3 atoms form Cooper pairs with p-wave symmetry. Multiple superfluid phases (A, B, A_1) with distinct order parameters are realized as the temperature, pressure, and magnetic field vary.
  • Particle accelerators. RF cavities at the LHC and at LCLS-II are made of niobium operating below 4.5 K. The lossless oscillation lets the cavities sustain the megavolts-per-meter accelerating gradients needed for high-energy beams.
  • SQUIDs. Superconducting quantum interference devices use Josephson junctions to detect magnetic fields with quantum resolution — widely used in magnetoencephalography, mineral surveying, and dark-matter searches.
  • Topological superconductors. Exotic Cooper pairing in semiconductor-superconductor heterostructures can host Majorana zero modes — candidate building blocks for fault-tolerant topological quantum computers.

The BCS ground state

Bardeen, Cooper, and Schrieffer's 1957 paper proposed a many-body wavefunction that puts every Fermi-surface state into a coherent superposition of being empty or doubly occupied by a paired electron. The variational ground state is parameterized by a self-consistently determined gap function Delta(k). The gap equation Delta = V N(0) integral tanh(E/2 k_B T) over E from Delta to omega_D solves implicitly. At zero temperature it gives Delta(0) approximately 1.76 k_B T_c. As T approaches T_c the gap closes continuously, signaling a second-order phase transition.

The wavefunction is a coherent superposition of pair states — not a localized molecule of two electrons. The number of electrons in the BCS state is not a sharp eigenstate; instead the BCS state is a phase-coherent superposition of states with different particle numbers. This indeterminacy of pair number gives the macroscopic phase that drives Josephson tunneling and flux quantization.

Beyond conventional BCS

  • High-T_c cuprates. La₂CuO₄-derived materials (Bednorz and Müller, 1986) and YBa₂Cu₃O₇ reach T_c above liquid nitrogen temperature. Pairs are d-wave; mechanism still debated — spin fluctuations leading candidate.
  • Iron-based superconductors. Discovered 2008 (Hosono group), reach T_c up to 55 K. Multiple Fermi surface sheets and interesting "s+/-" pairing structure.
  • Heavy-fermion superconductors. Compounds like UPt₃, CeCoIn₃, UTe₂. Effective masses up to 1000 times bare electron mass; some host triplet (spin-1) Cooper pairs.
  • Hydrogen sulfide and hydrides. H₃S at high pressure (155 GPa) reaches T_c above 200 K via phonon-mediated pairing. LaH₁₀ at over 170 GPa reaches T_c above 250 K.
  • Topological superconductors. Sr₂RuO₄ was a candidate p-wave superconductor; recent results favor d-wave. Semiconductor-superconductor nanowire structures pursue Majorana modes.
  • Fermi-degenerate cold atoms. Lithium-6 and potassium-40 in optical traps form atomic Cooper pairs in the BCS regime, with continuous tuning to BEC of tightly bound molecules via Feshbach resonance.

Common misconceptions

  • "Two electrons localized." Cooper pairs are spatially extended, with size 100 nm in lead. Pairs overlap heavily — many other Cooper pairs sit between the two members of any one pair. The picture of a tightly bound molecule is wrong.
  • "Always s-wave." Conventional superconductors are s-wave. Cuprates are d-wave; some heavy-fermion compounds and Sr₂RuO₄ have explored unconventional pairings; helium-3 is p-wave.
  • "Needs metal." Helium-3 forms Cooper pairs of neutral fermionic atoms. Cold-atom systems form Cooper pairs of fermionic alkali atoms in optical lattices. Any system of paired fermions with attractive interaction near a Fermi surface can pair.
  • "Pairs only at zero T." Cooper pairs exist at any temperature below T_c. The gap closes continuously to zero as T approaches T_c. Above T_c there are no stable pairs in conventional superconductors, though "preformed pairs" have been suggested in cuprates.
  • "Resistance is exactly zero." Truly zero in DC. AC superconductors have a finite (but very small) loss because uncondensed quasi-particles still exist at finite T. Lossless DC is the operational definition for magnets and power lines.
  • "BCS explains all superconductors." BCS works extremely well for elemental and alloy conventional superconductors. High-T_c cuprates, heavy fermions, and some hydrides require extensions or alternative mechanisms.

Frequently asked questions

How can two electrons (same charge) attract?

The bare Coulomb interaction is repulsive, but at low energies the lattice provides a phonon-mediated attractive contribution. An electron passing through the ion lattice draws nearby positive ions inward slightly. The displaced ions attract a second electron passing through later. The net effective interaction at energy scales below the Debye frequency is attractive in the s-wave channel. Above the Debye frequency only Coulomb survives. The net attractive shell near the Fermi surface is what binds Cooper pairs.

What is the role of phonons?

Phonons (quantized lattice vibrations) mediate the attractive electron-electron interaction in conventional superconductors. The isotope effect — T_c proportional to 1 over the square root of the ion mass — was an early clue. Heavier isotopes have lower phonon frequencies and slightly lower T_c, predicted by BCS and confirmed experimentally in mercury and tin. In high-T_c cuprates the phonon mechanism is questioned and spin fluctuations are leading candidates.

Why does pairing give zero resistance?

Cooper pairs are bosons (total spin zero) and condense into a coherent macroscopic wavefunction. Scattering an individual pair requires breaking it (energy cost 2 Delta) or scattering the entire condensate (effectively impossible). At temperatures below T_c, no available low-energy excitation can scatter electrons in the condensate, so a current flows without dissipation. The energy gap below the Fermi surface excludes single-particle scattering channels.

What is the BCS gap and how is it measured?

The energy gap Delta is the minimum energy needed to break a Cooper pair. BCS theory predicts 2 Delta at zero temperature equals about 3.53 k_B T_c, equivalent to Delta about 1.76 k_B T_c. Measured by tunneling spectroscopy (Giaever 1960): tunneling current shows a characteristic gap at low voltages. Also measured via infrared absorption, ultrasonic attenuation, and specific heat jumps at T_c. Strong-coupling superconductors (like lead) deviate slightly from BCS, with 2 Delta over k_B T_c about 4 instead of 3.53.

Why are Cooper pairs giant (about 100 nm)?

The pair size, called the Pippard coherence length xi, scales like hbar v_F divided by pi Delta. For lead v_F is about 10 to the 6 m per second and Delta is about 1.4 meV, giving xi about 80 nm. Many other electrons sit between the two members of one pair — pairs strongly overlap. The result is a coherent collective state where individual pairs cannot be localized as a tight bound state. High-T_c cuprates have shorter xi (1 to 2 nm) because Delta is much larger and v_F smaller.

What is d-wave pairing in cuprates?

Conventional BCS superconductors have an s-wave pair wavefunction: the gap is uniform around the Fermi surface. Cuprate high-T_c superconductors instead have a d-wave gap that vanishes along certain directions and changes sign across them. The pairing channel involves angular momentum L=2 rather than L=0. The d-wave nature has been confirmed by phase-sensitive experiments and angle-resolved photoemission, and it indicates a nonphononic pairing mechanism (likely magnetic or spin-fluctuation mediated).