A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
BCS theory explains superconductivity as a phase transition where electrons form Cooper pairs via a phonon-mediated attractive interaction. The ground state is a superfluid of paired electrons with energy gap Δ; excitations cost finite energy, yielding zero resistance. Theory predicts isotope effects and specific heat discontinuities observed experimentally.
From your study of the ideal Fermi gas and Fermi-Dirac statistics, you know that at low temperatures electrons fill states up to the Fermi energy and the system behaves as a degenerate quantum gas. The puzzle of superconductivity — discovered experimentally in 1911 but unexplained until 1957 — is that below a critical temperature T_c, metals suddenly acquire zero electrical resistance and expel magnetic fields. The key insight of Bardeen, Cooper, and Schrieffer is that the Fermi sea is unstable to even a tiny attractive interaction between electrons, causing them to pair up and condense into a qualitatively different ground state.
The phonon-mediated attraction works like this: electron 1 passes through the lattice and attracts the positive ions toward its path. The ions respond slowly (their mass is ~10⁴ times the electron mass), so by the time electron 2 arrives at the same spot a short time later, the lattice has relaxed and the local positive charge density is still elevated. Electron 2 is attracted to this residual positive polarization left by electron 1. The net effect is a weak, retarded, attractive interaction between the two electrons, mediated by the lattice vibrations (phonons). This attraction competes with the direct Coulomb repulsion; when the phonon-mediated term wins, pairing occurs.
Cooper's theorem (1956) showed that this pairing has a dramatic consequence: two electrons near the Fermi surface with opposite momenta (k, −k) and opposite spins (↑, ↓) form a bound state — a Cooper pair — no matter how weak the attractive interaction, because the filled Fermi sea below them blocks all scattering except those preserving total momentum k + (−k) = 0. BCS theory extends this to all electrons simultaneously: the ground state is a coherent superposition of paired states, and the many-body wavefunction has a definite quantum mechanical phase. This phase coherence is the essence of superconductivity — the paired electrons move as a collective quantum object that cannot scatter incoherently off impurities.
The energy gap Δ is the binding energy per electron in a Cooper pair, and it is the key observable prediction of BCS theory. To break a pair and create an excitation costs a minimum energy 2Δ; no excitations are available below this threshold. Because all scattering processes require creating excitations, and no excitations exist below 2Δ at low temperatures, the electrical resistance is exactly zero — current flows without dissipation. The gap also predicts a specific heat discontinuity at T_c (a jump, not a divergence) and an isotope effect: T_c ∝ M−1/2 where M is the atomic mass, because heavier atoms vibrate more slowly, weakening the phonon coupling. Both predictions were confirmed experimentally and provided strong evidence for the phonon-pairing mechanism before the full BCS theory was published.