The Pairing Instability
In 1956, Leon Cooper showed that the Fermi sea of electrons is unstable against the formation of even a weakly bound pair. When one electron slightly deforms the positive ion lattice, a second electron is attracted to the resulting positive charge concentration. This phonon-mediated attraction overcomes Coulomb repulsion at low temperatures, binding electrons into Cooper pairs with a characteristic size of hundreds of nanometers — the coherence length ξ₀.
The Energy Gap
BCS theory predicts that the superconducting ground state is separated from excited (normal) states by an energy gap Δ. At absolute zero, 2Δ₀ = 3.53 kBTc in the weak-coupling limit. As temperature rises toward Tc, thermal energy progressively breaks Cooper pairs, and the gap shrinks following the characteristic BCS curve until it vanishes continuously at the critical temperature — a second-order phase transition.
Macroscopic Quantum Coherence
All Cooper pairs occupy the same quantum state, described by a single complex order parameter Ψ = |Ψ|e^(iφ). This macroscopic wavefunction is what makes superconductivity extraordinary — it is quantum mechanics visible at human scales. The phase coherence across the entire sample enforces zero resistance: scattering would require breaking the global phase relationship, which costs energy Δ per pair.
Beyond Weak Coupling
While BCS theory works beautifully for conventional superconductors like aluminum (Tc = 1.2 K) and niobium (Tc = 9.2 K), strong-coupling materials push the gap ratio 2Δ₀/kBTc above 3.53. Lead (Tc = 7.2 K) has a ratio of 4.3, requiring the Eliashberg extension of BCS theory. High-temperature cuprate superconductors remain beyond the BCS framework entirely, with d-wave pairing symmetry and gap ratios exceeding 5.