Cooper pairing and the BCS gap

Cooper's 1956 question was deliberately small: freeze a filled Fermi sea, add exactly two electrons above it at k↑ and −k↓, and let them attract weakly. In free space a 3-D attraction must exceed a threshold before it binds, because the density of low-energy states vanishes as √E and there is nothing to sum. The sea removes that. Every state below E_F is occupied, so the two extra electrons may only scatter within a shell of thickness ħω_D above E_F, where the density of states is essentially the constant N(0). The bound-state condition then sets the integral of dξ/(2ξ − E) from 0 to ħω_D equal to 1/N(0)V; that integral is a logarithm, and a logarithm can be made to equal 1/N(0)V no matter how small V is. Out comes a bound state for every non-zero attraction, with binding energy 2ħω_D exp(−2/N(0)V). The normal metal is not a metastable minimum; it is not a minimum at all.

An electron moving through the lattice pulls the positive ions towards its track. The ions are some 10⁵ times heavier, so they arrive late: the lattice needs roughly one Debye period, 2π/ω_D ≈ 1.7 × 10⁻¹³ s for Θ_D = 275 K, to build the distortion. In that time an electron travelling at v_F ≈ 10⁶ m s⁻¹ has gone about 170 nm — hundreds of lattice spacings. A second electron arriving later sees a region of excess positive charge and is attracted to it, while the first electron is far too distant for the screened Coulomb repulsion, which reaches only about 0.1 nm, to matter. Repulsion and attraction are separated in time, not in strength. The exchange only works for electrons within ħω_D of E_F — 23.7 meV in niobium, against E_F ≈ 5 eV, so under one per cent of the sea is involved.

Cooper froze the sea; BCS let every electron pair at once and demanded consistency. The condition is 1 = N(0)V × ∫ dξ/√(ξ² + Δ²) taken from 0 to ħω_D, which is N(0)V⋅arcsinh(ħω_D/Δ), and for ħω_D ≫ Δ it inverts to Δ(0) = 2ħω_D exp(−1/N(0)V) — the same exponential as the single pair but with half the exponent, because the pairs are now condensing against a sea that is itself paired. The excitation spectrum becomes Eₖ = √(ξₖ² + Δ²), so the cheapest excitation costs Δ and breaking a pair costs 2Δ. Repeating the integral at finite T, with a tanh(E/2kBT) factor, gives kB Tc = 1.13 ħω_D exp(−1/N(0)V), where 1.13 is 2 exp(0.5772)/π = 1.134 and 0.5772 is Euler's constant. Divide the two: the exponential and ħω_D both cancel, leaving 2Δ(0)/kB Tc = 4/1.134 = 3.53, a pure number with nothing fitted in it.

Tunnelling is the most direct. In a normal–insulator–superconductor junction no quasiparticle current flows until eV = Δ; in a symmetric superconductor–insulator–superconductor junction a quasiparticle must be made on both sides, so the threshold is eV = 2Δ. For tin that is 1.13 mV, and Giaever's 1960 tunnelling measurements of such thresholds won him a share of the 1973 Nobel Prize. Heat capacity is the second: below about Tc/2 the electronic term is exponentially activated, C ∝ exp(−Δ/kBT), so ln C against 1/T is a straight line of slope −Δ/kB, and at Tc it jumps by 1.43 times the normal-state electronic heat capacity. Third, far-infrared absorption switches on when a photon can break a pair, at ħω = 2Δ — 1.13 meV for tin, which is 274 GHz. Three different apparatus, one number.

Nothing above proves the carrier has charge 2e; two 1961 experiments did. Deaver and Fairbank and, independently, Doll and Näbauer found the flux trapped in a superconducting ring quantised in units of h/2e = 2.068 × 10⁻¹⁵ Wb — half the h/e a single-electron condensate would give. Josephson added the phase in 1962. Link two superconductors weakly and they share a gauge-invariant phase difference φ, with I = Ic sin φ flowing at zero voltage; hold a voltage across the link and the phase winds, dφ/dt = 2eV/ħ, so the current oscillates at f = 2eV/h = 483.598 MHz per microvolt. A 25 µV bias gives 12.09 GHz. Irradiate the junction with microwaves of frequency ν and flat voltage steps appear at exactly V = nhν/2e, which is how the SI volt has been realised since 2019, with KJ = 483597.8484 GHz/V fixed by definition.

Put a cuprate into the formulae and they break. Hg-1223 superconducts at 133 K; with Θ_D ≈ 400 K the weak-coupling expression demands N(0)V = 0.82, far outside the N(0)V ≪ 1 that produced the exponential in the first place. Its measured 2Δ/kB Tc is 5.2, not 3.53, and its isotope exponent at optimal doping is about 0.02 rather than 0.5, so the lattice is not setting the scale. Phase-sensitive tricrystal experiments and ARPES nodes give a d(x²−y²) order parameter that changes sign around the Fermi surface, where BCS gives an isotropic s-wave gap. Pairs survive — flux is still quantised in h/2e — but the glue does not. The pressurised hydrides run the other way: H₃S at 203 K and 155 GPa behaves like strong-coupling BCS, with a large isotope shift, and there the live issue is independent replication, not mechanism.