Superconductivity: Zero Resistance and the Meissner Effect

Exactly zero, not very small

In 1911 Heike Kamerlingh Onnes cooled mercury to 4.2 K and watched its electrical resistance drop instantaneously to exactly zero. Not asymptotically small — zero. A current induced in a superconducting ring has been measured to persist for years with no detectable decay, with an experimental lower bound on the decay time exceeding 100,000 years. In an ordinary metal, resistance comes from electrons scattering off lattice vibrations (phonons) and impurities; cooling reduces phonon scattering but a residual resistance from impurities always remains. A superconductor does something categorically different below its critical temperature T_c: it enters a macroscopic quantum state in which scattering is quantum-mechanically forbidden.

Cooper pairs: electrons that attract

Electrons repel each other electrostatically, so how can they pair up? An electron moving through the crystal lattice briefly pulls nearby positive ions toward it, leaving a transient wake of excess positive charge; a second electron arriving slightly later is attracted into that wake. The net effect, mediated by a phonon (a quantum of lattice vibration), is a weak effective attraction. Because this interaction is retarded in time, the two electrons of a Cooper pair — with equal and opposite momenta and opposite spin, total spin zero — never need to be physically close; their average separation, the coherence length ξ, runs 10–1000 nm. Crucially, a pair of electrons is a boson, and bosons can all condense into the same quantum ground state.

BCS theory and the energy gap

Bardeen, Cooper and Schrieffer (1957) built the full theory on this insight: at T = 0, essentially all electrons near the Fermi surface pair up and condense into a single macroscopic wavefunction. This condensation opens an energy gap 2Δ around the Fermi energy, and breaking a pair to create a scattering state costs at least that much energy. At T ≪ T_c, thermal energy k_BT is far smaller than Δ, so scattering is frozen out entirely and resistance vanishes.

Δ = 2ħω_D · exp(−1 / N(0)V)          // BCS gap equation, T = 0
k_B·T_c ≈ 1.13 · ħω_D · exp(−1 / N(0)V)
// ω_D = Debye frequency, N(0) = density of states at E_F, V = pairing strength

Because Δ depends exponentially on the pairing strength N(0)V, conventional (phonon-mediated) superconductors have stubbornly low T_c — lead tops out at 7.2 K, niobium at 9.3 K — and modest improvements to the material don't move the number by much.

The Meissner effect

Cool a metal below T_c while it sits in a magnetic field, and you might expect the field to stay trapped inside. Instead the field is actively expelled — the superconductor spontaneously generates surface currents that exactly cancel the field in the bulk. This is the Meissner effect (Meissner & Ochsenfeld, 1933), and it proves superconductivity is a distinct thermodynamic phase, not merely "perfect conductivity" with zero resistance. The London equations describe it phenomenologically: B(x) = Bext · exp(−x/λL), where the penetration depth λ_L is typically 20–500 nm. This expulsion of flux is exactly what makes a magnet levitate above a superconductor.

Type I, Type II, and cuprates that break the rules

Type I superconductors (mercury, lead, tin) have one critical field H_c: below it, perfect Meissner expulsion; above it, superconductivity collapses abruptly. Type II superconductors (niobium, Nb₃Sn, MgB₂, YBCO) have two critical fields; between H_c1 and H_c2 the field penetrates as quantised flux tubes called Abrikosov vortices, each carrying exactly one flux quantum Φ₀ = h/2e. This mixed state is what lets Type II materials survive the enormous fields used in MRI magnets and the LHC's Nb₃Sn dipoles. In 1986 Bednorz and Müller discovered lanthanum barium copper oxide superconducting at 35 K — far above what phonon-mediated BCS pairing predicts — and within a year YBCO reached 93 K, above liquid nitrogen's boiling point. Nearly 40 years later, the pairing mechanism in these d-wave cuprate superconductors is still an open problem in condensed-matter physics.

Frequently asked questions

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Cool a material through T_c in Superconductivity and watch resistance vanish and magnetic flux get expelled, until a strong enough field kills the Meissner state — all in your browser.