Zero resistance, Meissner and critical fields

No experiment has ever measured zero; what a measurement gives is an upper limit. Push 100 mA through a strip of cross-section 1.0 mm × 0.10 mm and read the voltage across taps 10 mm apart: a 1 nV noise floor caps the resistance at R = 10⁻⁸ Ω and so the resistivity at ρ = RA/L = 10⁻¹³ Ω m. That is already five orders of magnitude below copper's 1.7×10⁻⁸ Ω m, and it is nowhere near the best bound available. Launch a current round a closed superconducting ring instead and watch its field decay as e^(−Rt/L). File and Mills followed such a current by NMR in 1963 and saw no decay in a run implying a time constant above 10⁵ years; for a ring of inductance 10⁻⁷ H that is R < 3×10⁻²⁰ Ω, and for a loop 0.1 m round of 1 mm² section, ρ below about 3×10⁻²⁵ Ω m. Seventeen orders below copper is still not zero, and the honest statement is the inequality.

Suppose only that the resistance vanishes. Then E = 0 inside, and Faraday's law ∇×E = −∂B/∂t forces ∂B/∂t = 0: whatever field threaded the sample when scattering stopped stays there for good. That is a perfect conductor, and its final state depends on its history — cool it in a 10 mT field, then switch the magnet off, and 10 mT of trapped flux remains, held by a surface current that never decays. Meissner and Ochsenfeld did the experiment on tin and lead in 1933 and found the opposite. Cooling through T_c in a steady field pushes the flux out as the transition passes, and the sample ends field-free whether the field was applied before or after cooling. Zero-field-cooled and field-cooled runs land on the same state, which is the signature of a thermodynamic phase rather than a frozen initial condition. No algebra takes you from R = 0 to that, so it has to be put in — and the second London equation, ∇×Jₛ = −B/μ₀λ², is where it goes in.

Take the London constitutive law ∇×Jₛ = −(1/μ₀λ²)B together with Ampère's law in its static form ∇×B = μ₀Jₛ. Curl the second, use ∇⋅B = 0, and the vector identity leaves ∇²B = B/λ². For a superconductor filling x > 0 with a uniform field B₀ parallel to the face, this reduces to d²B/dx² = B/λ², whose general solution is A e^(−x/λ) + C e^(+x/λ). The growing term is thrown away because B must stay finite deep inside, and matching B(0) = B₀ at the face fixes A, leaving B(x) = B₀ e^(−x/λ). Now read λ off the coefficient: λ = √(m/μ₀ nₛ e²). With nₛ = 4.0×10²⁸ m⁻³ — a sizeable fraction of the 10²⁹ m⁻³ or so conduction electrons in a typical metal — the denominator is μ₀ nₛ e² = 1.29×10⁻¹⁵, the quotient is 7.06×10⁻¹⁶ m², and λ = 26.6 nm. Measured depths run from tens of nanometres in elemental metals to a few hundred in the cuprates. Temperature enters only through nₛ(T), and it is put in by hand: the two-fluid model gives λ(T) = λ(0)/√(1 − (T/T_c)⁴), which diverges at T_c because the screening fluid has run out.

Expelling flux is not free. Holding B = 0 inside against an applied field B₀ costs magnetic energy density B₀²/2μ₀, and the ordered state can only pay for it out of the condensation energy it gained. The two balance at the thermodynamic critical field: B_c²/2μ₀ equals the condensation energy per unit volume. For lead, B_c(0) = 80.3 mT gives 0.0803²/(2 × 1.257×10⁻⁶) = 2.6 kJ m⁻³, and shared among 3.3×10²⁸ atoms per cubic metre that is 0.5 μeV each — over a thousand times less than k_BTc = 0.62 meV, so the ordered state lies only very slightly below the normal metal it replaces. The empirical curve B_c(T) = B_c(0)[1 − (T/T_c)²] carries it to zero at T_c. A transport current does the same damage as an external magnet because it makes its own field: a wire of radius a carrying I has surface field μ₀I/2πa, so it quenches when that reaches B_c. That is Silsbee's rule, I_c = 2πa B_c/μ₀, and it says the critical current of a type I wire is a surface property — double the radius and I_c doubles, it does not quadruple.

A normal-superconducting boundary carries an energy per unit area set by two lengths: the coherence length ξ, over which the order parameter recovers, and λ, over which the field dies. When ξ is long compared with λ the interface costs energy, subdividing is unfavourable, and the sample stays wholly flux-free up to B_c and then goes normal all at once — type I. When λ is long compared with ξ — precisely, when κ = λ/ξ exceeds 1/√2 — the interface energy is negative and the sample gains by making as much boundary as it can. Above B_c1 flux enters as vortices: a normal core of radius about ξ, circulating supercurrent spread over λ, and exactly one flux quantum Φ₀ = h/2e = 2.068×10⁻¹⁵ Wb through each. Their areal density is nᵥ = B/Φ₀, so at 5 T there are 2.4×10¹⁵ per square metre on a triangular lattice of spacing 22 nm. The material stays superconducting between the cores until they overlap at μ₀H_c2 = Φ₀/2πξ², which is why every high-field magnet is type II: NbTi at 4.2 K works to about 11 T and Nb₃Sn beyond 20 T, while lead has given up at 53 mT.

The London equations are a constitutive guess, and it is worth naming what they do not deliver. nₛ is fitted from the measured λ, not computed; there is no prediction of T_c, no reason the electrons should order, and no gap. The theory is also strictly local — Jₛ at a point is taken to respond to B at that point — and that fails whenever the coherence length exceeds λ, as it does in aluminium and lead, where the measured penetration depth comes out larger than the London value and needs Pippard's non-local kernel instead. The vortex picture has its own gap: it says where flux sits, not whether it stays. A vortex in a current-carrying wire feels a force per unit length J × Φ₀, and a vortex that moves dissipates, so the critical current of a real type II conductor is set by how well defects pin the lattice. That is metallurgy, and it is why the same NbTi alloy can differ tenfold in J_c between two heat treatments. Why the electrons pair at all is the next topic.