BCS Energy Gap & London Penetration Depth

Material

Controls

Temperature T 4.6 K

Applied field B₀ 40 mT

Slab depth shown 150 nm

State

Normal state

Energy gap Δ(T)

0.00 meV

Condensate nₛ/n

0 %

Penetration depth λ(T)

— nm

How it works

Below the critical temperature T꜀, electrons bind into Cooper pairs and a gap Δ opens in the excitation spectrum. The BCS interpolation formula (accurate to ~2% across the whole range) gives:

Δ(T) = Δ(0) · tanh( 1.74 · √(T꜀/T − 1) )     for T < T꜀
Δ(T) = 0                                      for T ≥ T꜀

The two-fluid (Gorter–Casimir) model splits the electron density n into a normal fraction and a superfluid (paired) fraction that condenses as T falls:

nₛ/n = 1 − (T/T꜀)⁴

Superfluid Cooper pairs carry a lossless screening current that expels an applied magnetic field from the bulk (the Meissner effect). The field decays exponentially with depth x from the surface, over a temperature-dependent London penetration depth:

B(x) = B₀ · exp(−x / λ(T))
λ(T) = λ(0) / √(nₛ/n) = λ(0) / √(1 − (T/T꜀)⁴)

As T → T꜀, nₛ/n → 0 and λ(T) → ∞ — the screening current vanishes and the field penetrates the whole sample, exactly the normal-state limit.

Material — swaps Δ(0), T꜀ and λ(0) for real elemental superconductors (BCS-derived tabulated gap values).
Temperature slider — sweeps T from 0 up to 1.2·T꜀; watch the gap close and the field arrows lengthen (penetrate deeper) as T꜀ is crossed.
Applied field — scales the field-arrow magnitude at the surface (x = 0); it does not change λ(T) itself.
Cooper-pair spheres — cyan dumbbells are paired (condensed) electrons, dim grey spheres are thermally-excited normal quasiparticles; their ratio tracks nₛ/n live.

Real-world relevance: this is the physics behind MRI-magnet and particle-accelerator superconducting coils (Nb-Ti, Nb₃Sn) — the coil must stay well below T꜀ or the Meissner screening collapses and the field (and the persistent current) dumps.

Applied field B(x) Cooper pair (condensed) Normal quasiparticle

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