Engineering & Science
London Magnetic Penetration Depth & BCS Superconducting Energy Gap Formula
Calculate the two-fluid temperature-dependent London penetration depth lambda_L(T) = lambda_0 / SQRT(1 - (T / Tc)^4) in nm and weak-coupling BCS Cooper-pair energy gap 2*Delta_0 = 3.528 * kB * Tc in meV.
Model Meissner magnetic screening and Cooper-pair breaking in superconducting qubits & RF cavities!
Below critical temperature Tc, a superconductor expels static magnetic fields from its interior (the Meissner effect) except within a thin surface screening layer of London penetration depth lambda_L! In the Gorter-Casimir two-fluid model, superfluid Cooper-pair density decreases with temperature as 1 - (T / Tc)^4, causing London penetration depth to diverge smoothly as T approaches Tc: lambda_L(T) = lambda_0 / SQRT(1 - (T / Tc)^4)!
Meanwhile, Bardeen-Cooper-Schrieffer (BCS, 1972 Nobel Prize) theory shows that breaking a Cooper pair at T -> 0 requires photon energy at least equal to the spectroscopic superconducting gap 2 * Delta_0 = 3.528 * kB * Tc (where kB = 0.08617333 meV/K)! For pure Niobium (Tc = 9.25 K, lambda_0 = 39.0 nm) cooled in liquid helium at T = 4.20 K, lambda_L(4.2 K) = 39.86 nm (barely above T=0!) and the pair-breaking gap is 2 * Delta_0 = 2.812 meV (~680 GHz)—explaining why 5 GHz superconducting transmon qubits operate far below the pair-breaking threshold, while warming to 8.50 K balloons lambda_L to 72.80 nm!
| Niobium Operating Temp (Tc = 9.25 K, lambda_0 = 39 nm) | Reduced Temp T / Tc | London Depth lambda_L(T) | BCS Energy Gap 2*Delta_0 |
|---|---|---|---|
| Liquid Helium Bath (T = 4.20 K) | 0.4541 ((T/Tc)^4 = 0.0425) | 39.86 nm | 2.812 meV (~680 GHz) |
| Warm Cryocooler Stage (T = 8.50 K) | 0.9189 ((T/Tc)^4 = 0.7133) | 72.80 nm | 2.812 meV (T=0 reference) |
| Above Critical Temp (T = 10.0 K >= Tc) | 1.0811 (>= 1!) | "Normal Metal State (T >= Tc)" | Superconductivity Quenched! |
Temperature-Dependent London Penetration Depth lambda_L(T) in Nanometers (nm)
Checks T < Tc so the Meissner screening state is active, then evaluates lambda_0 / SQRT(1 - (T / Tc)^4).
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0), "Invalid Superconductor Inputs", IF(B4 >= B2, "Normal Metal State (T >= Tc)", ROUND(B3 / SQRT(1 - ((B4 / B2) ^ 4)), 2)))BCS Weak-Coupling Zero-Temperature Pair-Breaking Gap 2 * Delta_0 (meV)
Evaluates 3.528 * kB_meV_K * Tc = 3.528 * 0.08617333 * Tc in milli-electron-volts.
=IF(B2 <= 0, "Invalid Superconductor Inputs", ROUND(3.528 * 0.08617333 * B2, 3))Try the example
Use the original values or change them to test your understanding.