Superconductor calculator: BCS gap from Tc, London penetration depth, coherence length, κ and critical fields

What it does

Below its critical temperature T_c a superconductor carries current with no resistance and pushes magnetic field out. A few numbers describe almost everything about it:

• the energy gap Δ, the energy needed to break a Cooper pair. BCS theory links it to T_c: Δ₀ ≈ 1.764 k_BT_c, and the gap shrinks to zero as T approaches T_c;
• the London penetration depth λ, how far a magnetic field leaks in;
• the coherence length ξ, the size of a Cooper pair;
• their ratio, the Ginzburg-Landau parameter κ = λ/ξ. Below 1/√2 the material is type I and expels field until it goes normal at B_c; above, it is type II and lets field in as vortices between B_c1 and B_c2.

This calculator gives the gap (solving the BCS gap equation at your temperature), the gap frequency (the photon frequency that breaks pairs, useful for detectors), λ(T), κ and the type, the critical fields and their temperature dependence, the Pauli limit set by spin splitting, the depairing current density, and a field-temperature phase diagram.

How to use it

1. Type T_c and the temperature you care about.
2. Keep the gap ratio 2Δ₀/k_BT_c at 3.528 (weak-coupling BCS) or raise it for a strong-coupling material.
3. Either type λ(0) and ξ(0) if you know them, or choose Estimate from the electrons and give the electron density, effective mass, Fermi velocity and, for a dirty (alloy) superconductor, the mean free path.
4. Read the results, look at the phase diagram, and download the table against temperature or a report.

Worked example

Take a made-up superconductor with T_c = 9.2 K, λ(0) = 40 nm and ξ(0) = 35 nm. The page shows Type II (κ = 1.14): Δ₀ = 1.4 meV, B_c2(0) ≈ 0.269 T. The gap frequency is 676 GHz. At 4.2 K the gap is still 97 % of its full value, 1.359 meV, and λ has grown to 40.9 nm. The thermodynamic field is 0.166 T and B_c1 is 13.7 mT. The Pauli limit, 17.1 T, is far above B_c2, so orbital effects limit the field here.

Switch to Estimate from the electrons with 10²⁸ electrons per m³ and v_F = 10⁶ m/s: λ_L = 53.1 nm and ξ₀ = 150 nm, so κ = 0.355 and the material is type I. Add a mean free path of 10 nm and ξ shrinks and λ grows until it becomes type II: this is why alloys are type II.

Tips

• B_c1 from ln κ is only rough when κ is close to 1; it is good for κ above about 2.
• The (1 − t²) law is a convenient approximation. Real B_c2(T) curves are often closer to straight near T_c.
• The gap frequency sets the upper limit for low-loss microwave resonators and the threshold of superconducting photon detectors.

FAQ

Does this work for high-T_c cuprates or iron-based superconductors? The formulas are BCS and Ginzburg-Landau estimates for conventional, isotropic superconductors. Unconventional materials have anisotropic gaps and need more careful models; use the numbers as rough guides only.

Why does a finite mean free path make λ larger? Scattering reduces how well the superfluid screens the field: λ ≈ λ_L √(1 + ξ₀/ℓ).

Is this good enough to design a magnet? No. It is for study and estimates, not a substitute for measured critical fields and professional judgement.

Last updated 2026-11-09.