How to use
The calculator is meant to be straightforward. Enter the critical temperature in Kelvin, enter the Fermi velocity in meters per second, and optionally supply a custom ratio if you want something other than the default weak-coupling value. Then press the compute button. The result line reports the zero-temperature gap in meV, the coherence length in nm, and the ratio used in the calculation.
- Enter the critical temperature Tc in Kelvin (K).
- Enter the Fermi velocity vF in meters per second (m/s).
- Optionally enter the ratio 2Δ0/(kBTc). Leave it blank to use 3.53.
- Click Compute Gap and Coherence.
- If you want to reuse the output in notes or an email, click Copy Result.
A practical way to use the page is to run it twice. First, leave the ratio blank so every material is judged against the same weak-coupling baseline. Second, repeat the calculation with an experimentally reported ratio if one exists. The difference between the two outputs often gives a quick sense of how strongly the material departs from the simplest BCS picture.
Practical tip: if you are comparing several samples, keep the ratio fixed for the first pass. That way, changes in Δ0 mainly follow Tc, and changes in ξ0 mainly reflect the competition between vF and the gap scale. After that, bring in custom ratios to see how strong coupling or anisotropy changes the story.
The calculator uses two standard clean-limit BCS relations. The first relation determines the zero-temperature gap parameter from the chosen ratio r:
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Gap from ratio:
Delta zero equals one half times ratio times k sub B times T sub c.
where r = 2Δ0/(kBTc).
The second relation estimates the BCS coherence length at zero temperature in the clean limit:
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BCS coherence length:
xi zero equals h bar v sub F divided by pi delta zero.
Those two lines already explain most of the scaling. A larger Tc or a larger ratio r makes Δ0 larger. Because ξ0 is inversely proportional to Δ0, that same increase tends to make the coherence length shorter. By contrast, a larger vF makes ξ0 larger at fixed gap. This is why a material with a modest gap but a large Fermi velocity can have a very long coherence length, while a stronger-coupling or higher-Tc material often ends up with a much shorter one.
Units and constants used internally are simple and explicit:
- Tc is entered in K, vF in m/s, and the ratio is dimensionless.
- kB = 1.380649×10−23 J/K and ħ = 1.054571817×10−34 J·s.
- Δ0 is calculated in joules and then converted to meV using 1 meV = 1.602176634×10−22 J.
- ξ0 is calculated in meters and displayed in nanometers.
One small but important convention is worth stating clearly. In many spectroscopic plots, authors talk about a gap width of 2Δ. This calculator outputs Δ0, the single-particle gap parameter, while the ratio input is defined using 2Δ0. That is standard, but it is easy to mix up if you are reading several papers quickly.
Worked examples
A worked example helps anchor the scales. These are not meant to be exact material databases. They are quick order-of-magnitude checks that show how the formulas behave.
Example A: a weak-coupling, low-Tc case
Suppose Tc = 1.2 K and vF = 2.0×106 m/s, and suppose you keep the default ratio r = 3.53. The gap is then roughly half of 3.53 times kBTc, which works out to about 0.18 meV. Plugging that gap into the coherence-length formula gives a value around 1600 nm. That is a long coherence length, exactly the sort of result you expect in a very clean, low-Tc elemental superconductor.
Example B: a stronger-coupling material with a larger gap ratio
Now take Tc = 7.2 K, vF = 1.8×106 m/s, and use a larger ratio such as r ≈ 4.3. The larger Tc already pushes the gap up, and the larger ratio pushes it up again. Since ξ0 is inversely proportional to the gap, the coherence length shortens dramatically. Instead of landing in the micron range, you move toward tens or hundreds of nanometers. Even without a full theory, that quick estimate tells you the superconducting state is much more tightly bound in space.
Example C: what changes what?
It is often helpful to think in terms of sensitivity. At fixed ratio, doubling Tc doubles Δ0. At fixed Δ0, doubling vF doubles ξ0. If you hold vF fixed and increase either Tc or the ratio, Δ0 rises and ξ0 falls. This simple competition between pairing energy and Fermi-surface velocity explains a surprising amount of qualitative behavior across conventional superconductors.
A short numerical check makes that relationship concrete. Imagine a sample with Tc = 9 K and vF = 3×105 m/s. With r = 3.53 you get a gap of order 1.37 meV. If you keep the same vF but switch to r = 4.3, the gap rises to about 1.67 meV. The coherence length must then shrink because the denominator in ξ0 = ħvF/(πΔ0) is larger. That is exactly the kind of change you can use this page to estimate in seconds.
Reference values (illustrative)
The table below gives a few commonly cited, order-of-magnitude examples. Treat it as a quick benchmark rather than a precise database. Exact values depend on sample purity, anisotropy, band structure, and which Fermi velocity is appropriate for the band that matters most in the experiment you care about.
It is also useful to distinguish this BCS coherence length from a Ginzburg–Landau coherence length quoted near Tc. The latter is temperature dependent and often extracted from critical-field measurements. The calculator here provides a T = 0 clean-limit BCS estimate. That distinction prevents many common comparison mistakes.
The page is intentionally honest about where the model stops. A quick estimate is valuable, but only if you remember what it assumes.
If you need a number for a specific experimental regime such as dirty films, proximity structures, strongly anisotropic superconductors, or unconventional pairing states, use the result here as a first estimate rather than a final answer. The more your sample departs from clean, isotropic, weak-coupling BCS assumptions, the more important a specialized model becomes.
Leave the field blank. The calculator will use 3.53, the classic weak-coupling BCS value for an isotropic s-wave superconductor. That gives you a consistent baseline and is usually the right first estimate.
