How this calculator works
An ideal three-dimensional electron gas fills momentum states up to kF at zero temperature. These equations use the electron rest mass, two spin states and no band interactions. Presets are rounded textbook free-electron Fermi energies, not precision predictions for real band structures. At finite temperature the occupation uses chemical potential approximately equal to the zero-temperature Fermi energy, appropriate for temperatures well below TF. At 0 K it shows the limiting step with half occupation at the boundary. Wave number here is angular wave number in reciprocal metres, not a length.
Formula
kF = (3pi² n)^(1/3); EF = hbar² kF²/(2me); vF = hbar kF/me; TF = EF/kB; f(E) = 1/[exp((E-EF)/(kB T))+1].
Worked example
With EF = 7 eV and T = 300 K, a state at 7 eV has occupation 0.5. A state at about 7.028401325 eV has occupation 0.25. The free-electron Fermi temperature is about 81232 K.
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Frequently asked questions
Is Fermi energy the same as chemical potential?
They coincide at zero temperature. This tool approximates chemical potential by EF at low finite temperature.
Why are the wave number units reciprocal metres?
A wave number measures spatial oscillation rate. The angular convention is k = 2pi/lambda.
Can I use these values for semiconductors?
A semiconductor needs band edges, effective masses, doping and charge neutrality. The simple metal electron-gas model does not include those effects.