Fermi Energy Calculator
Enter carrier density and effective mass to compute the Fermi energy, wavevector, velocity and Fermi temperature of a free-electron gas.
Input Data
Results
At a glance:The Fermi energy E_F is the highest occupied single-particle energy of a free-electron gas at T = 0 K. It relates to carrier density n by E_F = (ħ²/2m*)·(3π²n)^(2/3), with ħ the reduced Planck constant and m* the effective mass. The Fermi wavevector k_F = (3π²n)^(1/3), Fermi velocity v_F = ħk_F/m*, and Fermi temperature T_F = E_F/k (often enormous, e.g. ~10⁴–10⁵ K for metals, meaning electrons are highly degenerate at room temperature). The Fermi-Dirac distribution f(E) = 1/(e^((E−E_F)/kT)+1) describes occupancy; at T=0 all states below E_F are filled, above empty.
Formula
E_F = (ħ²/2m*)·(3π²n)^(2/3)
k_F = (3π²n)^(1/3)
v_F = ħk_F/m*
T_F = E_F/k
$$k_F = \left(3\,\pi^2\,n\right)^{1/3}$$$$E_F = \frac{\hbar^2}{2\,m^*}\,\left(3\,\pi^2\,n\right)^{2/3}$$$$T_F = \frac{E_F}{k_B}$$How to Use
- Enter the carrier density n (1/m³).
- Enter the effective mass m* (kg).
- The calculator returns E_F (J and eV), k_F, v_F and T_F.
Case Studies
Copper electrons
n ≈ 8.5×10²⁸ m⁻³, m* ≈ 9.1×10⁻³¹ kg.
E_F ≈ 7.0 eV, v_F ≈ 1.57×10⁶ m/s.
T_F ≈ 8.1×10⁴ K — degenerate at room T.
FAQ
Why is the Fermi temperature so high?
Because electrons are light and dense; even at room temperature kT ≪ E_F, so most electrons are 'frozen' in the degenerate sea and only those near E_F participate in conduction.
What is effective mass?
m* encodes how an electron responds to forces in a crystal lattice (including band curvature), differing from the free-electron mass. It can even be negative near band tops.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.