What does f(E) = 0.5 mean?

When E equals the Fermi energy E_f, the exponent (E - E_f)/kT is zero, so exp(0) = 1 and f(E_f) = 1/(1+1) = 0.5. This means there is always a 50 % chance that the state at the Fermi energy is occupied, regardless of temperature.

What happens at absolute zero?

At T = 0 K the distribution becomes a perfect step (Heaviside) function: f(E) = 1 for all E E_f. This calculator uses T = 1 K as the practical minimum to avoid division by zero.

How is this different from the Bose-Einstein distribution?

The Fermi-Dirac distribution applies to fermions (particles with half-integer spin, such as electrons) which obey the Pauli exclusion principle — at most one particle per state. The Bose-Einstein distribution applies to bosons (integer spin, such as photons) which have no such restriction and can have any occupation number.

What is the thermal energy kT?

kT is the product of the Boltzmann constant (k = 8.617 x 10^-5 eV/K) and the absolute temperature T. It sets the energy scale of thermal broadening. At room temperature (300 K), kT ≈ 0.0259 eV, so the Fermi edge is spread over roughly ±2kT ≈ ±0.05 eV.

What energy range does the plot cover?

The plot spans from E_f - 10kT to E_f + 10kT by default, which captures the full transition from near-1 occupation below the Fermi level to near-0 occupation above it. At very low temperatures this range narrows to ±0.02 eV.

Can I use this for semiconductors?

Yes. For intrinsic semiconductors the Fermi level sits near the middle of the band gap. You can set E_f to that midgap value and evaluate f(E) at the conduction band edge to estimate the fraction of thermally excited electrons.

Fermi-Dirac Distribution Calculator

Enter Fermi energy and temperature to compute occupation probability f(E) at any energy level and plot the full Fermi-Dirac distribution curve.

Parameters

Fermi Energy E_f (eV)

Temperature T (K)

Query Energy E (eV)

Energy at which f(E) is evaluated (can be any value).

Formula

f(E) = 1 / (exp((E - E_f) / kT) + 1)

k = 8.61733 × 10^-5 eV/K

f(E_f) = 0.5 at any temperature

Results at E = — eV

Enter parameters and click Calculate

f(E) vs Energy Curve

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f(E) at Selected Energies

Calculate to see the distribution table.

E (eV)	E - E_f (eV)	f(E)

Summary

The Fermi-Dirac distribution describes the probability that a quantum state at energy E is occupied by a fermion (such as an electron) at thermal equilibrium. It is defined as f(E) = 1 / (exp((E - E_f) / (kT)) + 1), where E_f is the Fermi energy, k is the Boltzmann constant, and T is the absolute temperature. At absolute zero the distribution is a perfect step function: all states below E_f are occupied and all above are empty. At higher temperatures the step softens, and states above E_f gain a non-zero occupation probability.

How it works

Enter the Fermi energy E_f in electron-volts (eV).
Enter the temperature in Kelvin (minimum 1 K).
Enter the energy E at which you want to evaluate f(E).
Click Calculate to see the occupation probability, the thermal energy kT, and a breakdown of the exponent.
The chart below updates automatically, showing f(E) vs E across a symmetric range around the Fermi level.
Use the preset temperature buttons (0 K limit, 300 K, 1000 K, 5000 K) for quick comparisons.

Use cases

Determining electron occupation probability in semiconductor band structures.
Visualizing how the Fermi edge sharpens at low temperatures.
Computing carrier concentration integrals in solid-state physics.
Teaching statistical mechanics and quantum statistics.
Estimating thermionic emission probabilities near the Fermi level.
Analysing metal conduction band filling in condensed matter research.

Frequently Asked Questions

Last updated: 2026-05-28 · Reviewed by Nham Vu