Fermi-Dirac Distribution Formula
The Fermi-Dirac distribution gives the probability that a quantum state at energy E is occupied by a fermion at thermal equilibrium.
The Formula
The Fermi-Dirac distribution gives the probability that a quantum state at energy E is occupied by a fermion (electron, proton, neutron, or any half-integer spin particle) at temperature T. It is the fundamental equation of quantum statistical mechanics for fermions and underpins the entire theory of metals, semiconductors, and insulators.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| f(E) | Probability of occupation of state at energy E (0 to 1) | dimensionless |
| E | Energy of the quantum state | Joules or eV |
| E_F | Fermi energy — the chemical potential at 0 K | Joules or eV |
| k_B | Boltzmann constant = 1.381 × 10⁻²³ J/K | J/K |
| T | Absolute temperature | Kelvin (K) |
Key behaviors:
- At T = 0 K: f(E) = 1 for all E < E_F, and f(E) = 0 for all E > E_F (perfect step function)
- At T > 0: the step "blurs" by about ±2k_BT around E_F
- At E = E_F: f(E) = ½ always, regardless of temperature
- For E − E_F ≫ k_BT: f(E) → e^(−(E−E_F)/k_BT) (reduces to Maxwell-Boltzmann)
Example 1 — Probability at Room Temperature
Copper has a Fermi energy E_F = 7.04 eV. At room temperature (T = 300 K), find the probability that a state 0.1 eV above E_F is occupied.
k_BT = 1.381 × 10⁻²³ × 300 = 4.14 × 10⁻²¹ J = 0.02585 eV
E − E_F = +0.1 eV
f = 1 / (e^(0.1/0.02585) + 1) = 1 / (e^3.868 + 1)
f = 1 / (47.8 + 1) = 1 / 48.8
f ≈ 0.0205 or about 2% chance the state is occupied — just above the Fermi level, states are mostly empty
Example 2 — At the Fermi Level
What is the occupation probability at E = E_F at any temperature?
f(E_F) = 1 / (e^((E_F − E_F)/k_BT) + 1) = 1 / (e^0 + 1)
f(E_F) = 1/(1 + 1) = 0.5 = 50% — the Fermi level is always half-occupied at any temperature above 0 K
When to Use It
Use the Fermi-Dirac distribution when:
- Calculating electron density in metals and semiconductors
- Understanding electrical conductivity — only electrons near E_F contribute to conduction
- Analyzing p-n junction behavior in diodes and transistors
- Studying white dwarf stars — their electron degeneracy pressure is described by Fermi-Dirac statistics
- Computing thermoelectric properties of materials
Compare with Bose-Einstein (+1 → −1 in denominator) for bosons, and Maxwell-Boltzmann (classical limit) for low-density, high-temperature situations.