Fermi-Dirac distribution

The Fermi-Dirac distribution gives the thermal-equilibrium probability that a quantum state at energy $E$ is occupied by an electron (or any other fermion). It is the fundamental statistical mechanics governing all electron populations in semiconductors and underpins every semiconductor device's electrical and optical behavior.

Definition.

$$f(E) \;=\; \frac{1}{1 + \exp[(E - E_F)/k_B T]},$$

where $E_F$ is the Fermi level, $k_B$ is the Boltzmann constant, and $T$ is absolute temperature. Properties:

• $f(E_F) = 1/2$ (the defining property of the Fermi level)
• $f(E) \to 1$ for $E \ll E_F$ (states well below $E_F$ are filled)
• $f(E) \to 0$ for $E \gg E_F$ (states well above $E_F$ are empty)
• Transition width: $\sim 4 k_B T$ (10% to 90% occupation)
• At $T = 0$: step function (Heaviside) at $E_F$

Thermal width. At room temperature, $k_B T = 25.9$ meV. The Fermi-Dirac distribution has full-width at half-maximum of the derivative (where most thermal action happens):

$$\text{FWHM} \;\approx\; 3.5 \, k_B T \;\approx\; 90 \text{ meV at 300 K}.$$

This $\sim 100$ meV is the natural energy scale of room-temperature semiconductor phenomena: how sharp band edges look in absorption, how wide the laser gain peak is, how quickly carrier concentrations change with energy.

Approximations.

Maxwell-Boltzmann (non-degenerate limit). When $E - E_F \gg k_B T$ (Fermi level several $k_B T$ below the band edge):

$$f(E) \;\approx\; \exp[-(E - E_F)/k_B T].$$

This is the standard approximation for moderately doped semiconductors. It is valid when the Fermi level is at least $3 k_B T$ from the band edge. Typical condition: $n < N_c / 3$ (carrier density less than ~1/3 of the effective DOS).

Degenerate limit. When $E_F$ is inside the conduction band (n-type degenerate) or valence band (p-type degenerate), $f(E)$ deviates significantly from Maxwell-Boltzmann. Full Fermi-Dirac integrals must be used:

$$n \;=\; N_c \, F_{1/2}(\eta), \quad \eta = (E_F - E_c)/k_B T,$$

where $F_{1/2}$ is the Fermi-Dirac integral of order 1/2. For $\eta > 0$ (degenerate), the integral is larger than $\exp(\eta)$ — meaning the Boltzmann approximation underestimates carrier density.

Crossover to degeneracy.

$n/N_c$ ratio	$E_F - E_c$	Regime
0.01	$-4.6 k_B T$	Strongly non-degenerate; Boltzmann OK
0.1	$-2.3 k_B T$	Non-degenerate; Boltzmann OK
0.3	$-1.2 k_B T$	Mildly non-degenerate; corrections begin
1	$\sim 0$	Onset of degeneracy
3	$\sim +1.2 k_B T$	Moderately degenerate
10	$\sim +3 k_B T$	Strongly degenerate

For $N_c = 5 \times 10^{17}$ cm⁻³ (typical for GaAs at 300 K), degeneracy onset is at $n \sim 5 \times 10^{17}$ cm⁻³ — exactly the regime where semiconductor lasers operate above threshold.

Holes and the complementary distribution. For holes (absence of electrons in valence band):

$$f_p(E) \;=\; 1 - f(E) \;=\; \frac{1}{1 + \exp[(E_F - E)/k_B T]}.$$

The hole distribution is the mirror image of the electron distribution; the Fermi level is the same energy reference.

Multi-band situations. In a real semiconductor, electrons populate multiple bands and valleys. The Fermi-Dirac distribution applies independently to each — with the same Fermi level (in equilibrium) but different effective densities of states.

For multi-valley semiconductors (Si, Ge, AlGaAs above the $\Gamma$-X crossover):

$$n_\text{total} \;=\; n_\Gamma + n_X + n_L,$$

where each valley contributes according to its own DOS and the same Fermi level.

Pauli exclusion and degeneracy pressure. The Fermi-Dirac form arises from Pauli's exclusion principle: each quantum state can hold at most one electron. As more electrons are added, they must occupy successively higher-energy states. This "degeneracy pressure" has macroscopic consequences:

• Density of degenerate electron gas: $n^{2/3}$ scaling in energy (not the thermal $n \cdot k_B T$)
• White dwarf stars: stability from electron degeneracy pressure
• Heat capacity of metals: $\propto T$ at low T (not $\propto T^{3/2}$ as Boltzmann would predict)
• Modulation-doped 2DEG: high mobility because of degenerate carrier statistics

Quantum statistical mechanics origin. The Fermi-Dirac distribution emerges from the grand canonical ensemble for fermions:

$$\langle n_i \rangle \;=\; \frac{1}{Z} \sum_n n_i \, e^{-(E_n - \mu N_n)/k_B T},$$

where $\mu$ is the chemical potential (= Fermi level for fermions) and the sum is over states with various occupation numbers. For fermions, each state can have $n_i = 0$ or $1$, giving the standard 2-term sum that produces the Fermi-Dirac form.

Bose-Einstein contrast. For bosons (photons, phonons), the analogous distribution is Bose-Einstein:

$$n_\text{BE}(E) \;=\; \frac{1}{\exp(E/k_B T) - 1}.$$

Photons have $\mu = 0$ in equilibrium with the walls of a cavity (blackbody radiation). Phonons follow the same form with $\mu = 0$. The key difference: bosons can pile up arbitrarily in a single state (no exclusion), so the distribution diverges at $E \to 0$.

Application: predicting carrier concentration. With known doping $N_D$ and temperature $T$, $E_F$ is determined by charge neutrality:

$$n - p - N_A^- + N_D^+ \;=\; 0.$$

Substituting Fermi-Dirac forms for $n$ and $p$ gives a transcendental equation for $E_F$. Solutions are tabulated or solved numerically; most TCAD device simulators use this approach.

Application: pseudopotential band-structure interpretation. ARPES (angle-resolved photoemission) measurements of band structure are interpreted via the Fermi-Dirac distribution: the experimentally observed Fermi edge is the convolution of the energy resolution with the Fermi-Dirac step.

At zero temperature. $f(E)$ becomes a step function:

$$f(E)|_{T=0} \;=\; \begin{cases} 1 & E < E_F \\ 0 & E > E_F \end{cases}.$$

All states below $E_F$ are filled; all above are empty. The Fermi level becomes the maximum occupied energy — the "Fermi surface" in metals, the "Fermi sea." In semiconductors, the conduction band is empty and the valence band is filled at $T = 0$.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 16; Ashcroft & Mermin, Solid State Physics (Saunders, 1976), Ch. 2 — canonical statistical-mechanics derivation; Sze & Ng, Physics of Semiconductor Devices (3rd ed., 2007), Ch. 1; Pierret, Semiconductor Device Fundamentals (1996) for the device-oriented treatment.