Fermi-Dirac Distribution

Fermi-Dirac Distribution

The Fermi-Dirac distribution gives the probability that a quantum state with energy E is occupied by an electron at temperature T:

$f(E) = \frac{1}{\exp\left(\frac{E - \mu}{k_B T}\right) + 1}$

Where:

• μ = chemical potential (≈ E_F at low temperatures)
• k_B = Boltzmann constant = 1.381 × 10⁻²³ J/K
• T = temperature in Kelvin

Key Behaviors

• At T = 0: f = 1 for E < E_F, f = 0 for E > E_F (sharp step)
• At T > 0: thermal smearing occurs over an energy range ~k_B T around E_F
• At E = μ: f = 0.5 always (regardless of temperature)

Density of States

The density of states g(E) counts available quantum states per unit energy per unit volume. For free electrons:

$g(E) = \frac{1}{2\pi^2} \left(\frac{2m_e}{\hbar^2}\right)^{3/2} \sqrt{E}$

In units of J⁻¹·m⁻³. The electron density is:

$n = \int_0^{E_F} g(E)\, dE$

Physical Constants (inside your functions)

• k_B = 1.381 × 10⁻²³ J/K
• ħ = 1.055 × 10⁻³⁴ J·s
• m_e = 9.109 × 10⁻³¹ kg
• 1 eV = 1.602 × 10⁻¹⁹ J