Fermi Energy

Fermi Energy and the Free Electron Model

The free electron model treats conduction electrons in a metal as a gas of non-interacting fermions. Electrons fill energy states from the ground state up to the Fermi energy E_F at absolute zero.

Fermi Energy:

$E_F = \frac{\hbar^2}{2m_e}(3\pi^2 n)^{2/3}$

Where:

• ħ = reduced Planck constant = 1.055 × 10⁻³⁴ J·s
• m_e = electron mass = 9.109 × 10⁻³¹ kg
• n = electron number density (electrons/m³)

Typical metals have E_F in the range of 1–12 eV, much larger than k_B T at room temperature (~0.025 eV).

Fermi Velocity

The speed of electrons at the Fermi surface:

$v_F = \sqrt{\frac{2E_F}{m_e}}$

Fermi velocities are ~10⁶ m/s — a significant fraction of the speed of light, despite being at "zero temperature."

Fermi Temperature

A characteristic temperature scale:

$T_F = \frac{E_F}{k_B}$

Since T_F ≫ room temperature for metals, conduction electrons are highly degenerate at normal conditions.

Physical Constants (inside your functions)

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