Nearly-free electron model — E(k) dispersion in a periodic potential
First band
Second band
Fermi level
Band gap
Nearly-free electron model: a weak periodic potential V(x) = V₀cos(2πx/a) mixes plane waves at Brillouin zone boundaries k = ±π/a, opening band gaps ΔE = 2V₀. Metals have the Fermi level inside a band; insulators have it in a gap; semiconductors have a small gap (~1 eV).