Density of States — 3D Parabolic Band

g(E) = (1/2π²)(2m*/ℏ²)^(3/2) √E — the square-root law of free-electron metals

Parameters

g(E) ∝ √E · (m*)^(3/2)
Free-electron DOS: In a 3D parabolic band E = ℏ²k²/2m*, counting states in k-space gives g(E) ∝ (m*)^(3/2)√E. Multiplied by the Fermi-Dirac distribution f(E,T) gives the electron density. The sharp Fermi edge broadens with temperature over ~kBT. The DOS determines specific heat, magnetic susceptibility, and conductivity.