Degeneracy test for a fully ionized light-mass donor gas ($n/N_c,N_c=2(m^{\ast }{k}_{B}T/2\pi {\hslash }^2{)}^{3/2}$)

Before using a Maxwell–Boltzmann chemical-potential formula in a doped semiconductor, compare the carrier density to its conduction-band effective density of states. When n is much larger than $N_c$, a nondegenerate logarithmic estimate is inconsistent. Use Fermi–Dirac statistics or a controlled degenerate expansion instead. A small effective mass strongly reduces $N_c$ and raises the Fermi energy at fixed density, so even an otherwise familiar donor concentration can put the chemical potential inside the conduction band.