= Degeneracy test for a fully ionized light-mass donor gas
{title2=$n/N_c,\quad N_c=2(m^*k_BT/2\pi\hbar^2)^{3/2}$}
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.
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