Boltzmann carrier density of a two-dimensional parabolic band ($n=gm{k}_{B}T{e}^{-(E_c-\mu )/(k_{B}T)}/(2\pi {\hslash }^2)$)

State counting with total degeneracy $g$ gives a constant two-dimensional density of states $gm/(2\pi {\hslash }^2)$ above a parabolic conduction band edge. Integrating the Boltzmann occupation yields the displayed areal carrier density. For one valley and two spin states the coefficient is $m{k}_{B}T/(\pi {\hslash }^2)$. Applying the same count to holes gives $np=N_c{N}_v{e}^{-E_g/(k_{B}T)}$ and the intrinsic density $n_i=\sqrt{N_c{N}_v}{e}^{-E_g/(2k_{B}T)}\propto T{e}^{-E_g/(2k_{B}T)}$, distinct from the three-dimensional $T^{3/2}$ prefactor.