Constant-density-of-states intrinsic semiconductor ($n=p=D{k}_{B}T\ln (1+e^{-E_g/(2k_{B}T)})$)

In a symmetric model with constant density of states $D$ per volume and energy in two broad bands, charge neutrality gives $\mu =(E_c+E_v)/2$ and $n=p=D{k}_{B}T\ln [1+e^{-E_g/(2k_{B}T)}]$. In the nondegenerate limit, $n\simeq D{k}_{B}T{e}^{-E_g/(2k_{B}T)}$. Thus the leading exponential activation law can coexist with an algebraic temperature prefactor.

For symmetric touching parabolic dispersions with constant density of states $D$ per energy band, including spin degeneracy, charge neutrality fixes $\mu =0$. The Fermi-Dirac distribution gives $N_c=D{k}_{B}T\ln 2$ and the thermal internal energy, including electrons and holes, is $\Delta U=({\pi }^2/6)D(k_{B}T{)}^2$. Thus $C=({\pi }^2/3)D{k}_B^{2}T$. Its large ratio to $N_c$ is compatible with the equipartition theorem because this excitation count grows with temperature and the heat capacity includes both species.