Intrinsic carrier extrapolation across a crossover

In a nondegenerate parabolic-band semiconductor with fixed density-of-states mass, $n(T)=C{T}^{3/2}{e}^{-\Delta (T)/(k_{B}T)}$, where $\Delta (T)=E_c(T)-\mu (T)$. ThusA single measured value of $\Delta$ does not determine its temperature dependence. A fixed-gap intrinsic model instead has $n_i=C_i{T}^{3/2}{e}^{-E_g/(2k_{B}T)}$, with the two effective density of states factors in $C_i$. Identifying a crossover by $n_i(T_c)\simeq n_{\mathrm{ext}}$ gives an order-of-magnitude dopant-density estimate once that activation law is specified. At exact charge neutrality with $n_i=n_{\mathrm{ext}}$, the total mobile-electron density is $(1+\sqrt{5})n_{\mathrm{ext}}/2$, so a crossover is not a sharp equality of every density definition.