= Intrinsic carrier extrapolation across a crossover
In a nondegenerate parabolic-band <semiconductor> with fixed density-of-states mass, $n(T)=C T^{3/2}e^{-\Delta(T)/(k_BT)}$, where $\Delta(T)=E_c(T)-\mu(T)$. Thus
$$
\frac{n(T_2)}{n(T_1)}=\left(\frac{T_2}{T_1}\right)^{3/2}\exp\left[-\frac{\Delta(T_2)}{k_BT_2}+\frac{\Delta(T_1)}{k_BT_1}\right].
$$
A single measured value of $\Delta$ does not determine its temperature dependence. A fixed-gap intrinsic model instead has $n_i=C_iT^{3/2}e^{-E_g/(2k_BT)}$, with the two <effective density of states> factors in $C_i$. Identifying a crossover by $n_i(T_c)\simeq n_{\rm ext}$ gives an order-of-magnitude dopant-density estimate once that activation law is specified. At exact charge neutrality with $n_i=n_{\rm ext}$, the total mobile-electron density is $(1+\sqrt5)n_{\rm ext}/2$, so a crossover is not a sharp equality of every density definition.
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