= Solution
<Thomas-Fermi screening> treats the carrier density locally as that of a uniform <electron> gas whose chemical potential is shifted by the slowly varying <electrostatic potential>. Linearizing gives $\delta n=(\partial n/\partial\mu)e\varphi$ for electrons. The resulting screened Poisson equation has inverse screening length
$$
k_s^2=\frac{e^2}{\varepsilon}\frac{\partial n}{\partial\mu}.
$$
For a degenerate gas, $\partial n/\partial\mu=g_{\rm vol}(E_F)$ including both spins, giving <Thomas-Fermi screening length> $k_s^{-1}$. A nondegenerate <semiconductor> instead has $\partial n/\partial\mu=n/(k_BT)$, giving the <Debye length> $\sqrt{\varepsilon k_BT/(ne^2)}$. Electrons and holes contribute additively to the charge susceptibility when both are present; the lattice permittivity replaces the vacuum value.
The approximation requires a potential varying slowly compared with microscopic electronic length scales and a small perturbation for the linearized result. It captures smooth electrostatic screening, but not the full nonlocal response or <Friedel oscillations>. Low carrier density or carrier freeze-out can make a <semiconductor>'s screening much weaker than a metal's.
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