 Solution

Use time dependence $e^{-i\omega t}$ and assume the dopants supply $N$ mobile electrons per unit volume. The Drude model equation isThe free-carrier electric polarization is ${\mathbf{P}}_c=-Ne\mathbf{x}$, so, after adding the bound-electron background,ThereforeThe symbol ${\omega }_p$ here is the background-screened plasma frequency; the unscreened carrier frequency is $\sqrt{{ϵ}_{\infty }}{\omega }_p$. For positive $\gamma$, the imaginary part of ${ϵ}_r$ is positive in this time convention, consistent with absorption. Incomplete dopant ionization would replace the nominal doping density by the actual mobile carrier density.