Optical Absorption for a Narrow-Gap Semiconductor

The intraband absorption of electromagnetic waves in doped narrow-gap semiconductors is calculated for waves whose energies $\hslash \omega$ are less than the gap energy $\Delta$. The treatment rests on the introduction of the temperature-dependent Green's function and Kubo's formula for the conductivity. We take into account the effects of dynamic screening of conduction electrons and virtual transitions from valence bands. Our result is in agreement with the result for a free-electron gas in the limit of $\hslash \omega \ll \Delta$.