Frequency- and wave-number-dependent dielectric function of semiconductors

An expression for the long-wavelength frequency-dependent dielectric function $\epsilon \left(\omega \right)$ of a semiconductor is derived for a two-band model using a tight-binding approach. The resulting expression for $\epsilon \left(\omega \right)$ contains two parameters, one of which is related to the average energy gap between the valence and conduction bands, and the other, to the widths of these bands. The frequency- and wave-number-dependent dielectric function $\epsilon (\overrightarrow{\mathrm{q}},\omega )$ is studied for frequencies below the band gap, and an approximate expression is obtained.