Why narrow gap semiconductors have large dielectric constant?
The dielectric constant measures how strongly the eletronic charge in the material can adjust to shield an external electric field, larger the dieletric constant, more effective is the screening. But for the electronic charge be able to adjust responding to the electric field, the electrons should be easily moved, which means that each electronic state will change assuming a polarized configuration. If the gap is large, the electric field can not mix the electronic states in the valence band with the electronic states in the conduction band, (think of perturbation theory, where the denominator will be of order of the band gap energy) so the electrons don’t polarize much and the dielectric constant is small. On the other hand, in a metal, where the gap is zero, any small electric field mix the occupied states with the unoccupied ones, and we have a strong screening.
