Excitons in Metals: Infinite Hole Mass

The optical conductivity is evaluated for interband transitions between a flat valence band and a parabolic conduction band. The conduction band is filled with electrons to a Fermi energy ${\mu }_F$. The conductivity is calculated assuming that the electron-hole interaction is attractive, static, and short range. The final-state interactions between the electron and hole cause a divergence in the conductivity at the interband threshold. This divergence appears to go as a power law. For this case of an infinite hole mass, the exciton binding energy vanishes, since the singularity in the scattering amplitude occurs just at threshold.