Electromagnetic Properties of Insulators. II

This paper is a continuation of an earlier study, from a many particle point of view, of the electromagnetic properties of insulators. Here the system of an insulator and one electron is treated. The Coulomb interactions between all the electrons in the system are allowed for to all orders of perturbation theory. The true effective mass $m^{*}$ of the extra particle is defined as the curvature in wave vector space of the energy surface connecting the ground state and the low-lying excited states of the interacting system. The central result then obtained is that the response of the system to long-wavelength, low-frequency electric fields is exactly that of a free electron of mass $m^{*}$ moving in a medium characterized by the dielectric constant of the perfect insulator. The energy levels of the system in a static magnetic field are also discussed. An alternative derivation of a single-particle effective-mass equation, previously obtained by Klein, is given. The eigenvalues of this equation are under certain conditions the energy levels of the interacting system in a magnetic field. In an Appendix a Kramers-Kronig relation connecting the difference in optical absorption of the present system and the perfect insulator with $m^{*}$ is derived. These results indicate that the usual effective-mass theory of semiconductors of low carrier concentration includes the effects of the electron-electron interactions to an excellent approximation.