Many-Particle Approach to the One-Electron Problem in Insulators and Semiconductors

Abstract

A formal theory of one-electron states in insulators and semiconductors is developed from a many-particle point of view. The techniques of second quantization are utilized for this purpose in a manner analogous to that introduced recently for the study of Fermi liquids, i.e., by the study of matrix elements of the electron field operator which describe the propagation of particles or holes. Both types of motion are described symmetrically by means of the one-particle Green's function or propagator. The utility of these constructs for the present study derives from the existence of a gap against single-particle excitation.The basic result of this paper is that the motion of electrons near the bottom of the conduction band in the presence of external electric and magnetic fields whose spatial variation over one lattice spacing is small and which contain no frequencies comparable with the gap frequency is governed by a simple Schr\"odinger equation. The latter contains as parameters only the effective mass (as measured in a cyclotron resonance experiment) and the static dielectric constant and magnetic permeability of the solid. Our proof, within the restriction of a fixed, perfect lattice, takes into account all many-body effects. A similar theorem obtains for the motion of holes.