Semiclassical Theory of Electron Transport Properties in a Disordered Material

We study the problem of an electron in a dilute gas of density ρ of hard spheres. The radius of the spheres is a. It is shown that a systematic expansion of the Green's function in a series involving the small parameter ρa³ leads to inconsistencies at low energies. This is attributed to the existence of localized states. We then assume that the problem can approximately be reduced to the motion of a classical particle through a suitably defined smooth, random, energy-dependent effective potential. Physical arguments are given to justify this reduction, although a rigorous proof is lacking. The resulting classical problem is in complete agreement with the Mott-Cohen-Fritzsche-Ovshinsky model of disordered materials, and is simple enough to permit detailed calculations of all quantities of interest. As a test for the validity of our approximations, we apply the theory to compute the mobility of excess electrons in gaseous He. By adjusting two parameters of order unity we get a good fit to the experimental data; the remaining small discrepancies admit satisfactory explanations.