Path-integral theory of an electron gas in a random potential

Abstract

An ideal gas of non-degenerate electrons of mass m in a Gaussian random potential is investigated. The potential is characterized by two parameters: y (whose square is the variance of the potential energy) and L (correlation length). Relevance to the case of a polycrystalline non-degenerate semiconductor is suggested. The autocorrelation function of the potential is taken Gaussian, W(r' — r") = exp [ — ( r' — r") 2 / L 2 ]. The averaged canonical density matrix (C B (r, r0) > is calculated by the use of Feynman’s path-integral formulation. Using the replacement W ( r - r" ) 2 ~ 1 — (r' —r") 2 /- L 2 (after discussing conditions under which it is possible), we derive an explicit formula for . A characteristic frequency W G = (n/L) (2/ mk B T)% is found and interpreted as being due to itinerant oscillators. A characteristic mass mG = coth (^/3hojG), ft = l/kBT is found; the ratio (mG —m)/m is interpreted as a measure of localization of the electrons in the random potential under consideration. A formula for the energy-level density is discussed with respect to conditions of applicability of the quasi-classical approximation. The function is a simple expression if the condition {hr//L J (2m)}» kBT is satisfied; moreover, if rj kBT, then the ‘quantal distribution function’ (averaged Wigner function) is shown to equal the Maxwell distribution function f(Ek) = f (0) exp (— f 3Ek) where Ek is the kinetic energy, Ek = h 2 k 2 /2m, and f (0) is practically independent of ?. Additional conditions of validity of the Boltzmann-equation theory are thus found.