Power law localisation in disordered systems

Abstract

A power law term in the eigenfunctions Psi of electrons in weakly disordered systems (for which kFl>>1, where l is the elastic mean free path) is derived from first-order perturbation theory. It is shown that psi =A psi ext(1)+(B/Rd-1) psi ext(2) for d>1 where d is the dimensionality, A is a constant, B is a function of the angle Theta between K and r, and psi ext(1), psi ext(2) are extended wavefunctions with different phases. This result confirms the recent conjecture of M. Kaveh and N.F. Mott (1983 Phil. Mag. B47 L9) which was based on a semiclassical diffusion model. The authors conclude that for any number of dimensions the corrections to the Boltzmann formula for the metallic conductivity in the weak disorder limit, which for d=3 can in some cases lead to a value of the conductivity tending continuously to zero, are due to this power law component of the wavefunction.