Power law localisation in disordered systems

A power law term in the eigenfunctions Psi of electrons in weakly disordered systems (for which k_Fl>>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/R^d-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.