Localization and space-time dispersion of the kinetic coefficients of a two-dimensional disordered system

A generalization of the Vollhardt-Wölfle self-consistent localization theory is proposed to take into account spatial dispersion of the kinetic coefficients of a two-dimensional disordered system. It is shown that the main contribution to the singular part of the collision integral of the Bethe-Salpeter equation in the limit ω→0 is from the diffusion pole iω=(p+p′)2D (|p+p′|,ω), which provides an anomalous increase in the probability of backscattering p→−p′. In this limit the dependence of the diffusion coefficient on q and ω exhibits localization behavior, D(q,ω)=−iωf(lDq), where |f(z)|⩽(0)=d2 (d is the localization length). According to the Berezinski\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} $$ \end{document}-Gor’kov criterion, D(q,0)=0 for all q. Spatial dispersion of D(q,ω) is manifested on a scale q ∝ 1/lD, where lD is the frequency-dependent diffusion length. In the localization state lD≪l, where l is the electron mean free path; lD ∝ ω as ω→0, suggesting the suppression of spatial dispersion of the kinetic coefficients down to atomic scales. Under the same conditions σ(q,ω) exhibits a strong dependence on q on a scale q ∝ 1/d, i.e., the nonlocality range of the electrical conductivity is of the order of the localization length d. At the microscopic level these results corroborate the main conclusions of Suslov (Zh. Éksp. Teor. Fiz. 108, 1686 (1995) [JETP 81, 925 (1995)]), which were obtained to a certain degree phenomenologically in the limit ω→0. A major advance beyond the work of Suslov in the present study is the analysis of spatial dispersion of the kinetic coefficients at finite (rather than infinitely low) frequencies.