acta physica slovaca

Numerical analysis of the Anderson localization

Peter Markoš
   Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia

Full text: ::pdf :: (submitted 15.09.2006, published 15.10.2006)

Abstract: The aim of this paper is to demonstrate, by simple numerical simulations, the main transport properties of disordered electron systems. These systems undergo the metal insulator transition when either Fermi energy crosses the mobility edge or the strength of the disorder increases over critical value. We study how disorder affects the energy spectrum and spatial distribution of electronic eigenstates in the diffusive and insulating regime, as well as in the critical region of the metal-insulator transition. Then, we introduce the transfer matrix and conductance, and we discuss how the quantum character of the electron propagation influences the transport properties of disordered samples. In the weakly disordered systems, the weak localization and anti-localization as well as the universal conductance fluctuation are numerically simulated and discussed. The localization in the one dimensional system is described and interpreted as a purely quantum effect. Statistical properties of the conductance in the critical and localized regimes are demonstrated. Special attention is given to the numerical study of the transport properties of the critical regime and to the numerical verification of the single parameter scaling theory of localization. Numerical data for the critical exponent in the orthogonal models in dimension d=2,3,4,5 are compared with theoretical predictions. We argue that the discrepancy between the theory and numerical data is due to the absence of the self-averaging of transmission quantities. This complicates the analytical analysis of the disordered systems. Finally, theoretical methods of description of weakly disordered systems are explained and their possible generalization to the localized regime is discussed. Since we concentrate on the one-electron propagation at zero temperature, no effects of electronelectron interaction and incoherent scattering are discussed in the paper.

PACS: 73.23.-b, 71.30.+h, 72.10.-d
Keywords: Electronic transport in disordered systems, Metal-insulator transition, Localization, Numerical simulations, Andreson model

About author: Peter Markoš, born in October 1958 in Kremnica, received his PhD degree in physics in 1989 from the Institute of Physics, Slovak Academy of Sciences, Bratislava. He is a researcher at Institute of Physics in Bratislava. He worked as Alexander von Humboldt Fellow at the Physikalisch-Technische Bundesanstalt Braunschweig (1991-1993) and as JSPS Fellow at the Sophia University Tokyo (2000). Since 2000, he collaborates with Ames Laboratory, Iowa. His current research interests include electronic transport in disordered systems, metal-insultor transition, and the theory of left-handed metamaterials.