Numerical analysis of the Anderson localization

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 2 < d ≤ 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.

Critical exponent of metal-insulator transition in doped semiconductors: The relevance of the Coulomb interaction

“…This estimate has been confirmed in subsequent work [10][11][12][13] . Clear demonstrations of the universality of this value include its confirmation in a non-interacting tight-binding model with random hopping of a doped semiconductor 14 , and its experimental measurement in a quasi-periodic atomic kicked rotor 15 .…”

Electron transport in strongly disordered structures

“…Generalized DMPK equation (GDMPK) contains new parameters K ab , which measure the spatial non-homogeneity of electron distribution [9]. Both approximate [9] and numerical [10] solutions of GDMPK equation agree very well with results of numerical simulations [5].…”

“…Although the typical transmission through the strongly disordered sample is small, we can find, thanks to large conductance fluctuations [5], the sample with relatively large transmission. Figures 3 and 4 show that the spatial electron distribution is not homogeneous inside the strongly disordered systems.…”

“…Numerically, it was shown [4,5] that, contrary to the well-established paradigm, the probability distribution of the logarithm of the conductance is not Gaussian. This was confirmed by recent numerical and analytical analysis [6,7] and by analytical formulation of the transport in strongly disordered systems [8].…”

Anderson Localization of Thermal Phonons Leads to a Thermal Conductivity Maximum

“…We found that at length scales corresponding to the transition from diffusive to localized transport, the probability distribution of the dimensionless conductance, g, develops non-analytic behavior around g ≈ 1 [19,20], suggesting transport governed by a single-channel [21][22][23].…”