Critical phenomena of the disorder driven localization-delocalization transition

The critical behavior of non-interacting electrons in disordered systems is investigated. The scaling functions of the localization lengths in two-dimensional systems with a magnetic field perpendicular to the plane and anisotropic hopping are rescaled to the isotropic one. The geometric mean of the critical values of the scaling functions equals the critical value of the isotropic scaling function. The critical exponent of the localization lengths in the two anisotropic directions is found to be equal and also equal to that of the isotropic system. The critical energy is independent of the direction of propagation. If disorder is strong enough, the critical energy does not coincide with the band center.;The probability distribution of the conductance at the critical energy of anisotropic systems is also rescaled to the corresponding distribution of the isotropic system. The ratio of side lengths of a two-dimensional anisotropic system should equal the square root of the ratio of the critical values of the scaling functions of the localization lengths.;The form of the critical distribution of the conductance is investigated for two-dimensional systems with magnetic fields or spin-orbit interaction, as well as three-dimensional systems, for both periodic as well as hard wall boundary conditions. The form for g \u3c 1 can be explained reasonably well, whereas the understanding of the distribution for systems with g \u3e 1 is still incomplete.;Finally, the fluctuations of the conductance for such systems are investigated from the ballistic regime to the localized regime, with an emphasis on the differences with respect to boundary conditions.;Parts of this dissertation have been published in Phys. Rev. B 63, 085102 (2001), Phys. Rev. B 64, 172202 (2001), Phys. Rev. B 64, 193103 (2001), and Phys. Rev. B (to be published)

The probability distribution of the conductance at the mobility edge

The probability distribution of the conductance p(g) of disordered 2d and 3d systems is calculated by transfer matrix techniques. As expected, p(g) is Gaussian for extended states while for localized states it is log-normal. We find that at the mobility edge p(g) is highly asymmetric and universal.Comment: 3 pages RevTeX, 6 figures included, submitted to Physica

Conductance fluctuations and boundary conditions

The conductance fluctuations for various types for two-- and three--dimensional disordered systems with hard wall and periodic boundary conditions are studied, all the way from the ballistic (metallic) regime to the localized regime. It is shown that the universal conductance fluctuations (UCF) depend on the boundary conditions. The same holds for the metal to insulator transition. The conditions for observing the UCF are also given.Comment: 4 pages RevTeX, 5 figures include

Symmetry, dimension and the distribution of the conductance at the mobility edge

The probability distribution of the conductance at the mobility edge, $p_c(g)$, in different universality classes and dimensions is investigated numerically for a variety of random systems. It is shown that $p_c(g)$ is universal for systems of given symmetry, dimensionality, and boundary conditions. An analytical form of $p_c(g)$ for small values of $g$ is discussed and agreement with numerical data is observed. For $g > 1$, $\ln p_c(g)$ is proportional to $(g-1)$ rather than $(g-1)^2$.Comment: 4 pages REVTeX, 5 figures and 2 tables include

Metal-insulator transitions in anisotropic 2d systems

Several phenomena related to the critical behaviour of non-interacting electrons in a disordered 2d tight-binding system with a magnetic field are studied. Localization lengths, critical exponents and density of states are computed using transfer matrix techniques. Scaling functions of isotropic systems are recovered once the dimension of the system in each direction is chosen proportional to the localization length. It is also found that the critical point is independent of the propagation direction, and that the critical exponents for the localization length for both propagating directions are equal to that of the isotropic system (approximately 7/3). We also calculate the critical value of the scaling function for both the isotropic and the anisotropic system. It is found that the isotropic value equals the geometric mean of the two anisotropic values. Detailed numerical studies of the density of states for the isotropic system reveals that for an appreciable amount of disorder the critical energy is off the band center.Comment: 6 pages RevTeX, 6 figures included, submitted to Physical Review

Critical phenomena of the disorder driven localization-delocalization transition

The critical behavior of non-interacting electrons in disordered systems is investigated. The scaling functions of the localization lengths in two-dimensional systems with a magnetic field perpendicular to the plane and anisotropic hopping are rescaled to the isotropic one. The geometric mean of the critical values of the scaling functions equals the critical value of the isotropic scaling function. The critical exponent of the localization lengths in the two anisotropic directions is found to be equal and also equal to that of the isotropic system. The critical energy is independent of the direction of propagation. If disorder is strong enough, the critical energy does not coincide with the band center.;The probability distribution of the conductance at the critical energy of anisotropic systems is also rescaled to the corresponding distribution of the isotropic system. The ratio of side lengths of a two-dimensional anisotropic system should equal the square root of the ratio of the critical values of the scaling functions of the localization lengths.;The form of the critical distribution of the conductance is investigated for two-dimensional systems with magnetic fields or spin-orbit interaction, as well as three-dimensional systems, for both periodic as well as hard wall boundary conditions. The form for g 1 is still incomplete.;Finally, the fluctuations of the conductance for such systems are investigated from the ballistic regime to the localized regime, with an emphasis on the differences with respect to boundary conditions.;Parts of this dissertation have been published in Phys. Rev. B 63, 085102 (2001), Phys. Rev. B 64, 172202 (2001), Phys. Rev. B 64, 193103 (2001), and Phys. Rev. B (to be published).</p