Effect of Anisotropy on the Localization in a Bifractal System

Bifractal is a highly anisotropic structure where planar fractals are stacked to form a 3-dimensional lattice. The localization lengths along fractal structure for the Anderson model defined on a bifractal are calculated. The critical disorder and the critical exponent of the localization lengths are obtained from the finite size scaling behavior. The numerical results are in a good agreement with previous results which have been obtained from the localization lengths along the perpendicular direction. This suggests that the anisotropy of the embedding lattice structure is irrelevant to the critical properties of the localization.Comment: 3 pages, source TeX file and 3 epsi figures. submitted to PR

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

Dimensional crossover and metal-insulator transition in quasi-two-dimensional disordered conductors

We study the metal-insulator transition (MIT) in weakly coupled disordered planes on the basis of a Non-Linear Sigma Model (NL$\sigma$M). Using two different methods, a renormalization group (RG) approach and an auxiliary field method, we calculate the crossover length between a 2D regime at small length scales and a 3D regime at larger length scales. The 3D regime is described by an anisotropic 3D NL$\sigma$M with renormalized coupling constants. We obtain the critical value of the single particle interplane hopping which separates the metallic and insulating phases. We also show that a strong parallel magnetic field favors the localized phase and derive the phase diagram.Comment: 16 pages (RevTex), 4 poscript figure

Multifractal analysis of the metal-insulator transition in anisotropic systems

We study the Anderson model of localization with anisotropic hopping in three dimensions for weakly coupled chains and weakly coupled planes. The eigenstates of the Hamiltonian, as computed by Lanczos diagonalization for systems of sizes up to $48^3$, show multifractal behavior at the metal-insulator transition even for strong anisotropy. The critical disorder strength $W_c$ determined from the system size dependence of the singularity spectra is in a reasonable agreement with a recent study using transfer matrix methods. But the respective spectrum at $W_c$ deviates from the ``characteristic spectrum'' determined for the isotropic system. This indicates a quantitative difference of the multifractal properties of states of the anisotropic as compared to the isotropic system. Further, we calculate the Kubo conductivity for given anisotropies by exact diagonalization. Already for small system sizes of only $12^3$ sites we observe a rapidly decreasing conductivity in the directions with reduced hopping if the coupling becomes weaker.Comment: 25 RevTeX pages with 10 PS-figures include

Dimensional Crossover of Weak Localization in a Magnetic Field

We study the dimensional crossover of weak localization in strongly anisotropic systems. This crossover from three-dimensional behavior to an effective lower dimensional system is triggered by increasing temperature if the phase coherence length gets shorter than the lattice spacing $a$. A similar effect occurs in a magnetic field if the magnetic length $L_m$ becomes shorter than $a(D_{||}/D_\perp)^\gamma$, where \D_{||}/D_\perp is the ratio of the diffusion coefficients parallel and perpendicular to the planes or chains. $\gamma$ depends on the direction of the magnetic field, e.g. $\gamma=1/4$ or 1/2 for a magnetic field parallel or perpendicular to the planes in a quasi two-dimensional system. We show that even in the limit of large magnetic field, weak localization is not fully suppressed in a lattice system. Experimental implications are discussed in detail.Comment: RevTeX, 11 pages, 4 figures; three references added and discusse

Localization Transition in Multilayered Disordered Systems

The Anderson delocalization-localization transition is studied in multilayered systems with randomly placed interlayer bonds of density $p$ and strength $t$. In the absence of diagonal disorder (W=0), following an appropriate perturbation expansion, we estimate the mean free paths in the main directions and verify by scaling of the conductance that the states remain extended for any finite $p$, despite the interlayer disorder. In the presence of additional diagonal disorder ($W > 0$) we obtain an Anderson transition with critical disorder $W_c$ and localization length exponent $\nu$ independently of the direction. The critical conductance distribution $P_{c}(g)$ varies, however, for the parallel and the perpendicular directions. The results are discussed in connection to disordered anisotropic materials.Comment: 10 pages, Revtex file, 8 postscript files, minor change

The three-dimensional Anderson model of localization with binary random potential

We study the three-dimensional two-band Anderson model of localization and compare our results to experimental results for amorphous metallic alloys (AMA). Using the transfer-matrix method, we identify and characterize the metal-insulator transitions as functions of Fermi level position, band broadening due to disorder and concentration of alloy composition. The appropriate phase diagrams of regions of extended and localized electronic states are studied and qualitative agreement with AMA such as Ti-Ni and Ti-Cu metallic glasses is found. We estimate the critical exponents nu_W, nu_E and nu_x when either disorder W, energy E or concentration x is varied, respectively. All our results are compatible with the universal value nu ~ 1.6 obtained in the single-band Anderson model.Comment: 9 RevTeX4 pages with 11 .eps figures included, submitted to PR

Energy-level statistics at the metal-insulator transition in anisotropic systems

We study the three-dimensional Anderson model of localization with anisotropic hopping, i.e. weakly coupled chains and weakly coupled planes. In our extensive numerical study we identify and characterize the metal-insulator transition using energy-level statistics. The values of the critical disorder $W_c$ are consistent with results of previous studies, including the transfer-matrix method and multifractal analysis of the wave functions. $W_c$ decreases from its isotropic value with a power law as a function of anisotropy. Using high accuracy data for large system sizes we estimate the critical exponent $\nu=1.45\pm0.2$. This is in agreement with its value in the isotropic case and in other models of the orthogonal universality class. The critical level statistics which is independent of the system size at the transition changes from its isotropic form towards the Poisson statistics with increasing anisotropy.Comment: 22 pages, including 8 figures, revtex few typos corrected, added journal referenc