The Study Of Electronic Localization In Low-dimensional Disordered Systems

For several low-dimensional systems such as one-dimensional disordered binary alloy, one-dimensional Fibonacci chain, DNA sequence, quasi-two-dimensional disordered systems with multi-chains and one-dimensional correlated sequences, based on the tight-binding model of the single electron, the properties of electronic localization are investigated systematically. With the help of many methods include in existence and created by authors, for example, the negative eigenvalue theory, the infinite order perturbation theory, the transfer-matrix approach, the renormalization-group method and the solving methods of three, five or seven diagonally symmetric matrixes, we study the densities of states, the eigen-energies, the eigen-vectors, the localization lengths, transmission coefficients and the conductance of above systems, and find the internal laws.For one-dimensional disordered binary alloy, using the tight-binding model of the single electron, where two different diagonal energiesε_A andε_B are assigned at random to each lattice site, we find that the electronic wave-functions are localized as expected and the densities of states and the localization lengths depend strongly on the eigen-energies, the concentrations and the order of hopping integral between sites. Under the special setting of the diagonal energies, we show the existence of the energy gaps. For finite systems, we confirm there exist localization-delocalization transition induced by concentration, but those extended states aren't stable and will disappear in the thermodynamic limit. If one of the sites energies is assigned at random to pairs of lattice sites, we obtain the random-dimer model and it can be shown to exhibit an absence of localization.For the Fibonacci chain constructed recursively as S_m+1={S_m|S_m-1}, using the one-dimensional random walk model, we investigate systematically the auto-correlation function, the standard deviation of displacement and the rescaled range function. The results show these statistical quantities can well reflect the natures of Fibonacci chain such as the correlation, scaling invariability, self-similarity and the trifurcating structure of the energy spectrum. Using the renormalization-group method together with the scattering theory, we study theoretically the localization length and the transmission coefficient of electronic states of this system and find there exit some extended states which have transmission coefficients equal or nearly equal to unit 1. But the electronic states corresponding to much more range of energy have very small localization length, which implies there are a number of localized states in the system. In addition, we find, as the length of chain increases, fewer states will present good transmission and both the number and the location of the resonant peaks will change to some extent.For the DNA sequences made up from four nucleotides: G, A, C, T or in other words from two pairs of nucleotides: AT, CG, we study the electronic properties of them by using the general disordered theories. Under the condition ofT = 0, we show that the random DNA sequence is insulator and that the localization length and conductivity strongly depend on energies and disordered degrees. Especially with finite size DNA sequence, the localization length and conductivity present obvious effect on nucleotides base pairs' concentration. Under the condition of T≠0, we discuss the effect of temperature by considering the relative twist angle from its equilibrium value between sites. The results show that the densities of states, the localization lengths and the transmission coefficients decrease with the temperature when T＜250K. In addition, for a DNA molecule with off-diagonal short-range correlation, we use the renormalization-group method to obtain an effective Hamiltonian and discuss the dependence of transmission coefficients on the energy eigen-value of DNA molecule under the condition of typically local correlation and golden correlation. The results show that both styles of correlation lead to extended states, and furthermore, at golden correlation condition for the hopping integrals, there exist much more extended states than at typically local correlation.Basing on the infinite order perturbation theory, we obtain the eigen-values of seven diagonally symmetric matrixes and investigate the electronic properties of disordered systems made up from three chains for three instances: diagonal disorder, non-diagonal disorder and full disorder. By comparing the results with that of systems with single chain or two chains, we fred that the densities of states in diagonal disordered systems depend strongly on the dimensionalities of the systems, while, with the non-diagonal disordered system, the dimensional effect becomes illegible. With the disordered systems made up from more than four chains, taking a special method to code the sites, we write the Hamiltonians of the systems as precisely symmetric matrixes, which can be transformed into three diagonally symmetric matrixes by using the Householder transformation. The densities of states, the localization lengths and the conductance of the systems are calculated numerically. We find, the distribution of the density of states changes obviously with the increase of the effective dimensionality. Especially, for the systems with multi-chains, at the energy band center, we find extended states, which indicate, with the increasing of the number of the chains, the correlated ranges expand and the systems present the similar behavior to that with off-diagonal long-range correlation. In order to study the correlated effect furthermore, we generated the long-range correlated random energy sequences with the power-law spectral density s(q)∝q^-p in the one-dimensional Anderson disordered chain, and then investigated the localization length. Our results show that after introduced the long-range diagonal correlations in the one-dimensional Anderson disordered chain, the properties of electronic states change greatly, and there exists a localization-delocalization transition at the energy band center.