Towards the Realization of Systematic, Self-Consistent Typical Medium Theory for Interacting Disordered Systems

This work is devoted to the development of a systematic method for studying electron localization. The developed method is Typical Medium Dynamical Cluster Approximation (TMDCA) using the Anderson-Hubbard model. The TMDCA incorporates non-local correlations beyond the local typical environment in a self-consistent way utilizing the momentum resolved typical-density-of-states and the non-local hybridization function to characterize the localization transition. For the (non-interacting) Anderson model, I show that the TMDCA provides a proper description of the Anderson localization transition in one, two, and three dimensions. In three-dimensions, as a function of cluster size, the TMDCA systematically recovers the re-entrance behavior of the mobility edge and obtains the correct critical disorder strength for the various disorder configurations and the associated \textit{universal order-parameter-critical-exponent} $\beta$ and in lower-dimensions, the well-knowing scaling relations are reproduced in agreement with numerical exact results. The TMDCA is also extended to treat diagonal and off-diagonal disorder by generalizing the local Blackman-Esterling-Berk and the importance of finite cluster is demonstrated. It was further generalized for multiband systems. Applying the TMDCA to weakly interaction electronic systems, I show that incorporating Coulomb interactions into disordered electron system result in two competing tendencies: the suppression of the current due to correlations and the screening of the disorder leading to the homogenizing of the system. It is shown that the critical disorder strength ($W_c^U$), required to localize all states, increases with increasing interactions ($U$); implying that the metallic phase is stabilized by interactions. Using the results, a soft pseudogap at the intermediate $W$ close to $W_c^U$ is predicted independent of filling and dimension, and I demonstrate in three-dimensions that the mobility edge is preserved as long as the chemical potential, $\mu$, is at or beyond the mobility edge energy ($\omega_\epsilon$). A two-particle formalism of electron localization is also developed within the TMDCA and used to calculate the direct-current conductivity, enabling direct comparison with experiments. Note significantly, the TMDCA benchmarks well with numerical exact results with a dramatic reduction in computational cost, enabling the incorporation of material\u27s specific details as such provide an avenue for the possibility of studying electron localization in real materials

First principle electronic, structural, elastic, and optical properties of strontium titanate

We report self-consistent ab-initio electronic, structural, elastic, and optical properties of cubic SrTiO$_{3}$ perovskite. Our non-relativistic calculations employed a generalized gradient approximation (GGA) potential and the linear combination of atomic orbitals (LCAO) formalism. The distinctive feature of our computations stem from solving self-consistently the system of equations describing the GGA, using the Bagayoko-Zhao-Williams (BZW) method. Our results are in agreement with experimental ones where the later are available. In particular, our theoretical, indirect band gap of 3.24 eV, at the experimental lattice constant of 3.91 \AA{}, is in excellent agreement with experiment. Our predicted, equilibrium lattice constant is 3.92 \AA{}, with a corresponding indirect band gap of 3.21 eV and bulk modulus of 183 GPa.Comment: 11 pages, 6 figures,Accepted for publication in AIP Advances (2012

First Principle Local Density Approximation Description of the Electronic Properties of Ferroelectric Sodium Nitrite

The electronic structure of the ferroelectric crystal, NaNO$_2$, is studied by means of first-principles, local density calculations. Our ab-initio, non-relativistic calculations employed a local density functional approximation (LDA) potential and the linear combination of atomic orbitals (LCAO). Following the Bagayoko, Zhao, Williams, method, as enhanced by Ekuma, and Franklin (BZW-EF), we solved self-consistently both the Kohn-Sham equation and the equation giving the ground state charge density in terms of the wave functions of the occupied states. We found an indirect band gap of 2.83 eV, from W to R. Our calculated direct gaps are 2.90, 2.98, 3.02, 3.22, and 3.51 eV at R, W, X, {\Gamma}, and T, respectively. The band structure and density of states show high localization, typical of a molecular solid. The partial density of states shows that the valence bands are formed only by complex anionic states. These results are in excellent agreement with experiment. So are the calculated densities of states. Our calculated electron effective masses of 1.18, 0.63, and 0.73 mo in the {\Gamma}-X, {\Gamma}-R, and {\Gamma}-W directions, respectively, show the highly anisotropic nature of this material.Comment: 13 Pages, 4 Figures, and 2 Table

Interpolation problems, the symmetrized bidisc and the tetrablock

PhD ThesisThe spectral Nevanlinna-Pick interpolation problem is to find, if it is possible, an analytic function f : D → Ck×k from the unit disc D = {z ∈ C : |z| < 1} to the space Ck×k of k × k complex matrices, which interpolates a finite number of distinct points in D to the target matrices in Ck×k subject to the spectral radius r(f(λ)) ≤ 1, for every λ ∈ D. For k = 2, this problem is connected to interpolation problem in Hol(D, Γ), where Hol(D, Γ) denotes the space of analytic functions from D to the closed symmetrized bidisc Γ = {(z1 + z2, z1z2) : z1, z2 ∈ D} ⊂ C 2 . In this thesis, we consider a special case of the three-point spectral Nevanlinna-Pick problem and give necessary and sufficient conditions for its solvability. We also study interpolation problems from D to the tetrablock. The closed tetrablock is defined to be E = {x ∈ C 3 : 1 − x1z − x2w + x3zw 6= 0 for all z, w ∈ D}. Given n distinct points λ1, · · · , λn in D and n points x 1 , · · · , x n in E, find, if is possible, an analytic function ϕ : D → E such that ϕ(λj) = x j for j = 1, · · · , n. This problem is closely connected to the µDiag-synthesis interpolation problem. For given data λj → Wj , 1 ≤ j ≤ n, where λj are distinct points in D and Wj are complex 2 × 2 matrices, find, if it is possible, an analytic matrix function F : D → C 2×2 such that F(λj) = Wj , 1 ≤ j ≤ n, and µDiag(F(λ)) ≤ 1 for all λ ∈ D. We give criteria for the solvability of such interpolation problems. Here Diag is the space of 2 × 2 diagonal matrices, and for A ∈ C2×2 , µDiag(A) := 1 inf{kXk : X ∈ Diag, 1 − AX is singular} . If 1 − AX is non-singular for all X ∈ Diag, then µDiag(A) = 0. In addition, we give a realization theorem for analytic functions from the disc to the tetrablock.government of Nigeria for the intervention, Tertiary Education Trust Fund, TETFUND, and my institution, Alex-Ekwueme Federal University, Ndufu-Alike Ikwo, (AEFUNAI) for financial suppor