Doctor of Philosophy Oral Defense:
Advisor: Truell W. Hyde, Ph.D.
Title: Spectral Approach to Transport Problems in
Two-Dimensional Disordered Lattices: Physical Interpretation and Applications
In condensed matter, a crystal without impurities at zero temperature acts like a perfect conductor for a travelling electron. As the amount of lattice disorder reaches a critical value, the electron wavefunction experiences a transition from extended to a localized state, called Anderson localization. Here we introduce the spectral approach to transport problems in infinite disordered lattices characterized by Anderson-type Hamiltonians. The spectral approach determines (with probability
) the existence of extended states for nonzero disorder in infinite lattices of any dimension and geometry. This presentation is focused on the critical two-dimensional (2D) case, where previous numerical and experimental results have shown disagreement with theory. Not being based on scaling theory, the proposed method avoids issues related to boundary conditions and provides an alternative approach to transport problems in which interaction with various types of disorder is considered.
We start with a discussion of the mathematical formulation and physical interpretation of the spectral approach. The method is then applied to the Anderson localization problem in the two-dimensional disordered honeycomb, triangular, and square lattices. Next, we investigate transport in the 2D honeycomb lattice with substitutional disorder using a spectral analysis of the quantum percolation problem. Finally, the applicability of the method is extended to the classical regime, where we examine diffusion of in-plane lattice waves in a 2D disordered complex plasma crystal.
Reception following in D.311 Conference Room
Hosted by CASPER