The theme of this thesis is the study of wave phenomena in complex systems. In particular, the following three topics constitute the foci of my research.

The first topic involves the generalization of an electronic transport mechanism commonly observed in disordered media, fluctuation induced tunneling conduction, by considering tunneling through not just insulating potential barriers, but also narrow conducting channels. Here the wave nature of the electron implies that a narrow conduction channel can act as an electronic waveguide, with a cutoff transverse dimension that is half the Fermi wavelength. My research involves the study of electronic transport through finite-length conducting channels with transverse dimensions below the cutoff. Such narrow conduction channel may be physically realized by chains of single conducting atoms, for example. At small voltage bias across the conduction channel, only tunneling transport is possible at zero temperature. But at finite temperatures some of the electrons with energies above the Fermi level can ballistically transport across the channel. By considering both tunneling and thermal activation mechanisms, with thermally-generated (random) voltage bias across the narrow channel, we obtained a temperature-dependent conductivity behavior that is in good agreement with the measured two-lead conductance of RuO₂ and IrO₂ nanowires. Furthermore, by considering high applied voltage across the nano conduction channels, our model predicts interesting electronic Fabry-Perot behavior whose experimental verification is presently underway.

The second topic involves the study of the Hall effect in mesoscopic samples. In particular, we are interested in the possibility of enhancing the Hall effect by nano-patterning samples of 2D electron gas. Through numerical solution of the Schrodinger equation in the presence of a magnetic field, mesoscopic transport behavior is obtained for samples with given geometric patterns of the inhomogeneities. By varying such geometric patterns, we attempt to optimize the ratio of the Hall voltage to the longitudinal voltage at the limit of small applied magnetic field. Several optimal nano-pattern designs have been found through the use of genetic algorithm, with enhancement of the Hall effect up to 500%.

The third topic involves the interaction of electromagnetic waves with crystals composed of fractal units. Here the fractal unit can possess multiple electromagnetic resonances, in exact analogy to atoms in a solid crystal. In particular, these resonances can be very subwavelength in character, and when the fractal units are periodically arranged, there can be coupling between the fractal local resonances and Bragg scattering, leading to hybrid modes with a rich array of interesting characteristics.

The study of the above wave phenomena requires a diverse assortment of numerical techniques. A section of the thesis is devoted to their exposition.