| In graphene the energy momentum dispersion has a point Fermi surface and is linear at the zone boundary. To actually discover a feature in a band structure that provides the quasiparticle dispersion of a new and unexpected type is rare, the discovery of a "semi Dirac'' dispersion pinned to the Fermi energy is a very recent example. Pardo et al. reported such a finding in ultrathin (001) VO2 layers embedded in TiO2. This new point Fermi surface system, dubbed 'semi-Dirac,' is a hybrid of conventional and unconventional: dispersion is linear ("massless'', Dirac-Weyl) in one of the directions of the two-dimensional (2D) layer, and is conventional quadratic ("massive'' Dirac) in the perpendicular direction.;In this dissertation the tightbinding model for the semi-Dirac dispersion is developed. The tightbinding model allows one to describe the semi-Dirac dispersion in terms of an electron hopping from one lattice site to another in a square lattice arrangement. Several low energy features of the semi-Dirac dispersion have been considered. That includes Hall coefficient, magnetic susceptibility and heat capacity. Results for the plasmon dispersion relation in the small momentum regime are obtained and the extreme anisotropy thereof is pointed out. Klein tunneling is considered for the case of normal impingement on a barrier oriented at an arbitrary angle. It is found that perfect transmission occurs under certain conditions. Comparisons of the semi-Dirac system are made throughout with other types of point Fermi surface systems. While for some properties semi-Dirac behavior is intermediate between these two, in some cases it displays several quite distinct properties. The Berry's phase for the semi-Dirac system is computed, and the result obtained is rather unexpected. In the dissertation Quantum Chaos and its relevance to the semi-Dirac dispersion is discussed. The energy level statistics for the semi-Dirac dispersion is obtained, which shows a novel and a very rich behavior compared to other conventional dispersions. |
