Electronic Phenomena in 2D Dirac-like Systems: Silicene and Topological Insulator Surface States
With the discovery of graphene, considerable attention has been given to two-dimensional (2D) physics. Over the past few years, several new systems have been synthesized. Of particular prominence are topologically nontrivial insulators. Here, metallic boundary states exist between regions of different topology. While the quantum Hall system has been known to exist since 1980, these systems display topological behaviour even in the absence of a magnetic field; additionally, their bound states carry a spin current as opposed to the electric current of the quantum Hall effect. In this thesis, we explore several novel phenomena in 2D topological Dirac-like systems. In particular, we consider a low-energy model for a buckled honeycomb lattice which maps onto the Kane-Mele Hamiltonian; we also examine the Bychkov-Rashba Hamiltonian for the surface states of a three-dimensional (3D) topological insulator. In the former system, we explore spin- and valley-polarized responses in the optical and Hall conductivities with particular attention to the effects of sublattice asymmetry which can be tuned by an external electric field. We also consider quantum oscillations and electron-electron interactions. For the 2D Dirac-like surface states of a 3D topological insulator, we restrict our attention to finite magnetic fields and emphasize the particle-hole asymmetry which is manifest in the optical and Hall conductivities as well as magnetic oscillations.