UC San Diego

Two-dimensional materials are one-atom-thick crystals, which are stable under ambient conditions. Heterostructures by stacking of two-dimensional (2D) crystals via the van der Waals force provide a versatile platform for investigation of emergent properties of composite materials. In this thesis, I studied three 2D materials, graphene, Bi$_2$Se$_3$ and hexagonal boron nitride (hBN), of which the first two materials host 2D Dirac fermions. The core of this thesis is to study the transport and optical properties of 2D Dirac fermions interacted with their three-dimensional (3D) environments.

In Chapter 2, we consider electron transport of graphene, adsorbing clusters of charged impurities. We model the clusters as circular barriers. We calculate the differential, total, and transport cross-sections for scattering of two-dimensional massless Dirac electrons by a circular barrier. For scatterer of a small radius, the cross-sections are dominated by quantum effects such as resonant scattering that can be computed using the partial-wave series. Scattering by larger size barriers is better described within the classical picture of reflection and refraction of rays, which leads to phenomena of caustics, rainbow, and critical scattering. Refraction can be negative if the potential of the scatterer is repulsive, so that a $p$-$n$ junction forms at its boundary. Qualitative differences of this case from the $n$-$N$ doping case are examined. Quantum interference effects beyond the classical ray picture are also considered, such as normal and anomalous diffraction, and also whispering-gallery resonances. Implications of these results for transport and scanned-probe experiments in graphene and topological insulators are discussed.

In Chapter 3, we consider how the Dirac plasmons of Bi$_2$Se$_3$ are coupled with its phonon polaritons. Layered topological insulators, for example, Bi$_2$Se$_3$ are optically hyperbolic materials in a range of THz frequencies. Such materials possess deeply subdiffractional, highly directional collective modes: hyperbolic phonon-polaritons. In thin crystals the dispersion of such modes is split into discrete subbands and is strongly influenced by electron surface states. If the surface states are doped, then hybrid collective modes result from coupling of the phonon-polaritons with surface plasmons. The strength of the hybridization can be controlled by an external gate that varies the chemical potential of the surface states. Momentum-dependence of the plasmon-phonon coupling leads to a polaritonic analog of the Goos-Hänchen effect. Directionality of the polaritonic rays and their tunable Goos-Hänchen shift are observable via THz nanoimaging.