Properties of the α-T3 Model
The α-T3 model interpolates between the pseudospin S = 1/2 honeycomb lattice of graphene and the pseudospin S = 1 dice lattice via the parameter α. In this thesis, we explore properties of the α-T3 model, while making connections to some of the famous results for graphene in the α → 0 limit of the model. A unique aspect of the α-T3 model is the variable Berry phase that changes contin- uously from 0 to π as we vary the parameter α. We examine the effect of this variable Berry phase on the density of states, the optical conductivity, the angular scattering probability, and the Hall conductivity. In particular, we study the Hall quantization as it evolves from a relativistic to a non-relativistic series and explicitly describe the Berry phase dependence of the dynamical longitudinal optical conductivity for the model. The quasiparticles of Dirac materials behave like massless Dirac fermions, afford- ing opportunities for testing high-energy physics phenomena, such as Klein tunnelling. We calculate Klein tunnelling for the α-T3 lattice, and examine transmission across sharp potential steps and barriers. We connect our results to previous calculationsfor the two limiting cases, and find a general trend of increased transmission as we increase the parameter α in the intermediate regime. In the presence of a magnetic field, the intermediate regime of the α-T3 model is characterized by differing Landau level energies in the K and K′ valleys. We calculate the magneto-optical response of the lattice and find signatures of these differing Landau levels in the form of doublets in the absorptive part of the magneto- optical conductivity. Also in a magnetic field, we calculate the Hofstadter butterfly spectra of the α-T3 lattice, and detail the evolution of the Hofstadter butterfly as it changes periodicity by a factor of three as we vary α between the two limiting cases.