The role of pseudospin in the optical and electronic properties of relativistic materials

This thesis focuses on the theoretical analysis of response functions, namely the optical conductivity and the dielectric function, of relativistic materials described by various values of pseudospin. Graphene, first theorized in 1947 but only discovered in 2004, is the hallmark two-dimensional relativistic material. The dynamics of low-energy quasiparticle excitations in graphene are described by the relativistic Dirac equation, despite the lack of any motion occurring at relativistic speeds. In addition to carrying the intrinsic spin of the electron, the Dirac fermions in graphene are imbued with an additional quantum spin-1/2 angular momentum referred to as pseudospin. Extending the mathematical theory behind graphene, it is possible to consider materials with pseudospin values higher than 1/2. The promise of graphene in future technologies and the remarkable behaviour in its quasiparticles prompts the search for other relativistic materials. Response functions are useful in this endeavour in that they describe the way that a specific system will interact with an experimental probe, allowing for the identification of new materials. These functions also carry a large amount of information about the system under study, more than can be surmised from the band structure alone. In this thesis, the magneto-optical conductivity of higher-pseudospin two-dimensional Dirac materials is analyzed. Signatures unique to each system are identified with the help of snowshoe diagrams. The same analysis is performed on the Kane system, a 3D model describing small-gap zincblende semiconductors. Under certain approximations, the Kane model exhibits massless excitations which are shown to be hybrid pseudospin-1/2 and pseudospin-1 Dirac fermions. This is then applied to a particular phase of the zincblende material HgCdTe in order to calculate its optical absorbance spectra and compare with experiment. Finally, the pseudospin-1 system is focused on specifically through the full derivation of the dynamical polarizability. This function describes all of the dielectric properties of the material, which in turn renormalizes the Coulomb interaction between charged species. The pseudospin-1 polarizability is compared to that of the pseudospin-1/2 system, showing novel differences due to the presence of a flat band fixed at zero energy in the former system. From the dielectric function, some of the collective behaviour of the pseudospin-1 system is analyzed (plasmon excitations and screening around electromagnetic impurities).