Optical Properties of Massive Topological Dirac Semimetals*

Most crystalline materials can be approximated as infinitely periodic. Our group is interested in understanding these infinitely periodic lattice structures using the tight-binding model approach, as this model allows us to better understand the dynamics of electrons within the material. We choose a basis where electrons are bound to localized atoms at different lattice sites, and then we consider the different hops that the electron can take to neighboring sites. We then use this information to calculate the band structure, which tells us the range of energy levels that the electrons may have within the material, and the density of states, which allows us to know the different ways electrons can exist in a certain energy window.

We expand upon our research last summer by analytically exploring a two-dimensional model for a Dirac fermion, where the electron energy linearly scales with its momentum, on a square lattice. Upon inclusion of a mass term, the system transitions from a topological semimetal, where electrons can flow in the bulk of the material, to a topological insulator, where electrons flow significantly along the edges. In this new state, the current transverse to an applied voltage (the Hall conductance) is a quantized number proportional to the band structure’s Chern number.