TRIGONAL WARPING EFFECTS ON OPTICAL PROPERTIES OF ANOMALOUS HALL MATERIALS - Mahayu Adhidewata, Jyesta
Many interesting phenomena in two-dimensional insulators are determined by the topological properties of the materials, such as the quantum Hall effect and the quantum spin Hall effect. The topological nature of these topological insulators is related to the symmetries present in the material, for example, quantum spin Hall effect can be observed in topological insulators with time reversal (TR) symmetry, while broken time reversal symmetry may give rise to the presence of anomalous quantum Hall effect, a formation of Hall current/voltage without any external magnetic field. In this thesis, we will consider the effects of broken rotational symmetry on a material with anomalous quantum Hall effect. We start with materials whose electrons can be described by the Dirac Hamiltonian. The simplest example of this system is graphene. By adding a broken TR band gap, we obtain quantum anomalous Hall effect with topological invariant described by the Chern number C= ±1/2. Interestingly, upon further breaking the rotational symmetry with a trigonal warping term we found that the Chern number morphs into C=?1. The origin of this topological transition is the emergence of three additional Dirac cones, with topological charges of opposite sign from the original one. To measure the efffect of this trigonal warping term, we will calculate the dynamical (AC) conductivity of our model. The presence of anomalous Hall conductivity can rotate the incoming electric field, and results in Kerr and Faraday rotations observed in reflected and transmitted waves, respectively. As the Kerr and Faraday rotations are approximately proportional to the optical Hall conductivity, we expect the Kerr and Faraday rotation to reflect the effects of trigonal warping on the Hall conductivity. The calculation of dynamical conductivity is done semi-analytically, using a pseudo-spin dynamics approach of the materials’ wave function. Numerical integration is done to find the numerical plot of the dynamic conductivity. Due to the C=?1 Chern number of the trigonal warping material, we found that the Hall conductivity for the trigonal warping material is larger, up to twice of the Dirac material’s Hall conductivity. Furthermore, we also observe a singularity in the longitudinal conductivity of our trigonal warping material. This singularity is related to the presence of the van Hove singularity in the trigonal warping Hamiltonian. We also calculate the second order optical conductivity of our material. We showed that due to zero Berry curvature dipole, there is no intraband contribution from the anomalous circular photocurrent to the Hall conductivity our material. However, the shift current is non-zero, and is directed to the ±y direction depending on the direction of the electric field. This anisotropy in the direction of the shift current is related to the symmetry properties of the trigonal warping Hamiltonian. Interestingly, we found that the direction of this shift current is unaffected by the chirality of the Dirac cone, which means even with Berry curvature dipole and time reversal symmetry we can obtain a nonzero nonlinear Hall current.