Anomalous Behaviour in the Magneto-Optics of a Gapped Topological Insulator

Abstract: The Dirac fermions at the surface of a topological insulator can be gapped by introducing magnetic dopants. Alternatively, in an ultra-thin slab with thickness on the order of the extent of the surface states, both the top and bottom surface states acquire a common gap value ($\Delta$) but with opposite sign. In a topological insulator, the dominant piece of the Hamiltonian ($\hat{H}$) is of a relativistic nature. A subdominant non-relativistic piece is also present and in an external magnetic field ($B$) applied perpendicular to the surface, the $N=0$ Landau level is no longer at zero energy but is shifted to positive energy by the Schr{\"o}dinger magnetic energy. When a gap is present, it further shifts this level down by $-\Delta$ for positive $\Delta$ and up by $|\Delta|$ for a negative gap. This has important consequences for the magneto-optical properties of such systems. In particular, at charge neutrality, the lowest energy transition displays anomalous non-monotonic behaviour as a function of $B$ in both its position in energy and its optical spectral weight. The gap can also have a profound impact on the spectral weight of the interband lines and on corresponding structures in the real part of the dynamical Hall conductivity. Conversely, the interband background in zero field remains unchanged by the non-relativistic term in $\hat{H}$ (although its onset frequency is modified).