Gap properties of helical edge states in two-dimensional topological insulators with time-dependent magnetic impurities

Abstract: The quantum mechanical equivalent of the classical Hall effect can lead to interesting results in solid state physics. A similar effect, that has received attention in recent years, occurs when large spin-orbit coupling is present in a material, the so-called quantum-spin Hall effect. In two-dimensional materials it leads to so-called helical edge states that exhibit counter-propagating electron states located at the edge with spin-momentum locking protected by time-reversal symmetry. That means, different spin species travel in different directions along the edge and cannot be scattered into each other unless time-reversal is broken by a magnetic contribution. Two-dimensional materials hosting these quantum spin Hall states are commonly called two-dimensional topological insulators. In this thesis we investigate the electronic structure of the edge states in the presence of magnetic impurities with rotating magnetic moments. Since they break time reversal symmetry, the impurities lead to backscattering and the density of states can be altered drastically in their presence. To calculate the time-averaged density of states, a Floquet-like approach is applied to the single-particle Green's function of the 2x2 effective edge Hamiltonian of the Bernevig-Hughes-Zhang model with impurities. The rotation of the impurities in the x-z-plane leads to the density of states transitioning between a gapped and an ungapped state, which in turn leads to drastically different shapes of the density of states depending on the driving frequency. A numerical model is derived and criteria for choosing reasonable numerical cut-offs are given. The resulting density of states looks different for different driving frequencies. Slow driving, compared to a time scale defined by the magnetic impurity strength, leads to a density of states comparable to the average over static impurities for different impurity orientations. Fast driving effectively does not alter the low energy density of states, leaving it constant around the center, with distinct resonances at energies related to the driving frequency. Driving with frequencies around the time scale defined by the impurities leads to different results, exhibiting additional resonances, broadened due to the impurity nature of the system.