Electron dynamics on the surface of a three-dimensional topological insulator

In topological insulators momentum and spin lock in a well-defined way. With offset in this spin–momentum locking mechanism eigenenergies for Landau levels are investigated. In particular, the Landau-level eigenenergies with the warping effect included are numerically computed. The density of states for the topological surface-state electrons with and without the warping effect is found and compared. It is seen that the warping effect decreases the density of states, and that Landau levels are clearly seen in the density of states. The total density of states for the topological surface-state electrons and the bulk conduction band-induced surface-state electrons is also found numerically. By semiclassical considerations, it is found that as the energy spectrum of Dirac electrons becomes gapped, the magnitude of Berry’s phase is reduced from the non-gapped value of π to 0 as the gap strength becomes very large. Including warping has the effect of increasing the magnitude of Berry’s phase. The connection between Berry’s phase and the anomalous velocity is discussed and the associated anomalous quantum Hall conductivity is found to be quantized in units of (e/~)/(4π). Finally, the Bohr–Sommerfeld quantization condition is utilized to obtain the energies of the surface-state electrons and these energies are compared to energies computed in a finite-size Hilbert space approximation.