Thin topological insulator film in a perpendicular magnetic field

We report on a study of an ultrathin topological insulator film with hybridization between the top and bottom surfaces, placed in a quantizing perpendicular magnetic field. We calculate the full Landau-level spectrum of the film as a function of the applied magnetic field and the magnitude of the hybridization matrix element, taking into account both the orbital and the Zeeman spin splitting effects of the field. For an undoped film, we find a quantum phase transition between a state with a zero Hall conductivity and a state with a quantized Hall conductivity equal to $e^2/h$, as a function of the magnitude of the applied field. The transition is driven by the competition between the Zeeman and the hybridization energies.

Figure 1

LL density of states of a thin TI film (broadening added by hand). Dashed line represents the zero of energy, set at the Dirac point position in the absence of the magnetic field. (a) ${\Delta }_z=0$, ${\Delta }_t>0$ case. Only the $n=0$ LL is split and the spectrum is particle-hole symmetric. (b) ${\Delta }_z>{\Delta }_t>0$ case. Both $n=0$ LL are holelike and particle-hole symmetry of the spectrum is broken. All LL splittings in the figure are exaggerated relative to the main LL spacing.