High Chern number phase in topological insulator multilayer structures

The high Chern number phases with a Chern number $Cg1$ have been observed in a recent experiment that was performed on the topological insulator (TI) multilayer structures, consisting of the alternating magnetic-doped and undoped TI layers. In this paper, we develop an effective method to determine the Chern numbers in the TI multilayer structures and then make a systematic study on the Chern number phase diagrams that are modulated by the magnetic doping and the middle-layer thickness. We point out that in the multilayer structure, the high-$C$ behavior can be attributed to the band inversion mechanisms. Moreover, we find that the lowest bands may be multifold degenerate around the $\mathrm{\ensuremath{\Gamma}}$ point, and when they are inverted, the Chern number change will be larger than 1. Besides the TI multilayer structures implemented in the experiment, we also explore the high-$C$ phase realizations in two other kinds of the TI multilayer structures. The implications of our results for experiments are discussed.