Quantum anomalous Hall effect with tunable Chern number in magnetic topological insulator film

We study the possibility of realizing quantum anomalous Hall effect (QAHE) with tunable Chern number through doping magnetic elements in the multi-layer topological insulator film. We find that high Chern number QAHE phases exist in the ideal neutral samples and it can make transition to another QAHE phase directly by means of tuning the exchange field strength or sample thickness. With the help of an extended Haldane model, we demonstrate the physical mechanism of the tunable Chern number QAHE phase. Further, we study the phenomena in the realistic samples with the Fermi energy being gating-modulated, where the multiple-integer QAHE plateaus remain and two separated QAHE phase are connected by metallic phase. The stability of the high Chern number QAHE phase is also discussed.