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

We study the possibility of realizing quantum anomalous Hall (QAH) effect with tunable Chern number through doping magnetic elements in a multilayer topological insulator film. We find that high Chern number QAH phases exist in ideal neutral samples and can make transition to another QAH phase directly by means of tuning exchange field strength or sample thickness. With the help of an extended Haldane model, we demonstrate the physical mechanism of the tunable Chern number QAH phase. We show that the high Chern number QAH phases are robust against weak magnetic and nonmagnetic disorders.

Figure 1

Schematic plot of the setup: a magnetically doped topological insulator on a substrate with its Fermi energy being tuned by the attached external gate. Blue arrows indicate the magnetized dopants. $N_z$ measures the sample layers along the $Z$ direction.

Figure 2

The bulk band spectra along the high-symmetry lines of the magnetic element doped topological insulator film in the presence of different exchange field strength $m=0$ (a), 0.037 (b), 0.3 (c), 0.525 (d), 0.7 (e), 0.819 (f), 0.9 (g), 1.089 (h), 1.15 (i). The layer thickness is set to be $N_z=12$.

Figure 3

Edge spectral functions $A(k_x,\omega )$ of the semi-infinite sample along the $y$ direction for different exchange field strength $m=0$.0 (a), 0.3 (b), 0.7 (c), 0.9 (d). The sample thickness is set to be $N_z=12$.

Figure 4

Edge spectral functions $A(k_x,\omega )$ of the semi-infinite sample along the $y$ direction for different sample thicknesses $N_z=6$ (a), 12 (b), 18 (c), 24 (d). The exchange field strength is set to be $m=0.7$.

Figure 5

(a) Phase diagram of the Hall conductance ${\sigma }_{xy}$ in the plane ($m$, $N_z$). (b) and (c) plot the Hall conductance for fixed $N_z=6/9/12$ and $m=0.3/0.5/0.7/0.9$. The phenomena beyond the parameter space in panel (a) are also shown.

Figure 6

(a), (b) The Hall conductance ${\sigma }_{xy}$ as a function of the exchange field strength $m$ under different Fermi energies $E_F=0.00,0.05,0.10$ at fixed sample thickness $N_z=9$ (a) and $N_z=12$ (b), respectively. (c), (d) The Hall conductance ${\sigma }_{xy}$ as a function of the layer thickness $N_z$ under different Fermi energies $E_F=0.00,0.05,0.10$ at fixed exchange field $m=0.5$ (a) and $m=0.7$ (b), respectively.

Figure 7

The longitudinal conductance ${\sigma }_{SD}$ as a function of the exchange field strength $m$ under different white noise disorder strength $W$ and the nonuniform doping factor $n_W$ combinations. The error bar is used to denote the conductance fluctuation $\delta {\sigma }_{SD}$. The Fermi energy is set to be $E_F=0.05$. Inset: the schematic plot of the two-terminal device. Disorders are only considered in the central scattering regime, which is modeled by a $N_x\times N_y\times N_z=30\times 40\times 9$ cubic lattice models.