Quantum Anomalous Hall Effect with Higher Plateaus

The quantum anomalous Hall (QAH) effect in magnetic topological insulators is driven by the combination of spontaneous magnetic moments and spin-orbit coupling. Its recent experimental discovery raises the question if higher plateaus can also be realized. Here, we present a general theory for a QAH effect with higher Chern numbers and show by first-principles calculations that a thin film magnetic topological insulator of Cr-doped ${\mathrm{Bi}}_2(\mathrm{Se},\mathrm{Te}{)}_3$ is a candidate for the $C=2$ QAH insulator. Remarkably, whereas a higher magnetic field leads to lower Hall conductance plateaus in the integer quantum Hall effect, a higher magnetic moment leads to higher Hall conductance plateaus in the QAH effect.

Figure 1

Evolution of the subband structure upon increasing the exchange field. The solid lines denote the subbands that have even parity at the $\Gamma$ point, and dashed lines denote subbands with odd parity at the $\Gamma$ point. The blue color ($\uparrow$) denotes the spin-up electrons, while red ($\downarrow$) denotes spin-down electrons. The superscript $+$ and $-$ represent even and odd parity, respectively. (a) The initial ($E_1$, $H_1$) subbands are already inverted, while the ($E_2$, $H_2$) subbands are not inverted. The exchange field ${\Delta }_1$ releases the band inversion in one pair of ($E_1$, $H_1$) subbands and increases the band inversion in the other pair, while the ($E_2$, $H_2$) subbands are still not inverted. With a stronger exchange field ${\Delta }_2$, a pair of inverted ($E_2$, $H_2$) subbands appears, while keeping only one pair of ($E_1$, $H_1$) subbands inverted. (b) Schematic drawing of a Hall bar device of the $C=2$ QAH effect. (c) Expected chemical potential dependence of zero magnetic field ${\rho }_{xx}$ (in red) and ${\rho }_{xy}$ (in blue).

Figure 2

The phase diagram of the QAH effect in thin films of magnetic TIs with two variables: the exchange field $\Delta$ and thickness of thin film $d$. All parameters are taken from Ref. 21 for ${\mathrm{Bi}}_2({\mathrm{Se}}_{0.4}{\mathrm{Te}}_{0.6}{)}_3$. (a),(b) Phase diagrams without and with the $A_1$ term, respectively. The different QAH phases are denoted by corresponding Chern numbers. The particle-hole-asymmetric term is neglected, for it does not change the topology of the phase diagram. The width of each Hall plateau in the QAH effect depends on the band parameters and the thickness of the material, which is distinct from that in the IQHE.

Figure 3

Band structure, Brillouin zone, and edge states. (a) Band structure for 12 QLs ${\mathrm{Bi}}_{1.78}{\mathrm{Cr}}_{0.22}({\mathrm{Se}}_{0.4}{\mathrm{Te}}_{0.6}{)}_3$ without an exchange field. The dashed line indicates the Fermi level. (b) The top view of a 2D thin film with two in-plane lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. The 1D edges are indicated by the dashed lines, edge $A$ along the $[1\overline{1}]$ direction and edge $B$ along the [01] direction. The inset shows the 2D Brillouin zone, in which the high-symmetry $\mathbf{k}$ points $\Gamma (0,0)$, $K(\pi ,\pi )$, and $M(\pi ,0)$ are labeled. (c)–(f) Energy and momentum dependence of the LDOS along edge $A$ for the ${\mathrm{Bi}}_{1.78}{\mathrm{Cr}}_{0.22}({\mathrm{Se}}_{0.4}{\mathrm{Te}}_{0.6}{)}_3$ film with a thickness of 6 QLs and an exchange field of (c) 0.0 eV and (d) 0.08 eV and with a thickness of 12 QLs and an exchange field of (e) 0.08 eV and (f) 0.14 eV. Here, the warmer colors (lighter gray in gray scale) represent a higher LDOS. The red (light gray) and blue (dark gray) regions indicate 2D bulk energy bands and energy gaps, respectively. The gapless chiral edge states can be clearly seen around the $\Gamma$ point as red (light gray) lines dispersing in the 2D bulk gap. In (c)–(f), the number of chiral edge states is $C=0$,1,1,2.