Colloquium: Quantum anomalous Hall effect

The quantum Hall (QH) effect, quantized Hall resistance combined with zero longitudinal resistance, is the characteristic experimental fingerprint of Chern insulators—topologically nontrivial states of two-dimensional matter with broken time-reversal symmetry. In Chern insulators, nontrivial bulk band topology is expressed by chiral states that carry current along sample edges without dissipation. The quantum anomalous Hall (QAH) effect refers to QH effects that occur in the absence of external magnetic fields due to spontaneously broken time-reversal symmetry. The QAH effect has now been realized in four different classes of two-dimensional materials: (i) thin films of magnetically (Cr- and/or V-) doped topological insulators in the $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ family, (ii) thin films of the intrinsic magnetic topological insulator ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, (iii) moiré materials formed from graphene, and (iv) moiré materials formed from transition-metal dichalcogenides. In this Colloquium, the physical mechanisms responsible for each class of QAH insulator are reviewed, with both differences and commonalities highlighted, and potential applications of the QAH effect are commented upon.

Figure 1

QAH effects in magnetic TI and moiré superlattice systems. In both cases gapped Dirac cones contribute ${\sigma }_{xy}=\pm e^2/2h$ to the Hall conductance. The sign of the contribution is distinguished by the color (red or blue) of the lower energy occupied band. The Fermi level [green (light gray) plane] must be in a gap for the Hall effect to be quantized. (a) Schematic of a magnetic TI thin film. The TR symmetry of a TI is broken by the appearance of ferromagnetism in the sample. The total Hall conductance of a magnetic TI film is ${\sigma }_{xy}=(1/2+1/2)e^2/h=e^2/h$ when the two surfaces have parallel magnetization directions. (b) The BZ and moiré BZ of a triangular lattice moiré material. The moiré minibands located near the $K$ and $K^{\prime }$ valleys are related by TR. A QAH effect occurs when the two 2D Dirac cones in the same valley contribute half-quantized Hall conductances of the same sign and the moiré miniband filling factors in the two valleys differ. The green (light gray) planes in (a) and (b) are constant energy surfaces in gaps between occupied and empty bands.

Figure 2

Haldane model for the QAH effect. (a) The Hall conductivity ${\sigma }_{xy}$ as a function of the mass $m$ in 2D Dirac equation. When $m$ changes from a negative value to a positive value, ${\sigma }_{xy}$ jumps from $-e^2/2h$ to $e^2/2h$. (b) Honeycomb lattice of the Haldane model. Here ${\mathbf{a}}_i$ and ${\mathbf{b}}_i$, respectively, label the NN and NNN translation vectors. (c) Phase diagram of the Haldane model. $\nu$ is the Chern number. From [99].

Figure 3

Magnetically doped TI films. (a) Schematic structure of Cr-doped $(\mathrm{Bi}/\mathrm{Sb}{)}_2(\mathrm{Se}/\mathrm{Te}{)}_3$. Hall traces of eight QL (b) ${\mathrm{Bi}}_{1.78}{\mathrm{Cr}}_{0.22}{\mathrm{Se}}_3$, (c) ${\mathrm{Bi}}_{1.78}{\mathrm{Cr}}_{0.22}{\mathrm{Te}}_3$, and (d) ${\mathrm{Sb}}_{1.78}{\mathrm{Cr}}_{0.22}{\mathrm{Te}}_3$ films. STM images of Cr-doped (e) ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, (f) ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, and (g) ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. From [42][43], and [413].

Figure 4

Experimental realization of the QAH effect in five QL Cr-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ films. Magnetic-field ${\mu }_{0}H$ dependence of (a) the Hall resistance ${\rho }_{yx}$ and (c) the longitudinal resistance ${\rho }_{xx}$ at different $V_g$’s. (b) Gate dependence of the zero-magnetic-field Hall resistance ${\rho }_{yx}(0)$ (blue square) and the zero-magnetic-field longitudinal resistance ${\rho }_{xx}(0)$ (red circle). (d) Gate dependence of the zero-magnetic-field Hall resistance ${\sigma }_{xy}(0)$ (blue square) and the zero-magnetic-field longitudinal resistance ${\sigma }_{xx}(0)$ (red circle). From [41].

Figure 5

Higher temperature QAH effects in TI heterostructures with modulation magnetic doping. Schematics are shown for different Cr-doping geometries: (a) uniformly doped, (b) Cr-doped TI trilayer, and (c) pentalayer. Gate dependence of (d) the Hall resistance $R_{yx}$ and (e) the longitudinal resistance $R_{xx}$ of the three samples shown in (a)–(c) measured at $T=0.5\mathrm{K}$. From [222].

Figure 6

QAH effect in V-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ films and $\mathrm{Cr}/\mathrm{V}$ codoped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ films. (a) Magnetic-field dependence of the Hall resistance ${\rho }_{yx}$ (blue curves) and the longitudinal resistance ${\rho }_{xx}$ (red curves) of four QL V-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ films at the charge neutral point $V_{\mathrm{g}}=V_{\mathrm{g}}^0$ and temperature $T=25\mathrm{mK}$. Magnetic-field dependence of (b) ${\rho }_{yx}$ and (c) ${\rho }_{xx}$ of five QL $\mathrm{Cr}/\mathrm{V}$ codoped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ films at the charge neutral point $V_{\mathrm{g}}=V_{\mathrm{g}}^0$ and $T=300\mathrm{mK}$. From [39], and [251].

Figure 7

Zero Hall conductance plateaus and axion insulator states in magnetic TI films and heterostructures. Top panels: schematics of a uniformly Cr-doped TI, a magnetic TI sandwich with the asymmetric Cr doping near the top and bottom surfaces, and a magnetic TI sandwich with magnetic doping with Cr and V. Bottom panels: magnetic-field ${\mu }_{0}H$ dependence of the Hall conductance ${\sigma }_{xy}$ and the longitudinal conductance ${\sigma }_{xx}$ of (a) six QL Cr-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ films, (b) a two QL Cr-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3/\mathrm{five}\mathrm{QL}$ $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3/\mathrm{two}\mathrm{QL}$ Cr-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3/\mathrm{one}\mathrm{QL}$ $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ heterostructure, and (c) a three QL V-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3/\mathrm{five}\mathrm{QL}$ $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3/\mathrm{three}\mathrm{QL}$ Cr-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ heterostructure. From [153], [221], and [368].

Figure 8

Scaling behaviors of the QAH effect in uniformly magnetically doped TI films, and in sandwiches with V and Cr surface doping. (a) $({\sigma }_{xy},{\sigma }_{xx})$ flow diagram of uniformly V-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ with thicknesses of nine QLs (green curve), eight QLs (orange curve), and six QLs (dark yellow curve). (b) $({\sigma }_{xy},{\sigma }_{xx})$ flow diagram of a sample with an axion insulator doping profile: three QL V-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3/\mathrm{five}\mathrm{QL}$ $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3/\mathrm{three}\mathrm{QL}$ Cr-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$. Two semicircles of radius $0.5e^2/h$ centered at $(0.5e^2/h,0)$ and $(-0.5e^2/h,0)$ are plotted as black dashed lines in (a) and (b) for comparison. A semicircle of the radius $e^2/h$ centered at $(0,0)$ is shown as a red dashed line in (a). From [96], and [368].

Figure 9

Current-induced breakdown of the QAH effect. (a) Bias current dependence of the voltage $V_{xx}$ for three Hall bars with different widths $W=50$ (blue line), $100$ (green line), and $200\mathit{\mu }\mathrm{m}$ (red line). The QAH samples used here are nine QL Cr-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ films. (b) Bias current $I_x$ dependence of the longitudinal resistance ${\rho }_{xx}$ of a six QL Cr-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ film. All curves are measured at the charge neutral point $V_{\mathrm{g}}=V_{\mathrm{g}}^0$. From [142], and [81].

Figure 10

(a)–(e) High-Chern-number QAH effect in magnetic TI/TI multilayer structures. Top images: schematic (LEGO blocks) multilayer structures for the QAH effect with Chern numbers ($C$) ranging from 1 to 5. The red and gray LEGOs correspond to three QL Cr-doped $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$ and four QL $(\mathrm{Bi},\mathrm{Sb}{)}_2{\mathrm{Te}}_3$, respectively. Bottom panels: magnetic-field ${\mu }_{0}H$ dependence of the longitudinal resistance ${\rho }_{xx}$ (red curve) and the Hall resistance ${\rho }_{yx}$ (blue curve) measured at the charge neutral point $V_{\mathrm{g}}=V_{\mathrm{g}}^0$ and $T=25\mathrm{mK}$. From [426].

Figure 11

Properties of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ single crystals. (a) Crystal structure of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. One SL of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ is composed of Te-Bi-Te-Mn-Te-Bi-Te. The red arrows indicate the magnetic moments on the Mn layer. (b) Calculated electronic band structure of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. (c) Temperature dependence of the longitudinal resistivity ${\rho }_{xx}$. The hump feature occurs at the Néel temperature $T_N$ of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ single crystals. Inset: magnetic-field ${\mu }_{0}H$ dependence of ${\rho }_{xx}$ at different temperatures. (d) ARPES band structures of ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ single crystals measured at different photon energies. From [45], [55], [402], and [193].

Figure 12

Relationship between RMCD and Hall data. (a) RMCD measurements on ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ flakes (from one to nine SLs) at $T=1.6\mathrm{K}$. The shaded areas highlight the thickness dependence of the low-field spin-flip and spin-flop phase transitions in odd- and even-number SL samples. (b),(c) Magnetic-field ${\mu }_{0}H$ dependence of the RMCD signal and the Hall resistance $R_{yx}$ of a five SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ device. The RMCD and transport characteristics in (b) and (c) are measured in the same sample at $V_{\mathrm{g}}=0$ and at the charge neutral point $V_{\mathrm{g}}=V_{\mathrm{g}}^0$, respectively. From [252], and [387].

Figure 13

Realization of the $C=1$ QAH insulator and a zero Hall plateau in exfoliated ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ flakes. (a) $C=1$ QAH insulator observed in five SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. (b) Zero Hall plateau, taken as evidence for an axion insulator state, observed in six SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. From [67], and [193].

Figure 14

$C=2$ Chern insulator state in ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ devices. (a) Schematics for the $C=2$ Chern insulator state with two chiral edge states propagating along the edges of the sample. (b) Magnetic-field ${\mu }_{0}H$ dependence of the Hall resistance $R_{yx}$ of a ten SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ device at fixed bottom gate voltages $V_{\mathrm{bg}}=-17$ and $-58\mathrm{V}$. (c) ${\rho }_{yx}$ as a function of the magnetic field ${\mu }_{0}H$ and the gate injected carrier density $n_G$ at a fixed electric field $D/{ϵ}_0=-0.2\mathrm{V}/\mathrm{nm}$ in a seven SL ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ device. Black, green, and yellow dashed lines enclose $C=1,2,3$, respectively. The marked densities are $(n_1,n_2,n_3)=(2.23,1.87,1.29)\times {10}^{12}{\mathrm{cm}}^{-2}$. From [88], and [28].

Figure 15

Orbital magnetism in twisted bilayer graphene systems. (a) Schematic of twisted bilayer graphene. The relative orientation angle $\theta$ determines the periodicity of the moiré superllattice. (b) Magnetic-field ${\mu }_{0}H$ dependence of the longitudinal resistance $R_{xx}$ and the Hall resistance $R_{yx}$ with $n/n_s=3/4$ and $D/D_0=-0.62\mathrm{V}/\mathrm{nm}$ at $T=30\mathrm{mK}$ demonstrating the hysteretic AH effect resulting from the emergent orbital magnetic order. From [293].

Figure 16

$C=1$ QAH effect in TBG and $C=2$ Chern insulator state in trilayer graphene. (a) Magnetic-field ${\mu }_{0}H$ dependence of the longitudinal resistance $R_{xx}$ and the Hall resistance $R_{yx}$ in TBG measured at $n=2.37\times {10}^{12}{\mathrm{cm}}^{-2}$. (b) Magnetic-field ${\mu }_{0}H$ dependence of the longitudinal resistance ${\rho }_{xx}$ and the Hall resistance ${\rho }_{yx}$ in a rhombohedral trilayer graphene-$h$-BN moiré superlattice. A $C=2$ Chern insulator state is realized under a finite external magnetic field. From [49], and [292].

Figure 17

$C=1$ QAH effect in an $AB$-stacked ${\mathrm{MoTe}}_2/{\mathrm{WSe}}_2$ heterobilayer. (a) Left image: triangular moiré superlattice with the high-symmetry stacking sites $MM$, $MX$, and $XX$ ($M=\mathrm{Mo},\mathrm{W}$; $X=\mathrm{Se},\mathrm{Te}$). Right image: schematic of out-of-plane electric-field-induced band inversion in an $AB$-stacked ${\mathrm{MoTe}}_2/{\mathrm{WSe}}_2$ heterobilayer. (b) Magnetic-field ${\mu }_{0}H$ dependence of the Hall resistance $R_{yx}$ in an $AB$-stacked ${\mathrm{MoTe}}_2/{\mathrm{WSe}}_2$ heterobilayer with the filling factor $v=1$ at different temperatures. A $C=1$ QAH insulator state is realized. From [181].

Figure 18

Absence of the chiral Majorana edge modes in millimeter-size QAH-superconductor devices. (a) A millimeter-size superconductor strip covering a QAH device induces an electrical short that circumvents a chiral MEM signal. (b) Magnetic-field ${\mu }_{0}H$ dependence of the two-terminal conductance ${\sigma }_{1,2}$ of a QAH-superconductor device. ${\sigma }_{1,2}\sim 0.5e^2/h$ for the entire ${\mu }_{0}H$ range when the magnetization is accurately aligned. No change in ${\sigma }_{1,2}$ is observed when the Nb strip transitions from the superconducting state to the normal state. From [143].