Observation of Integer and Fractional Quantum Anomalous Hall Effects in Twisted Bilayer ${\mathrm{MoTe}}_2$

The interplay between strong correlations and topology can lead to the emergence of intriguing quantum states of matter. One well-known example is the fractional quantum Hall effect, where exotic electron fluids with fractionally charged excitations form in partially filled Landau levels. The emergence of topological moiré flat bands provides exciting opportunities to realize the lattice analogs of both the integer and fractional quantum Hall effects without the need for an external magnetic field. These effects are known as the integer and fractional quantum anomalous Hall (IQAH and FQAH) effects. Here, we present direct transport evidence of the existence of both IQAH and FQAH effects in small-angle-twisted bilayer ${\mathrm{MoTe}}_2$. At zero magnetic field, we observe well-quantized Hall resistance of $h/e^2$ around moiré filling factor $\nu =-1$ (corresponding to one hole per moiré unit cell), and nearly quantized Hall resistance of $3h/2e^2$ around $\nu =-2/3$, respectively. Concomitantly, the longitudinal resistance exhibits distinct minima around $\nu =-1$ and $-2/3$. The application of an electric field induces topological quantum phase transition from the IQAH state to a charge transfer insulator at $\nu =-1$, and from the FQAH state to a topologically trivial correlated insulator, further transitioning to a metallic state at $\nu =-2/3$. Our study paves the way for the investigation of fractionally charged excitations and anyonic statistics at zero magnetic field based on semiconductor moiré materials.

Popular Summary

The interplay between strong correlations among electrons and the topology of a system can lead to intriguing quantum states of matter. One well-known example is the fractional quantum Hall effect, where exotic electron fluids form with fractionally charged excitations. Moiré materials—where two atomic layers are stacked slightly askew to create a moiré interference pattern—offer an intriguing platform to realize analogs of the fractional quantum Hall effect, and the related integer effect, without the usual need for an external magnetic field. These analogs are known as the integer and fractional quantum anomalous Hall (IQAH and FQAH) effects. Here, we present direct evidence of both effects in small-angle-twisted bilayer ${\mathrm{MoTe}}_2$.

At zero magnetic field, we observe a well-quantized resistance that is transverse to the direction of electrical current—a key signature of the quantum Hall effect—at a moiré filling factor of $-1$, corresponding to one positively charged hole per moiré unit cell. We also see a nearly quantized resistance at a filling factor of $-2/3$, or a fraction of a hole per cell. Concomitantly, the longitudinal resistance exhibits distinct minima around these two filling factors. The application of an electric field induces a quantum phase transition from the IQAH state to a charge transfer insulator at filling factor $-1$. For a filling factor of $-2/3$, the same application changes the FQAH state to a type of correlated insulator and then on to a metallic state.

Our study paves the way for the investigation of fractionally charged excitations and anyonic statistics—that is, behaviors attributed to quasiparticles that are neither fermions nor bosons—at zero magnetic field based on semiconductor moiré materials.

Figure 1

Phase diagram of AA-stacked $t{\mathrm{MoTe}}_2$. (a) Left: schematic of the Hall bar device used for transport measurements. Right: side view of the device structure. The carrier density of the Hall bar channel is only controlled by the top gate and the back gate, whereas the carrier density of ${\mathrm{TaSe}}_2/{\mathrm{MoTe}}_2$ contact regions and Hall bar arms are mainly controlled by the global $\mathrm{Si}/{\mathrm{SiO}}_2$ gate. The inset shows the schematic moiré superlattices of $t{\mathrm{MoTe}}_2$. High symmetry stackings are highlighted by circles. (b) Longitudinal sheet resistance and (c) Hall resistance as a function of $\nu$ and $D$. ${\rho }_{xx}$ is measured under zero magnetic field at 900 mK; ${\rho }_{xy}$ is the antisymmetrized results under an out-of-plane magnetic field of $\pm 0.1\mathrm{T}$ at 1.2 K. The arrows mark $\nu =-1$, $-2/3$, $-1/2$, and $-1/3$, where correlated insulating states (topologically trivial) formed at relatively large $D$ field. The black regions are very insulating and experimentally inaccessible.

Figure 2

IQAH effect at $\nu =-1$ and FQAH effect at $\nu =-2/3$. (a),(b) Magnetic-field dependence of ${\rho }_{xy}$ (a) and ${\rho }_{xx}$ (b) of the IQAH effect at $\nu =-1$ and $D=-115\mathrm{mV}/\mathrm{nm}$ at varying temperatures. Quantized ${\rho }_{xy}$ of $h/e^2$ and vanishing ${\rho }_{xx}$ are observed below 2 K at zero magnetic field. (c),(d) Magnetic-field dependence of ${\rho }_{xy}$ (c) and ${\rho }_{xx}$ (d) of the FQAH effect at $\nu =-2/3$ and $D=5\mathrm{mV}/\mathrm{nm}$ at varying temperatures. Nearly quantized ${\rho }_{xy}$ of $3h/2e^2$ can be observed. (e) Hall conductivity ${\sigma }_{xy}$ as a function of $\nu$ at $T=600\mathrm{mK}$, 900 mK, and 1.2 K with $D=5\mathrm{mV}/\mathrm{nm}$. The ${\sigma }_{xy}$ is obtained using the antisymmetrized ${\rho }_{xy}$ and symmetrized ${\rho }_{xx}$ at $B=\pm 0.3\mathrm{T}$. Distinct ${\sigma }_{xy}$ plateaus can be observed around $\nu =-1$ and $-2/3$, with values closely approaching the quantized values of $e^2/h$ and $2e^2/3h$, respectively. (f) ${\sigma }_{xy}$ as a function of $D$ measured at 900 mK at $\nu =-1$ and $\nu =-2/3$. The ${\sigma }_{xy}$ remains at the quantized values within certain $D$ range, then rapidly vanishes as $D$ reaches $D_{\mathrm{c}}$. For $\nu =-1$, ${\sigma }_{xy}$ is derived from measured ${\rho }_{xy}$ and ${\rho }_{xx}$ at $B=0\mathrm{T}$, and for $\nu =-2/3$, ${\sigma }_{xy}$ is obtained from antisymmetrized ${\rho }_{xy}$ and symmetrized ${\rho }_{xx}$ at $B=\pm 0.3\mathrm{T}$.

Figure 3

$D$-field tuned topological phase transitions. (a),(b) ${\rho }_{xx}$ map as a function of $D$ and $T$ at $\nu =-1$ (a) and $\nu =-2/3$ (b). The ${\rho }_{xx}$ of $\nu =-1$ is measured at $B=0$, and the ${\rho }_{xx}$ of $\nu =-2/3$ is symmetrized at $B=\pm 0.3\mathrm{T}$. (c),(d) Extracted energy gap $\mathrm{\Delta }$ (circles) by thermal activation fitting, as a function of $D$, at $\nu =-1$ (c) and $\nu =-2/3$ (d). Distinct phases are differentiated by various colors. The error bars smaller than the size of the circles are not displayed.

Figure 4

(a) Optical micrograph of device I. The Hall bar geometry is defined by standard EBL and RIE processes. The scale bar is $10\mathrm{\mu }\mathrm{m}$. We found the middle region of the device (between contact 22, 6 and 19, 5) is relatively uniform. (b) Schematic figure of the measurement configuration. For most results presented in the main text, electrode 8 is grounded; electrodes 24 and 3 are used as a source electrode. The longitudinal voltage drop is measured between 22 and 19, and the transverse voltage drop is measured between 22 and 6. (c) Contact resistance versus filling factors at $T=1.5\mathrm{K}$ and $D=5\mathrm{mV}/\mathrm{nm}$. During the measurement of contact resistance, only the specific contact is biased with a 2 mV dc voltage while all other contacts are grounded. The dc current versus filling factors is measured using a source meter.

Figure 5

${\rho }_{xy}$ (top) and ${\rho }_{xx}$ (bottom) as a function of $\nu$ at $T=600\mathrm{mK}$, 900 mK, and 1.2 K with $D=5\mathrm{mV}/\mathrm{nm}$. The ${\rho }_{xy}$ (${\rho }_{xx}$) is antisymmetrized (symmetrized) at $B=\pm 0.3\mathrm{T}$. ${\rho }_{xx}$ shows clear dips around $\nu =-1$ and $\nu =-2/3$ due to the chiral edge transport. ${\rho }_{xy}$ quantized at $h/e^2$ approximately from $\nu =-1.1$ to $-0.95$, and the quantized range becomes narrower with increasing temperature. Around $\nu =-2/3$, a narrower ${\rho }_{xy}$ plateau can be roughly identified above 900 mK, and the ${\rho }_{xy}$ value of the plateau is significantly larger than $h/e^2$ but still about 5%–15% smaller than the expected quantized value of $3h/2e^2$. Below 900 mK, the ${\rho }_{xy}$ plateau around $\nu =-2/3$ becomes indistinguishable, presumably due to the contact issue and sample inhomogeneities.

Figure 6

(a)–(d) Longitudinal resistance $R_{xx}$ measured with contacts 22, 19, 5, and 6 under different configurations at 600 mK. The specific current bias and voltage measurement configurations are indicated in the insets. (e) Symmetrized $R_{xx}$ as a function of $B$ obtained from (a) and (b) using Onsager reciprocal relations $R_{xx}=(R_{19-22,5-6}+R_{5-6,19-22})/2$. (f) Symmetrized $R_{xx}$ obtained from (c) and (d) using Onsager reciprocal relations $R_{xx}=(R_{22-6,19-5}+R_{19-5,22-6})/2$. (g) Longitudinal sheet resistance ${\rho }_{xx}$ (symmetrized at $B=\pm 0.3\mathrm{T}$) as a function of $\nu$ measured at 600 mK with another pair of contacts, as illustrated in the inset. (h) Onsager symmetrized $R_{xx}=(R_{19-22,5-6}+R_{5-6,19-22})/2$ as a function of $\nu$ at $B=0\mathrm{T}$ and $T=600\mathrm{mK}$. Distinct $R_{xx}$ minima can be identified around $\nu =-1$ and $-2/3$ in both (g) and (h).

Figure 7

(a) Hall conductivity ${\sigma }_{xy}$ as a function of $\nu$ at $T=600\mathrm{mK}$, 900 mK, and 1.2 K measured from contacts 19–5 with $D=5\mathrm{mV}/\mathrm{nm}$. The ${\sigma }_{xy}$ is obtained using the antisymmetrized ${\rho }_{xy}$ and symmetrized ${\rho }_{xx}$ at $B=\pm 0.3\mathrm{T}$. (b) Magnetic-field dependence of ${\rho }_{xy}$ at $\nu \approx -2/3$ at varying temperatures measured from contacts 19–5 with $D=5\mathrm{mV}/\mathrm{nm}$. Overall, the results from contacts 19–5 are essentially the same as those from contact 22–6 (shown in Fig. 2).

Figure 8

The ${\rho }_{xx}$ map as a function of $\nu$ and $B$ at 600 mK with contacts 22–19 (a) and contact 6–5 (b), respectively. Empty circles mark the positions of ${\rho }_{xx}$ dips around $\nu =-1$ and $-2/3$. Three red lines around $\nu =-1$ in (a) represent possible unbiased fitting lines based on the vanishing ${\rho }_{xx}$ region, with Chern numbers of 1.13, 1.09, and 0.97, from left to right, respectively. (c),(d) ${\rho }_{xx}$ (symmetrized at $B=\pm 0.1\mathrm{T}$) map as a function of $\nu$ and $D$ using (c) contacts 22–19 and (d) contacts 6–5 as voltage probes, measured at 1.5 K. The data from contacts 6–5 exhibit lower quality compared to contacts 22–19, which may be attributed to the characteristics of contact 5.