Abstract
Chern insulators, which are the lattice analogues of the quantum Hall states, can potentially manifest high-temperature topological orders at zero magnetic field to enable next-generation topological quantum devices1,2,3. Until now, integer Chern insulators have been experimentally demonstrated in several systems at zero magnetic field3,4,5,6,7,8, whereas fractional Chern insulators have been reported in only graphene-based systems under a finite magnetic field9,10. The emergence of semiconductor moiré materials11, which support tunable topological flat bands12,13, provides an opportunity to realize fractional Chern insulators13,14,15,16. Here we report thermodynamic evidence of both integer and fractional Chern insulators at zero magnetic field in small-angle twisted bilayer MoTe2 by combining the local electronic compressibility and magneto-optical measurements. At hole filling factor ν = 1 and 2/3, the system is incompressible and spontaneously breaks time-reversal symmetry. We show that they are integer and fractional Chern insulators, respectively, from the dispersion of the state in the filling factor with an applied magnetic field. We further demonstrate electric-field-tuned topological phase transitions involving the Chern insulators. Our findings pave the way for the demonstration of quantized fractional Hall conductance and anyonic excitation and braiding17 in semiconductor moiré materials.
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Acknowledgements
We thank L. Fu for the discussions. This work was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, under award no. DE-SC0019481 (thermodynamic studies), the Air Force Office of Scientific Research under award no. FA9550-20-1-0219 (magneto-optical studies), and the Cornell University Materials Research Science and Engineering Center under award no. DMR-1719875 (device fabrication). This work was also funded in part by the Gordon and Betty Moore Foundation (grant doi: https://doi.org/10.37807/GBMF11563) and performed in part at the Cornell NanoScale Facility, an NNCI member supported by the NSF grant NNCI-2025233. The growth of the hBN crystals was supported by the Elemental Strategy Initiative of MEXT, Japan, and CREST (JPMJCR15F3), JST. We also acknowledge support from the David and Lucille Packard Fellowship (K.F.M.) and the Swiss National Science Foundation (P.K.).
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Y.Z. and Z.X. fabricated the devices, performed the measurements and analysed the data. K.K., P.K., J.Z. and C.V. provided data from additional devices. K.W. and T.T. grew the bulk hBN crystals. Y.Z., Z.X., K.F.M. and J.S. designed the scientific objectives. K.F.M. and J.S. oversaw the project. All authors discussed the results and commented on the paper.
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Extended data figures and tables
Extended Data Fig. 1 Calibration of the moiré density.
a, Dependence of the optical reflectance spectrum of the sensor on the sample-sensor bias voltage Vsample (sensor grounded); the top and bottom gate voltages are kept constant; the sample is heavily hole-doped; the magnetic field is fixed at 8.8 T. Clear quantum oscillations in the attractive polaron resonance energy and amplitude are observed with hole doping due to the formation of spin-valley-polarized Landau levels at high magnetic fields. b, Dependence of the integrated attractive polaron amplitude (over the spectral window bound by the dashed lines in a) on Vs (blue curve). The orange curve represents the smooth background. We show data between −0.6 and −1 V, for which the sensor is heavily hole-doped and the Landau levels are two-fold degenerate. c, Oscillation amplitude (after removal of the smooth background) as a function of Vs. The average distance between adjacent amplitude peaks at high hole doping densities is 82.5 mV; this corresponds to a change in the sensor doping density of 4.26 × 1011 cm−2 based on the known Landau level degeneracy (= 2) in this density range. The data allow us to accurately determine the sample-to-sensor geometrical capacitance Cs ≈ 8.3 ± 0.3 μFcm−2. d, Centre-of-mass filling factor for the ν = 1 state as a function of the bottom gate voltage (Vbg) and the sample-sensor bias voltage (Vs). The slope determines the capacitance ratio \(\frac{{C}_{s}}{{C}_{{bg}}}=2.17\pm 0.01\). Combined with the calibration of Cs, the moiré density can be determined nM = (3.2 ± 0.2) × 1012 cm−2 (see Methods for details).
Extended Data Fig. 2 Determination of the chemical potential jump and incompressibility peak in filling factors.
a, Filling-factor dependent incompressibility at 1.6 K, zero electric field and B = 1 T. The incompressibility peak above the baseline (blue dashed lines) is integrated to obtain the chemical potential jump at ν = 2/3 and 1. This procedure effectively removes the negative incompressibility background in the chemical potential measurements. The same incompressibility peak (above the background) is used to calculate the centre-of-mass filling factor for the ν = 2/3 and 1 states (green arrows). Gaussian fit to the data (red dashed lines) near the local incompressibility peak at ν = 2/3 and 1 yields nearly identical peak fillings (red arrows). The vertical dashed lines denote the filling range for both analyses, the centre of which is chosen at the local incompressibility maximum (black arrows). b, Peak position of the incompressible states extracted from the Gaussian fit as a function of magnetic field (the analysis used the same data set as that in Fig. 2). The error bars are the fit uncertainty. Similar quantum numbers are obtained from the linear fits as in the main text.
Extended Data Fig. 3 Determination of the quantum numbers at small magnetic fields.
a–c, Electronic incompressibility versus the filling factor and magnetic field (0–3 T) near ν = 1 (a) and ν = 2/3 (b) at zero interlayer potential difference, and near ν = 1 at large interlayer potential difference (c). Data were obtained from the same location of the sample as in the main text. d–f, Incompressibility peak position versus magnetic field extracted from the centre-of-mass analysis. Linear fits in d and e yield a slope of 0.67 ± 0.08 for ν = 2/3 and 1.1 ± 0.1 for ν = 1, consistent with the emergence of fractional and integer CIs, respectively, in the zero-magnetic-field limit. The red line in f marks a non-topological Mott insulator that does not disperse with applied magnetic field. Error bars denote the combined systematic and random uncertainties of the measurements.
Extended Data Fig. 4 Thermodynamic equation of state at varying magnetic fields.
The filling factor dependent chemical potential near zero electric field is shown. The curves are vertically displaced for clarity. We subtract the B = 0 T curve from the B = 3 T curve to obtain the plot in Fig. 2d.
Extended Data Fig. 5 Electric-field dependent optical reflectance and spontaneous MCD spectra of tMoTe2 at ν = 1.
a, Optical reflectance spectrum (unpolarized) as a function of vertical electric field. The bonding (about 1.12 eV) and anti-bonding (about 1.14 eV) features in the layer-hybridized region evolve into the neutral exciton feature (about 1.13 eV) of one layer in the layer-polarized region. The red dashed lines denote the critical electric fields that separate the layer-hybridized and layer-polarized regions. We trace the highest reflectance amplitude to obtain the 2D map in Fig. 3c. b, Strong spontaneous MCD is observed only in the layer-hybridized region. The MCD spectrum between the black dashed lines is integrated to obtain the 2D map in Fig. 3b.
Extended Data Fig. 6 Dual-gated tMoTe2 devices for chemical potential measurements.
a, Schematic structure of dual-gated devices with a monolayer WSe2 sensor inserted between the top gate and the twisted bilayer MoTe2 sample. In addition to the structure shown in Fig. 1b, the device has contact gates to both the metal-sample and metal-sensor contacts. In this experiment, a large negative voltage is applied to the contact gates to heavily hole-dope the contact regions in order to achieve good electrical contacts for the chemical potential measurements. b, Optical micrograph of the device studied in the main text. The red and blue lines outline the boundaries of the twisted bilayer MoTe2 sample and the WSe2 sensor, respectively. The results presented in the main text and in Extended Data Fig. 5 were obtained at the black spot. Results obtained at the red and blue locations are shown in Extended Data Fig. 7.
Extended Data Fig. 7 Integer and fractional CIs at other sample locations.
a,b, Electronic incompressibility versus the filling factor and the out-of-plane electric field at zero magnetic field for the red (a) and blue (b) spots in Extended Data Fig. 6b. Similar to the black spot, incompressible states at ν = 1 and 2/3 are observed in the layer-hybridized region. The gate capacitances were measured locally using the sensor’s quantum oscillations as described in Methods and Extended Data Fig. 1, allowing accurate determination of the local twist angle. c–f, Linear fits of the magnetic-field dependence of the incompressibility peak filling factor at the red spot yield a slope of 0.63 ± 0.08 for ν = 2/3 and 1.08 ± 0.09 for ν ± 1, consistent with the emergence of FCI and CI.
Extended Data Fig. 8 Spontaneous MCD in samples with different twist angles.
Integrated MCD as a function of filling factor and electric field in 3 different samples with twist angles of 2.1 degrees (a), 2.7 degrees (b) and 4.6 degrees (c). All data were acquired at T = 1.6 K and B = 20 mT. The incident light was kept below 20 nW on the sample. Ferromagnetism is observed over a wide range of twist angles.
Extended Data Fig. 9 Integer and fractional CIs in other samples.
a,b, Incompressibility as a function of ν and B in samples with twist angles of 2.1 degrees (a) and 2.7 degrees (b). The nearly horizontal stripes in b are the quantum oscillations of the graphite gate electrodes. c,d, Linear fits of the extracted centre-of-mass fillings for the CIs in a,b. Integer CIs at ν = 1 are observed in all samples while FCIs at ν = 2/3 are observed only in samples with twist angles of 2.7 and 3.4 degrees.
Extended Data Fig. 10 Independent calibration of the ν = 1 state quantum numbers.
Electric-field and filling-factor dependent incompressibility at 8.8 T and 4 K. A clear upshift in the filling factor for the ν = 1 incompressible state is observed only in the layer-hybridized region (red dashed lines denoting the boundaries between the layer-hybridized and layer-polarized regions). The upshift reflects the emergence of the CI (the correlated insulator in the layer-polarized region is non-topological). The average centre-of-mass filling shift from the non-topological state (marked by the black dashed lines) is 0.070 ± 0.006 at 8.8 T. This corresponds to \(t=\frac{h}{e}\frac{dv}{dB}{n}_{M}=\frac{h}{e}\frac{0.070\pm 0.006}{8.8}{n}_{M}=1.1\,\pm \,0.1\) (see Methods). No non-topological insulating state at ν = 2/3 can be identified.
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Zeng, Y., Xia, Z., Kang, K. et al. Thermodynamic evidence of fractional Chern insulator in moiré MoTe2. Nature 622, 69–73 (2023). https://doi.org/10.1038/s41586-023-06452-3
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DOI: https://doi.org/10.1038/s41586-023-06452-3
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