The strong electron interactions in the minibands formed in moiré superlattices of van der Waals materials, such as twisted graphene and transition metal dichalcogenides, make such systems a fascinating platform with which to study strongly correlated states1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19. In most systems, the correlated states appear when the moiré lattice is filled by an integer number of electrons per moiré unit cell. Recently, correlated states at fractional fillings of 1/3 and 2/3 holes per moiré unit cell have been reported in the WS2/WSe2 hetero-bilayer, hinting at the long-range nature of the electron interaction16. Here we observe a series of correlated insulating states at fractional fillings of the moiré minibands on both electron- and hole-doped sides in angle-aligned WS2/WSe2 hetero-bilayers, with certain states persisting at temperatures up to 120 K. Simulations reveal that these insulating states correspond to ordering of electrons in the moiré lattice with a periodicity much larger than the moiré unit cell, indicating a surprisingly strong and long-range interaction beyond the nearest neighbours.
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Data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.
The source code for the Monte Carlo simulations is available from the corresponding author upon reasonable request.
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We thank D. Chen, L. Yan, L. Ma and K. Li for help with device fabrication. We are grateful to R. Swendsen and M. Widom for their help with the Monte Carlo simulation. C.W. and D.X. thank W. Duan for providing part of the computational resources. X.H. and Y.-T.C. acknowledge support from the NSF under award no. DMR-2004701, a Hellman Fellowship award and a seed fund from SHINES, an EFRC funded by the US Department of Energy (DOE), Basic Energy Sciences (BES) under award no. SC0012670. S.M., Z. Li and S.-F.S. acknowledge support by AFOSR through grant no. FA9550-18-1-0312. T.W. and S.-F.S. acknowledge support from ACS PRF through grant no. 59957-DNI10. Z. Lian and S.-F.S. acknowledge support from NYSTAR through Focus Center-NY–RPI contract C150117. Device fabrication was supported by the Micro and Nanofabrication Clean Room (MNCR) at Rensselaer Polytechnic Institute (RPI). S.-F.S. also acknowledges support from the NSF through Career grant no. DMR-1945420. The research by S.O. is supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. C.W. and D.X. acknowledge support from DOE, BES grant no. DE-SC0012509. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT (Japan; grant no. JPMXP0112101001), JSPS (KAKENHI grant no. JP20H00354) and the CREST (JPMJCR15F3), JST. We acknowledge computing time provided by BRIDGES at the Pittsburgh Supercomputing Center (award no. TG-DMR190080) under the Extreme Science and Engineering Discovery Environment (XSEDE) supported by the NSF (ACI-1548562).
The authors declare no competing interests.
Peer review information Nature Physics thanks Fengcheng Wu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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a,b Optical microscope images of devices (a) D1 and (b) D2 with different flakes outlined. c,d Angular dependence of the SHG signal in (c) device D1 and (d) device D2.
a, MIM-Im vs gate voltage traces for both sweeping directions in device D1 at T = 4 K. b, MIM-Im vs gate voltage traces with extended gate range from −7 V to 6 V taken at T = 3 K. The feature near −5 V likely correponds to n = −2 state. However, due to poor electrical contact at high hole doping, this feature is not stable and its gate position is not repeatable at different spots.
a,b Optical image and atomic force microscopy (AFM) image of device D1. c, MIM-Im image at Vg = −2 V scanned over the region marked by the rectangles in (a) and (b). d, MIM-Im images taken at different gate voltages marked along the top axis in (e). e, MIM-Im vs gate traces taken at spots A-F as indicated in (c). All MIM data are taken at T = 10 K. Scale bars are 2 μm.
Extended Data Fig. 4 Ordering patterns simulated on a 48 by 48 lattice for n = 1/3, 1/4, 1/7, and 1/9.
The column on the right plots the fast Fourier transform (FFT) image of the ordering pattern.
Extended Data Fig. 5 Ordering patterns simulated on a 48 by 48 lattice for n = 1/2, 1/6, 4/9, and 2/9.
The column on the right plots the FFT image of the ordering pattern.
The temperature scale, kBT, is normalized by the nearest neighbor Coulomb interaction. The identified ordering temperatures, kBTc, are listed in the figure.
Extended Data Fig. 7 The Fourier transformed configurations in the reciprocal space as kBT varies for filling 1/7.
The lowest kBT is 0.017; the highest kBT is 0.031; kBTc = 0.021 is labeled in the picture. To visualize the transition around the critical temperature, we select eight configurations from the simulations of 1/7 filling, on which we perform Fourier transformations. The absolute values of the eight Fourier amplitudes are added and plotted in the reciprocal space. It is clear the Fourier amplitudes changes abruptly around Tc, which justifies our approach of identifying the peak in the specific heat as the transition temperature.
Extended Data Fig. 8 Temperature dependence of MIM-Im for gate sweeping down from 5 V to -5 V in device D2.
The black markers indicate the positions of the n = −1/3 state, which can be resolved at temperatures of 20 K and above.
a, Optical microscope of D2. b, AFM scan around the area indicated by the solid square in (a). The scale bar is 5 μm. c, MIM spectra taken at spots A-E as marked in (b). d, Sweep up and down traces at spot B. e, Sweep up and down traces at spot E. All data taken at T = 2.8 K.
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Huang, X., Wang, T., Miao, S. et al. Correlated insulating states at fractional fillings of the WS2/WSe2 moiré lattice. Nat. Phys. 17, 715–719 (2021). https://doi.org/10.1038/s41567-021-01171-w
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