Abstract
Quantum particles on a lattice with competing long-range interactions are ubiquitous in physics; transition metal oxides1,2, layered molecular crystals3 and trapped-ion arrays4 are a few examples. In the strongly interacting regime, these systems often show a rich variety of quantum many-body ground states that challenge theory2. The emergence of transition metal dichalcogenide moiré superlattices provides a highly controllable platform in which to study long-range electronic correlations5,6,7,8,9,10,11,12. Here we report an observation of nearly two dozen correlated insulating states at fractional fillings of tungsten diselenide/tungsten disulfide moiré superlattices. This finding is enabled by a new optical sensing technique that is based on the sensitivity to the dielectric environment of the exciton excited states in a single-layer semiconductor of tungsten diselenide. The cascade of insulating states shows an energy ordering that is nearly symmetric about a filling factor of half a particle per superlattice site. We propose a series of charge-ordered states at commensurate filling fractions that range from generalized Wigner crystals7 to charge density waves. Our study lays the groundwork for using moiré superlattices to simulate a wealth of quantum many-body problems that are described by the two-dimensional extended Hubbard model3,13,14 or spin models with long-range charge–charge and exchange interactions15,16.
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Data availability
The data that support the plots within this paper, and other findings of this study, are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank L. Fu for fruitful discussions. Research was primarily supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award number DE-SC0019481 (optical spectroscopy and growth of WSe2 crystals). Device fabrication was supported by US Office of Naval Research under award number N00014-18-1-2368. The growth of the hBN crystals was supported by the Elemental Strategy Initiative of MEXT, Japan and CREST (JPMJCR15F3), JST. K.F.M. also acknowledges support from David and Lucille Packard Fellowship.
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Y.X. fabricated the devices, performed the measurements and analysed the data. V.E. performed theoretical calculations. S.L., D.A.R. and J.H. grew the bulk WSe2 crystals, and K.W. and T.T. grew the bulk hBN crystals. Y.X., K.F.M. and J.S. designed the scientific objectives and oversaw the project. Y.X., V.E., K.F.M. and J.S. co-wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Additional doping-dependence data and analysis for the control device.
a, Gate-dependent reflection contrast (ΔR/R0) spectrum near the WSe2 (sensor) 2s transition energy (same data as in Fig. 1d, middle). The WS2 sample is electron-doped above Vg = 1.5 V (white dashed line). b, Representative linecuts of a. Curves are shifted vertically for clarity. The red and blue dashed curves highlight the redshifted and blueshifted branches, respectively. c, Spectral weight of the two branches as a function of Vg (calculated after removing the background taken at 1 V and 4 V for the redshifted and blueshifted branches, respectively). The blueshifted branch has been multiplied by a factor of 0.3. The redshifted feature emerges while the blueshifted feature quickly goes to zero at Vg ≈ 1.5 V.
Extended Data Fig. 2 Assignment of the filling factor of the insulating states.
For each insulating state, the position (filled squares) and the FWHM (horizontal bars) in gate voltage are determined through a Lorentzian fit of the 2s exciton resonance energy as a function of gate voltage. The red lines are linear fits using four established states v = 2, v = 1, v = 2/3 and v = 1/3 on the electron and hole sides independently. Both slopes correspond to 0.25 filling per volt. The filling factor of the other states is assigned using the slope as described in Methods.
Extended Data Fig. 3 Results from different regions of the device.
a–e, Gate-dependent reflection contrast spectrum measured at different regions of the device P1 (a), P2 (b), P3 (c), P4 (d) and P5 (e) at 1.6 K. All share the same x axis, given at the bottom. The correlated states that can be identified are labelled on the top axes.
Extended Data Fig. 4 Results from a different device.
a, Gate-dependent reflection contrast spectrum of a different sample as descried in the Methods at 1.6 K. The filling factor is shown on the right axis. A fixed back gate voltage of 8 V is applied to dope the contact region to form good electrical contact between the TMD moiré superlattice and the contact electrode. The top gate voltage Vg is swept to tune the electron doping density. Below Vg = −9 V (dashed white line), the sample is charge neutral. b, As in a, focusing on the 2s transition in the sensor. The identified insulating states are marked with their corresponding filling factors.
Extended Data Fig. 5 Analysis of the −1/3 and −2/3 states.
a, b, Two horizontal line cuts of Fig. 2a at 1.8403 eV (a) and 1.8383 eV (b). These energies correspond to the 2s exciton peak energy for the v = −1/3 and v = −2/3 states, respectively. They appear as peaks in −ΔR/R0. Results at different temperatures (in ascending order from bottom to top, 1.6 K, 4 K, 6 K, 10 K, 13 K, 15 K, 17 K, 19 K, 22 K, 26 K, 29 K, 32 K, 35 K, 38 K, 40 K and 50 K) are vertically displaced for clarity. The red curves are Lorentzian fits to the peaks for the corresponding states. c, Reflection contrast spectrum (−ΔR/R0) for Vg = − 3.3 V (v = −2/3) at 1.6 K. The red area underneath the 2s peak is integrated to obtain the 2s spectral weight. d, The 2s spectral weight as a function of temperature for state v = −1/3 (black symbols) and v = −2/3 (red symbols). The v = −1/3 state has a slightly higher TC than the v = −2/3 state. The lines are guides to the eye.
Extended Data Fig. 6 First derivative of data shown in Fig. 2a with respect to energy.
This is used to evaluate the gate voltage and width of the less well-developed insulating states such as those with |v| > 1. The black arrows highlight the enhanced features for the 1/7 and 6/7 states.
Extended Data Fig. 7 Optical response of the WSe2/WS2 moiré superlattice.
a, b, Gate-dependent reflection contrast spectrum in regions of the device without (a) and with (b) the WSe2 sensor at 1.6 K. The Fermi level is inside the WSe2/WS2 bandgap between the dashed lines.
Extended Data Fig. 8 Contour plots of additional gate-dependent reflection contrast spectrum at higher temperatures.
From top to bottom, T = 67 K, 80 K and 150 K. All share the same x axis, given at bottom.
Extended Data Fig. 9 Transition temperature to the charge-ordered state (simulation).
Transition temperatures are determined from the peak in the temperature dependence of the heat capacity Cp for fillings 1/7, 1/4, 1/3, 2/5 and 1/2. Temperature is given in units of the energy e2/(4πεε0a) and d/a is fixed to be 10.
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Xu, Y., Liu, S., Rhodes, D.A. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature 587, 214–218 (2020). https://doi.org/10.1038/s41586-020-2868-6
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DOI: https://doi.org/10.1038/s41586-020-2868-6
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