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# Exciton density waves in Coulomb-coupled dual moiré lattices

## Abstract

Strongly correlated bosons in a lattice are a platform that can realize rich bosonic states of matter and quantum phase transitions1. While strongly correlated bosons in a lattice have been studied in cold-atom experiments2,3,4, their realization in a solid-state system has remained challenging5. Here we trap interlayer excitons–bosons composed of bound electron–hole pairs, in a lattice provided by an angle-aligned WS2/bilayer WSe2/WS2 multilayer. The heterostructure supports Coulomb-coupled triangular moiré lattices of nearly identical period at the top and bottom interfaces. We observe correlated insulating states when the combined electron filling factor of the two lattices, with arbitrary partitions, equals $$\frac{1}{3},\frac{2}{3},\frac{4}{3}$$ and $$\frac{5}{3}$$. These states can be interpreted as exciton density waves in a Bose–Fermi mixture of excitons and holes6,7. Because of the strong repulsive interactions between the constituents, the holes form robust generalized Wigner crystals8,9,10,11, which restrict the exciton fluid to channels that spontaneously break the translational symmetry of the lattice. Our results demonstrate that Coulomb-coupled moiré lattices are fertile ground for correlated many-boson phenomena12,13.

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## Data availability

Source data are provided with this paper. Additional data that support the findings of this study are available from the corresponding authors upon reasonable request.

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## Acknowledgements

We thank A. H. MacDonald, E. J. Mueller, Y.-H. Zhang and A. Vishwanath for fruitful discussions. This study was supported by the National Science Foundation (NSF) under DMR-2114535 (optical measurements), the US Office of Naval Research under award no. N00014-21-1-2471 (device fabrication) and the US Department of Energy, Office of Science, Basic Energy Sciences, under award number DE-SC0019481 (analysis). The study was also funded in part by the Gordon and Betty Moore Foundation. It made use of the Cornell NanoScale Facility, a National Nanotechnology Coordinated Infrastructure (NNCI) member supported by NSF grant NNCI-2025233. The growth of the hBN crystals was supported by the Elemental Strategy Initiative of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, and the CREST programme (JPMJCR15F3) of the Japan Science and Technology Agency (JST).

## Author information

Authors

### Contributions

Y.Z., Z.X. and R.D. fabricated the devices. Y.Z. and Z.X. performed the measurements and analysed the data. K.W. and T.T. grew the bulk hBN crystals. Y.Z., Z.X., K.F.M. and J.S. designed the scientific objectives, oversaw the project and cowrote the manuscript. All authors discussed the results and commented on the manuscript.

### Corresponding authors

Correspondence to Jie Shan or Kin Fai Mak.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Materials thanks David Ruiz-Tijerina and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Extended data

### Extended Data Fig. 1 Reflectance contrast (RC) spectra.

a,b, Representative RC spectra of the sensor 2s exciton (a) and the fundamental moiré exciton in WS2 (b) of device S1 with lattice filling (νt, νb). The spectral line shape is given by the optical interference effect in the multiple layer structure on the Si/SiO2 substrate. We use the peak-to-peak amplitude of the RC between 1.836 and 1.853 eV (R2s) as an indicator of the 2s exciton spectral weight (a), and between 1.95 and 2.05 eV (RMX) for the moiré exciton (b).

### Extended Data Fig. 2 Optical reflectance contrast spectrum of device S1.

a–c, Gate voltage dependence of the reflectance contrast spectrum covering the 1 s and 2s resonances of the sensor and the WS2 moiré excitons. Electrons are located in the bottom lattice solely (a), the top lattice solely (b) and equally in two lattices (c). The total filling factors ν = 0, 1 and 2 are determined according to the sensor 2s exciton resonance.

### Extended Data Fig. 3 Magnetic-field dependence.

Filling dependence of R2s at representative perpendicular magnetic fields. The curves are displaced vertically for clarity. The perpendicular electric field is 2 mV/nm. The black dashed lines mark total filling factor ν = 0, 2/3, 1 and 2, at which insulating states are observed in the absence of the magnetic field. These states do not disperse with magnetic field and are therefore non-topological. The orange line at each magnetic field denotes the peak positions if the state were topological with Chern number 1.

### Extended Data Fig. 4 Determination of band alignment in the heterostructure.

a, Left: Simplified band diagram of natural bilayer WSe2. While the valence bands are from the K/K’ valley, the conduction band minimum is from the Q valley. The indirect optical gap is 1.565 eV as determined by the emission peak in the PL spectrum (right). b, Left: Type-II band alignment in the dual-moiré heterostructure. The dashed and solid lines represent spin-down and spin-up bands from the K valley of the monolayer TMDs. The energy scales are determined by PL measurements. Right: PL spectrum of the heterostructure showing the interlayer exciton emission at 1.43 eV and the bilayer WSe2 indirect optical gap at 1.56 eV.

### Extended Data Fig. 5 Additional sample regions of device S1.

a,b, Dependence of R2s on the total filling factor and electric field at two additional regions of the same device as in the main figures. Electrons are in both moiré lattices in the region enclosed by the dashed lines. The insulating state at ν = 1 is robust in all regions. Fewer and less robust insulating states are observed here particularly in a. The features appear curved at certain electric fields because the large contact resistance causes nonlinear gating effects. The electric field offset at νt = νb is likely caused by the layer asymmetry in these regions.

### Extended Data Fig. 6 Additional devices.

a,b, Dependence of R2s on the total filling factor and electric field for device S2 (a) and S3 (b). Electrons are in both moiré lattices in the region enclosed by the dashed lines. The insulating state at ν = 1 is robust in both devices. The insulating states at total fractional fillings are observed only in device S3, including an insulating state at $$\nu = \frac{1}{2}$$. The features appear curved at certain electric fields because the large contact resistance causes nonlinear gating effects.

### Extended Data Fig. 7 Exciton density waves in a particle–hole transformation picture.

a,–c, Schematic representation of the correlated insulating states in coupled moiré lattices at total filling ν = 1 (a), $$\nu = \frac{1}{3}$$ (b) and $$\nu = \frac{2}{3}$$ (c). Top, electrons in the top lattice (red) and bottom lattice (blue). Electrons in the top lattice are bound to the empty sites in the bottom lattice directly below them to minimize the Coulomb interactions. Middle, particle–hole transformation performed on the bottom lattice generates interlayer excitons (red-blue circles) and excess holes (empty blue circles). Bottom, the excess holes form generalized Wigner crystals (or charge-ordered states) because of the hole-hole repulsion V. The excitons are guided to the channels defined by the hole Wigner crystals by the exciton–hole repulsion V′. The exciton density distribution shows an exciton fluid at ν = 1 and exciton density waves at $$\nu = \frac{1}{3},\frac{2}{3}$$. The latter breaks the translational symmetry of the lattice.

## Source data

### Source Data Fig. 1

Statistical source data.

### Source Data Fig. 2

Statistical source data.

### Source Data Fig. 3

Statistical source data.

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Zeng, Y., Xia, Z., Dery, R. et al. Exciton density waves in Coulomb-coupled dual moiré lattices. Nat. Mater. (2023). https://doi.org/10.1038/s41563-022-01454-4

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• DOI: https://doi.org/10.1038/s41563-022-01454-4