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# Enhanced interactions of interlayer excitons in free-standing heterobilayers

## Abstract

Strong, long-range dipole–dipole interactions between interlayer excitons (IXs) can lead to new multiparticle correlation regimes1,2, which drive the system into distinct quantum and classical phases2,3,4,5, including dipolar liquids, crystals and superfluids. Both repulsive and attractive dipole–dipole interactions have been theoretically predicted between IXs in a semiconductor bilayer2,6,7,8, but only repulsive interactions have been reported experimentally so far3,9,10,11,12,13,14,15,16. This study investigated free-standing, twisted (51°, 53°, 45°) tungsten diselenide/tungsten disulfide (WSe2/WS2) heterobilayers, in which we observed a transition in the nature of dipolar interactions among IXs, from repulsive to attractive. This was caused by quantum-exchange-correlation effects, leading to the appearance of a robust interlayer biexciton phase (formed by two IXs), which has been theoretically predicted6,7,8 but never observed before in experiments. The reduced dielectric screening in a free-standing heterobilayer not only resulted in a much higher formation efficiency of IXs, but also led to strongly enhanced dipole–dipole interactions, which enabled us to observe the many-body correlations of pristine IXs at the two-dimensional quantum limit. In addition, we firstly observed several emission peaks from moiré-trapped IXs at room temperature in a well-aligned, free-standing WSe2/WS2 heterobilayer. Our findings open avenues for exploring new quantum phases with potential for applications in non-linear optics.

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## Data availability

The datasets generated during the current study are available from the corresponding author upon request.

## References

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## Acknowledgements

The authors acknowledge funding support from ANU PhD student scholarship, Australian Research Council (grant no. DP220102219, DP180103238, LE200100032) and ARC Centre of Excellence in Quantum Computation and Communication Technology (project number CE170100012), and National Heart Foundation (ARIES ID: 35852). The authors thank C. Jagadish, D. MacDonald and H. Nguyen from the Australian National University for their facility support. T.L. acknowledges T. Wu and S. Fang from the Information and Network Centre of Xiamen University for their help with the computing.

## Author information

Authors

### Contributions

Y.L. conceived and supervised this study. Y.Z. fabricated sample 2 and X.S. fabricated all the other samples used in this study. X.S. and Y.Z. performed the optical experiments. X.S., H.Q., B.L. and S.R. contributed to second-harmonic generation measurements. H.Q., Y.T. and Y.Z. contributed to experimental setup. T.L. did simulations using the quantum-electrostatic heterostructure method. T.Y. contributed to simulation and fitting. Y.L. analysed the data, developed the models and interpreted the results. X.S. processed experimental raw data with fittings and created all the figure plots. Y.L., X.S. and Y.Z. drafted the manuscript, and all authors participated in manuscript editing and discussion.

### Corresponding author

Correspondence to Yuerui Lu.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature thanks Christopher Gies and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Extended data figures and tables

### Extended Data Fig. 1 Angle-resolved second-harmonic-generation (SHG).

The polarization-dependent SHG signal measured on the monolayer WSe2 (blue circles) and WS2 (red circles) regions of samples and the corresponding fittings (blue and red curves), which confirm the twist angles of 0 ± 0.5° (a), 3 ± 0.5° (b), 45 ± 0.5° (c) and 51 ± 0.5° (d), between WSe2 and WS2 layers. Samples in (a) and (b) are R-type stacked heterobilayers, which are confirmed by the enhanced SHG signal in experiments from WSe2/WS2 hetero-structure region comparing with the corresponding WSe2 and WS2 monolayer regions. Samples in (c) and (d) are H-type stacked heterobilayers, which are confirmed by the quenched SHG signal in experiments from WSe2/WS2 hetero-structure region comparing with the corresponding WSe2 and WS2 monolayer regions.

### Extended Data Fig. 2 Temperature-dependent PL spectra of IX from sample 1.

Zoomed in temperature dependent PL spectra with fitting curves for interlayer exciton peaks: IX1 (blue), IX2 (red) and IX3 (green), from free-standing heterobilayer sample (sample 1, close-to-zero twist angle). Black and dark yellow curves are experimental data and accumulative fitting curve, respectively. A 532 nm CW laser was used for excitation and the power used was 8.83 µW.

### Extended Data Fig. 3 Dielectric screening dependent IX emissions in WS2/WSe2 heterobilayers.

ac, Measured PL spectra at 83 K from: (a) sample 1. Inset: schematic cross-section diagram for free-standing (region A) and SiO2-supported (region C) WS2/WSe2. (b) Sample 5 (twist angle 52 ± 0.5°). A WS2/WSe2 heterobilayer sample was firstly deposited on a polydimethylsiloxane (PDMS) substrate and then it was mechanically transferred to a SiO2/Si substrate (with a pre-patterned circular hole). But during the transfer process, the heterobilayer was broken, and the hole-region sample remained on the PDMS (as shown by inset). PDMS-supported part (region B) and the SiO2-supported part (region C) have the same twist angle. (c) Sample #6 (twist angle 5 ± 0.5°). Inset: SiO2-supported part (region C), hBN-SiO2 encapsulated part (region D) and hBN-hBN encapsulated (region E) have the same twist angle. d, Extracted beta ratio (βXC) of samples with different dielectric screening, using the measured PL spectra displayed in (a–c), as a function of the top and bottom dielectric constants. βX is the β value (defined in main text) measured from sample in region X (X = A, B, C, D, E), and βC is the β value measured from the corresponding SiO2-supported sample part (from the same heterobilayer flake with region X and thus has the same twist angle with X). This beta ratio can exclude the contribution of sample twist angles, when we compare the beta values from various heterobilayer samples as shown in (a–c). The mean of βC was set to be 1. The error bars represent the experimental variation (standard deviations) observed from multiple (>5) measurements. Note: The dielectric constants of air, PDMS46, SiO2 and hBN47 (Extended Data Fig. 3) are 1, 2.3, 3.9 and 5.1, respectively. The dielectric constant of hBN was calculated by using $$\sqrt{{\varepsilon }_{i}^{\perp }\cdot {\varepsilon }_{i}^{||}}$$, where $${\varepsilon }_{i}^{\perp }$$ and $${\varepsilon }_{i}^{||}$$ are the out-of-plane and in-plane dielectric constants47.

### Extended Data Fig. 4 Twist angle dependence.

a, Measured IX formation efficiency β as a function of twist angle, for both suspended and SiO2-supported WSe2/WS2 heterobilayer sample at 83K. The error bars represent variations (standard deviations) from multiple measurements (>5). The data points without error bars showed a beta value (β) of zero, which means the IX emissions were not detected from those samples. b, the highest temperature where we started to observe IX emissions as a function of twist angle, for the free-standing WSe2/WS2 heterobilayers.

### Extended Data Fig. 5 Temperature-dependent PL spectra from the free-standing WS2/WSe2 heterobilayer (sample 2).

a, Measured temperature-dependent PL spectra from the suspended WS2/WSe2 heterobilayer (sample 2, H-type stacking, measured twist angle 51°) with fitting curves for interlayer exciton peaks: IX1 (blue), IX2 (red) and IX3 (green). The excitation power is around 3.5 µW. Black and dark yellow curves are experimental data and accumulative fitting curve, respectively. b, Extracted PL peak energy as a function of temperature, for three interlayer exciton peaks, labelled as IX1, IX2 and IX3. c, Extracted integrated PL intensity as a function of temperature, for three interlayer exciton PL peaks. d, Extracted full width at half maximum (FWHM) of interlayer exciton peaks as a function of temperature. The error bars in (b)–(d) represent the fitting uncertainty (standard deviations) obtained from multiple (>3) fitting analysis.

### Extended Data Fig. 6 Power-dependent PL spectra from the free-standing WS2/WSe2 heterobilayer (sample 2) at 83 K.

a, Zoomed in PL spectra with fitting curves for two interlayer peaks: IX1 (blue) and IX2 (red). Black and dark yellow curves are experimental data and accumulative fitting curve, respectively. b, Extracted full width at half maximum (FWHM) of interlayer excitons as a function of excitation power. c, Extracted PL peak energy as a function of excitation power, for WS2 exciton. The negligible energy shift of the A excitons from the WS2 over the entire power range used (< 1 meV) suggests that the laser induced thermal effect in the power-dependent PL measurements was negligible and it does not affect the analysis of our power dependent PL results. The error bars in (b) and (c) represent the fitting uncertainty (standard deviations) obtained from multiple (>3) fitting analysis.

### Extended Data Fig. 7 Calculation of interaction energy and dielectric function.

a, Calculated interaction energy of two IXs $${E}_{{XX}}$$ (in unit of Rydberg energy $${R}_{y}^{* }$$) as a function of interlayer exciton distance $$\triangle {\rho }^{{XX}}$$ (in unit of Bohr radius $${a}_{B}$$). $${E}_{{XX}}$$ shows a minimum energy at $${\triangle \rho }_{0}^{{XX}}$$. b, Zoom in plot of calculated dielectric function $$\varepsilon \left(q\right)$$ for free-standing WSe2/WS2, as shown in Supplementary Fig. 10(a). The solid vertical line is $$q=1/{\triangle \rho }_{0}^{{XX}}$$.

### Extended Data Fig. 8 Repulsive and attractive dipolar interactions in free-standing WS2/WSe2 heterobilayer with twist angle of 53 degree (sample 3).

a, Measured excitation power dependent PL spectra from a free-standing WS2/WSe2 heterobilayer (H-type stacking, measured twist angle 53 ± 0.5°) at 83 K with 532 nm CW laser. Two interlayer exciton PL peaks are highlighted by blue and red arrows. b, Measured PL peak energy of interlayer exciton IX1 (blue balls) and IX2 (red balls) as a function of exciton density at 83 K. c, Extracted integrated PL intensity of IX1 and IX2 as a function of pumping power. From the fitting curve, integrated PL intensities of IX 1 and IX2 grew sub-linearly (α = 0.80) and super-linearly (α = 1.39), respectively, with the increase of excitation power. The error bars in (b) and (c) represent the fitting uncertainty (standard deviations) obtained from multiple (>3) fitting analysis.

### Extended Data Fig. 9 Power-dependent PL at 83K from a WS2/WSe2 heterobilayer with twist angle of 45 degree (sample 4).

a, Measured excitation power-dependent PL spectra from a WS2/WSe2 heterobilayer (H-type stacking, measured twist angle 45 ± 0.5°) at 83 K using 532 nm CW laser. b, Zoomed in PL spectra for IX emissions, with fitting curves for three IX peaks labelled as IXX (red), IX 1 (green), IX 2 (purple). Black and dark yellow curves are experimental data and accumulative fitting curve, respectively.

### Extended Data Fig. 10 Repulsive and attractive dipolar interactions in a WS2/WSe2 heterobilayer with twist angle of 45 degree (sample 4).

ac, Extracted PL peak energy as a function of excitation power, for three interlayer exciton peaks, labelled as IXX (a), IX 1 (b) and IX 2 (c). The inset in (b) shows the energy difference between IX1 and IXX as a function a excitation power. d, Extracted full width at half maximum (FWHM) of IXX, IX 1 and IX 2 as a function of excitation power. e, Extracted integrated PL intensity of three interlayer peaks as a function of excitation power, the fitting curve give rise to a slope of 1.58 for IXX (red), a slope of 1.01 for IX 1 (green) and a slope of 1.18 for IX 2 (purple). f, Extracted PL peak energy of the A excitons from the WS2 layer in the heterobilayer, as a function of excitation power. The negligible energy shift of the A excitons from the WS2 over the entire power range used (< 0.5 meV) suggests that the laser induced thermal effect in the power-dependent PL measurements was negligible and it does not affect the analysis of our power dependent PL results. The error bars in all the panels represent the fitting uncertainty (standard deviations) obtained from multiple (>3) fitting analysis.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–10, Notes 1–11, Table 1 and references.

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Sun, X., Zhu, Y., Qin, H. et al. Enhanced interactions of interlayer excitons in free-standing heterobilayers. Nature 610, 478–484 (2022). https://doi.org/10.1038/s41586-022-05193-z

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