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# Strongly correlated excitonic insulator in atomic double layers

## Abstract

Excitonic insulators (EIs) arise from the formation of bound electron–hole pairs (excitons)1,2 in semiconductors and provide a solid-state platform for quantum many-boson physics3,4,5,6,7,8. Strong exciton–exciton repulsion is expected to stabilize condensed superfluid and crystalline phases by suppressing both density and phase fluctuations8,9,10,11. Although spectroscopic signatures of EIs have been reported6,12,13,14, conclusive evidence for strongly correlated EI states has remained elusive. Here we demonstrate a strongly correlated two-dimensional (2D) EI ground state formed in transition metal dichalcogenide (TMD) semiconductor double layers. A quasi-equilibrium spatially indirect exciton fluid is created when the bias voltage applied between the two electrically isolated TMD layers is tuned to a range that populates bound electron–hole pairs, but not free electrons or holes15,16,17. Capacitance measurements show that the fluid is exciton-compressible but charge-incompressible—direct thermodynamic evidence of the EI. The fluid is also strongly correlated with a dimensionless exciton coupling constant exceeding 10. We construct an exciton phase diagram that reveals both the Mott transition and interaction-stabilized quasi-condensation. Our experiment paves the path for realizing exotic quantum phases of excitons8, as well as multi-terminal exciton circuitry for applications18,19,20.

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## Data availability

The source data that support the findings of this study are available with the paper. Source data are provided with this paper.

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## Acknowledgements

We thank E. J. Mueller and C. Dean for fruitful discussions. Research was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award numbers DE-SC0019481 (device fabrication and theoretical analysis) and DE-SC0022058 (optical characterization), the US Office of Naval Research under award number N00014-21-1-2471 (data analysis) and the National Science Foundation (NSF) under DMR-2004451 (capacitance measurement). Growth of the hBN crystals was supported by the Elemental Strategy Initiative of MEXT, Japan and CREST (JPMJCR15F3), JST. This work was performed in part at the Cornell NanoScale Facility, an NNCI member supported by NSF Grant NNCI-2025233. K.F.M. acknowledges support from the David and Lucille Packard Fellowship.

## Author information

Authors

### Contributions

L.M. and P.X.N. fabricated the devices, performed the measurements and analysed the data. Z.W. provided assistance in device fabrication. Y.Z. and A.H.M. provided theoretical support for the measurements. K.W. and T.T. grew the bulk hBN crystals. L.M., P.X.N., 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 manuscript.

### Corresponding authors

Correspondence to Kin Fai Mak or Jie Shan.

## Ethics declarations

### Competing interests

The authors declare no competing financial interests.

Peer review information Nature thanks Raymond Ashoori, Denis Golez and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Optical images of devices 1 and 2.

a, b, Schematic cross-section (upper panels) and optical micrograph (lower panels) of device 1 (a) and device 2 (b). Constituent layers in the stack include WSe2 monolayer (red line), MoSe2 monolayer (yellow line), top gate (TG, white dashed line), bottom gate (BG, white dash-dotted line) and local gate (LG Mo, white dotted line). The region of interest is shaded brown (‘1’). In device 1, the TMD double layers are separated by a thick hBN spacer (blue line) in the contact region (‘2’); the thin hBN spacer in the region of interest is not marked. Both electrons and holes are injected into ‘1’ from ‘2’ through one edge (solid black line). In device 2, electrons and holes are injected into ‘1’ from the two long sides separately. The local gate is not shown in the schematics; it heavily dopes the isolated MoSe2 region to form good contact to the metal electrode. The rectangular yellow bars are Pt electrodes. The scale bar is 5 μm.

### Extended Data Fig. 2 Dependence of interlayer bandgap on anti-symmetric gating.

a, Gate dependence of penetration capacitance at 15 K under varying anti-symmetric gating Δ and Vb = 0 V. The step falls signify electron doping into the MoSe2 conduction band (CB) or hole doping into the WSe2 valence band (VB). The separation between the rising and falling edges determines the bandgap EG. An additional (small) step on the electron-doping side arises from the presence of a small non-overlapped MoSe2 monolayer inside the dual-gated device that affects the penetration capacitance. b, Interlayer bandgap EG (symbols) extracted from a as a function of . The linear fit (dashed line) has a slope of ~0.21e. The gap energy extrapolated for  = 0 corresponds to the intrinsic bandgap energy $${E}_{{\rm{G}}}^{0}\approx 1.6{\rm{eV}}$$.

### Extended Data Fig. 3 Tunnelling current, exciton lifetime and electroluminescence (EL).

a, Bias dependence of interlayer tunnelling current of device 1 under anti-symmetric gating Δ = 4.6 V (red), 4.25 V (blue), 4.00 V (black) and 3.75 V (green). The current onset is observed approximately when eVbEG. b, Interlayer tunnelling current over a large bias range at Δ = 4.3 V. The insets illustrate the non-resonant (left) and resonant (right) tunnelling regimes. c, Estimated exciton lifetime as a function of Vb from the tunnelling data at Δ = 4.6 V in a. d, Bias dependence of the EL spectrum (upper) and spectrally integrated EL intensity (lower). The data are acquired simultaneously with the tunnelling current in b at equal electron and hole densities at 30 K. The EL at 1.60–1.65 eV arises from intralayer exciton emission in MoSe2; the feature below 1.60 eV is likely to originate from impurity bound states. Intralayer exciton emission from WSe2 is not observed, presumably due to quenching from resonant energy transfer. The EL starts to emerge at bias voltages of roughly 1.7 V (red dashed line), when the device enters the resonant tunnelling regime. Both the tunnelling current and the EL intensity increases drastically above 1.7 V.

### Extended Data Fig. 4 Determination of the charge gap.

a, b, The penetration capacitance (a) and the charge chemical potential of the double layer (b) at 15 K as a function of Vg at varying exciton densities. The capacitance peak shows the presence of a charge-incompressible state. The integrated area of the peak gives the chemical potential jump (or the charge gap) at equal electron–hole density. The zero point of the chemical potential shift in b has been shifted to Vg = 0 V for comparison of different exciton densities. The charge gap closes near the Mott density. c, d, Similar to a, b, at fixed exciton density nX = 2.4 × 1011 cm−2 for different temperatures. The charge gap closes at the ionization temperature.

### Extended Data Fig. 5 Exciton compressibility in the high-temperature limit.

a, Inverse interlayer capacitance (or exciton compressibility) as a function of temperature at varying exciton densities. The lines are linear fits in the high-temperature limit, that is, gnX/kBTTs. b, Amplitude of the extracted x intercept in a as a function of exciton density. A linear fit (blue) gives g = (2.6 ±0.1) × 10−14 eV cm2. The inset shows the density dependence of the slope extracted from a (symbols). The red line is the mean-field result described in the main text. The density range is the same as in the main panel. The vertical error bars are the uncertainty of the linear fit in a. The horizontal error bar is the typical uncertainty of the exciton density from the analysis shown in Fig. 1f.

### Extended Data Fig. 6 Main results from device 2.

a, b, Normalized penetration capacitance (a) and interlayer capacitance (b) as a function of bias and gate voltages at 20 K. The Mo layer is grounded and the bias voltage is applied to the W layer. The two red dashed lines denote the conduction band edge of MoSe2 (vertical line) and the valence band edge of WSe2 (line with slope +1). The contour plot is flipped horizontally compared to Fig. 2 of the main text for device 1, in which the W layer is grounded and the bias voltage is applied to the Mo layer. There the valence band edge of WSe2 corresponds a vertical line, and the conduction band edge of MoSe2, a line with slope −1. The two white dashed lines denote the bias voltage at which the charge gap closes (I) and the exciton fluid becomes compressible (II). The difference between the two values (~40 mV) corresponds to the exciton binding energy in the limit of zero exciton density. The exciton binding energy in device 2 with a slightly thinner hBN spacer (roughly 5-layer) is larger than that in device 1 (approximately 25 mV).

### Extended Data Fig. 7 Penetration and interlayer capacitances of device 3.

a, b, Penetration capacitance CP (a) and interlayer capacitance CI (b) as a function of bias and gate voltages at 10 K. The device is a MoS2/WSe2 double layer with the electron and hole contacts not close by. The Mo layer is grounded. The two red dashed lines denote the conduction band edge of MoS2 (vertical line) and the valence band edge of WSe2 (line with slope +1). The charge incompressible region is enclosed by red dashed lines. White dashed lines correspond to the bias voltages at which the charge gap closes and the exciton population is expected to appear, respectively. Exciton injection without creating free charges is not observed, corresponding to negligible CI in the triangular region bound by the red and white dashed lines. c, d, Same as b at 30 K and 50 K, respectively. Thermally assisted exciton injection is observed.

### Extended Data Fig. 8 Circuit diagram.

a, Circuit diagram for the penetration capacitance measurement. The red dashed line encloses the sample area. The reference (Vref and Cref) is used to cancel the parasitic background capacitance. The HEMT is biased at voltage VH. b, Equivalent circuit model of CP in a. Here Cg ≈ 2Cgg is the sample-to-gate geometrical capacitance, which is about twice the gate-to-gate geometrical capacitance Cgg.

### Extended Data Fig. 9 Equivalent device circuit model for electrostatics simulation.

CQ,M and CQ,W are the quantum capacitances of the MoSe2 and WSe2 monolayers, respectively. For details see Methods.

### Extended Data Fig. 10 Exciton entropy analysis.

a, Equation of state (chemical potential µX versus density nX) for excitons at varying temperatures T. The data are obtained by integrating the measured interlayer capacitance with respect to exciton density. b, Exciton chemical potential of a, replotted as a function of temperature at varying exciton densities. c, Entropy change per exciton, $${\left(\frac{\partial S}{\partial {n}_{{\rm{X}}}}\right)}_{T}$$, as a function of exciton density at varying temperatures (S denoting the 2D exciton entropy density). The data are obtained from b by applying the Maxwell’s relation, $${\left(\frac{\partial {\mu }_{{\rm{X}}}}{\partial T}\right)}_{{n}_{{\rm{X}}}}={-\left(\frac{\partial S}{\partial {n}_{{\rm{X}}}}\right)}_{T}$$. As density increases, $${\left(\frac{\partial S}{\partial {n}_{{\rm{X}}}}\right)}_{T}$$ drops rapidly, followed by a slow decrease. The crossover between the two regimes is correlated with the location of the compressibility peak in Fig. 4b of the main text. It corresponds to a crossover from a non-degenerate to a degenerate exciton fluid. In the latter, the density fluctuations are suppressed and the entropy change per exciton is substantially reduced. No entropic signature of the 2D superfluid transition is observed. A broad entropy peak is observed near the Mott density, reflecting the presence of a low-temperature Mott critical point. Only a high-temperature crossover is accessed here.

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Ma, L., Nguyen, P.X., Wang, Z. et al. Strongly correlated excitonic insulator in atomic double layers. Nature 598, 585–589 (2021). https://doi.org/10.1038/s41586-021-03947-9

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• DOI: https://doi.org/10.1038/s41586-021-03947-9

• ### Common microscopic origin of the phase transitions in Ta2NiS5 and the excitonic insulator candidate Ta2NiSe5

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