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Ruishi Qi, Andrew Y. Joe, Zuocheng Zhang, Yongxin Zeng, Tiancheng Zheng, Qixin Feng, Jingxu Xie, Emma Regan, Zheyu Lu, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Sefaattin Tongay, Michael F. Crommie, Allan H. MacDonald, Feng Wang

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[Thermodynamic behavior of correlated electron-hole fluids in van der Waals heterostructures](https://mdr.nims.go.jp/datasets/ed47b389-89dd-4878-a835-460907fcdc84)

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Thermodynamic behavior of correlated electron-hole fluids in van der Waals heterostructuresArticle https://doi.org/10.1038/s41467-023-43799-7Thermodynamic behavior of correlatedelectron-hole fluids in van der WaalsheterostructuresRuishi Qi 1,2,10, Andrew Y. Joe 1,2,10 , Zuocheng Zhang 1, Yongxin Zeng3,Tiancheng Zheng1,4, Qixin Feng1,2, Jingxu Xie 2,5, Emma Regan 1,2,5,Zheyu Lu2,5, Takashi Taniguchi 6, Kenji Watanabe 7, Sefaattin Tongay 8,Michael F. Crommie 1,2, Allan H. MacDonald 3 & Feng Wang 1,2,9Coupled two-dimensional electron-hole bilayers provide a unique platform tostudy strongly correlated Bose-Fermimixtures in condensedmatter. Electronsand holes in spatially separated layers can bind to form interlayer excitons,composite Bosons expected to support high-temperature exciton con-densates. The interlayer excitons can also interact strongly with excess chargecarriers when electron and hole densities are unequal. Here, we use opticalspectroscopy to quantitatively probe the local thermodynamic properties ofstrongly correlated electron-hole fluids in MoSe2/hBN/WSe2 heterostructures.We observe a discontinuity in the electron and hole chemical potentials atmatched electron and hole densities, a definitive signature of an excitonicinsulator ground state. The excitonic insulator is stable up to aMott density of~0.8 × 1012 cm−2 and has a thermal ionization temperature of ~70 K. The densitydependence of the electron, hole, and exciton chemical potentials revealsstrong correlation effects across the phase diagram. Compared with a non-interacting uniform charge distribution, the correlation effects lead to sig-nificant attractive exciton-exciton and exciton-charge interactions in theelectron-hole fluid. Ourwork highlights the unique quantumbehavior that canemerge in strongly correlated electron-hole systems.Two-dimensional (2D) electron gases have been extensively studied inthe last few decades, leading to fascinating discoveries such as theinteger and fractional quantum Hall effects and 2D Wigner crystals1–5.Electron-hole bilayers6,7 composed of 2D electron and hole gases thatare in close proximity to each other while remaining electrically iso-lated have been difficult to realize experimentally and are thereforemuch less explored. The strong attractive interactions between elec-trons and holes are expected to play a significant role in the thermo-dynamic behavior of the coupled system. In the limit of low andmatched electron andhole densities, the layer-separated electrons andholes can form bound pairs to create interlayer excitons. Interlayerexcitons are composite bosons and are therefore an attractiveReceived: 5 October 2023Accepted: 20 November 2023Check for updates1Department of Physics, University of California, Berkeley, CA 94720, USA. 2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA94720, USA. 3Department of Physics, University of Texas at Austin, Austin, TX 78712, USA. 4School of Physical Sciences, University of Chinese Academy ofSciences, Beijing, China. 5Graduate Group in Applied Science and Technology, University of California at Berkeley, Berkeley, CA 94720, USA. 6InternationalCenter for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 7Research Center for FunctionalMaterials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 8School for Engineering of Matter, Transport and Energy, ArizonaState University, Tempe, AZ 85287, USA. 9Kavli Energy NanoSciences Institute, University of California Berkeley and Lawrence Berkeley National Laboratory,Berkeley, CA 94720, USA. 10These authors contributed equally: Ruishi Qi, Andrew Y. Joe. e-mail: andrew.joe@ucr.edu; fengwang76@berkeley.eduNature Communications |         (2023) 14:8264 11234567890():,;1234567890():,;http://orcid.org/0009-0000-1305-1104http://orcid.org/0009-0000-1305-1104http://orcid.org/0009-0000-1305-1104http://orcid.org/0009-0000-1305-1104http://orcid.org/0009-0000-1305-1104http://orcid.org/0000-0003-4376-7386http://orcid.org/0000-0003-4376-7386http://orcid.org/0000-0003-4376-7386http://orcid.org/0000-0003-4376-7386http://orcid.org/0000-0003-4376-7386http://orcid.org/0000-0001-7851-6101http://orcid.org/0000-0001-7851-6101http://orcid.org/0000-0001-7851-6101http://orcid.org/0000-0001-7851-6101http://orcid.org/0000-0001-7851-6101http://orcid.org/0009-0000-9691-2474http://orcid.org/0009-0000-9691-2474http://orcid.org/0009-0000-9691-2474http://orcid.org/0009-0000-9691-2474http://orcid.org/0009-0000-9691-2474http://orcid.org/0000-0002-9100-6031http://orcid.org/0000-0002-9100-6031http://orcid.org/0000-0002-9100-6031http://orcid.org/0000-0002-9100-6031http://orcid.org/0000-0002-9100-6031http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8246-3444http://orcid.org/0000-0001-8246-3444http://orcid.org/0000-0001-8246-3444http://orcid.org/0000-0001-8246-3444http://orcid.org/0000-0001-8246-3444http://orcid.org/0000-0003-3561-3379http://orcid.org/0000-0003-3561-3379http://orcid.org/0000-0003-3561-3379http://orcid.org/0000-0003-3561-3379http://orcid.org/0000-0003-3561-3379http://orcid.org/0000-0001-8369-6194http://orcid.org/0000-0001-8369-6194http://orcid.org/0000-0001-8369-6194http://orcid.org/0000-0001-8369-6194http://orcid.org/0000-0001-8369-6194http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43799-7&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43799-7&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43799-7&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-43799-7&domain=pdfmailto:andrew.joe@ucr.edumailto:fengwang76@berkeley.educandidate for studying correlated quantum Bosonic states in con-densed matter8–11, whose phase diagrams are expected to includeexciton Bose-Einstein condensates (BECs)12–14, excitonic insulators15,16,and exciton crystals17. Although these correlated excitonphysics can inprinciple be achieved in bulkmaterials18,19, interlayer excitons in such abilayer system have unique advantages such as easily tunable energygap and separate electrical access to the electrons and holes. In thegeneral case of unequal electron and hole densities, the system pro-vides a platform to study a tunable mixture of equilibrium Fermi andBose gases that contain charge-exciton complexes.Previous studies of correlated excitonic phases have focused onbilayer systems with semiconductor double quantum wells20–22 andgraphene bilayers23,24. However, in these systems the exciton bindingenergy is relatively weak (a few meV or less), so it usually requires astrong magnetic field to form quasi-electrons/holes in the Landaulevels or requires a low temperature for exciton formation. Semi-conducting transition metal dichalcogenide (TMD) heterostructures,due to their strong Coulomb interaction and large exciton bindingenergies (hundreds of meV)25–27, provide a model system to explorecorrelated 2D electron-hole (e-h) fluids in the strong coupling regime.By inserting a thin tunnel barrier such as hexagonal boron nitride(hBN) between two TMD layers, ultralong exciton lifetimes, highexciton density, and equilibrium e-h and exciton fluids can berealized15,28. Theoretical studies of such heterostructures have pre-dicted record-high BEC critical temperatures with the tunability toexplore strong and weak coupling regimes6,13,29,30. Recently, electricalcapacitance measurements have been employed to probe interlayerexcitons in MoSe2/hBN/WSe2 heterostructures and revealed the pre-sence of correlated excitonic insulators15, defined as a charge-incompressible but exciton-compressible state. However, thermo-dynamic behavior of the e-h fluid across the electron and hole dopedphase diagram has not been established.Here, we develop quantitative optical spectroscopy to accuratelydetermine the local electron and hole density and use it to determinethe thermodynamic behavior of correlated e-h fluids. It is wellknown that optical resonances in TMD layers depend sensitively on thecarrier density due to their enhanced light-matter and Coulombinteractions31,32. With careful in-situ calibration of this dependence, wecan determine the electron, hole, and exciton densities, and then theelectron, hole, and exciton chemical potentials of our gated TMDheterostructures. We observe electrically tunable and highly corre-lated excitons and exciton-charge complexes in this system: (1) Anexcitonic insulator ground state can appear at equal electron and holedensity, which features strong exciton-exciton correlation. The exci-ton insulator ionizes at both increased exciton density (Mott transi-tion) and elevated temperature (thermal melting). (2) The interlayerexcitons interact stronglywith additional electrons or holes, leading toan exciton-density dependent charge chemical potential for a fixed netcharge density.ResultsFigure 1a shows a schematic cross-section and electrical circuitrepresentation of our device. We use MoSe2 and WSe2 as the activeelectron and hole layers and separate them by a 3 nm thin hBN layer.The heterostructure is encapsulated with dielectric hBN and graphitetop (VTG) and bottom (VBG) gates. To create reliable electrical con-tacts, we follow ref. 15 to incorporate a thicker hBN spacer between theTMD layers in part of the heterostructure (region 0). The TMD layersare then contacted by few-layer graphene in region 0. The verticalelectric field generated by VΔ � VTG � VBG can control the electricpotential difference between the MoSe2 and WSe2 layers. It enables aheavily electron doped MoSe2 layer and a heavily hole doped WSe2layer in region 0, which serve as carrier reservoirs for the area ofinterest (region 1) (details in Methods and Supplementary Figs. 1, 2).This improved contact method allows simultaneous doping of bothelectrons and holes in the active region. The combination of the thinhBN layer suppressing interlayer tunneling and efficient charge injec-tion from the contacts ensures an equilibrium state and accuratemeasurements of thermodynamic quantities. An optical image of sucha device is shown in Fig. 1b.We control the band alignment and electron (ne) and hole (nh)densities in the active region using gate voltages (VTG andVBG) and theWSe2 voltage (Vh) while grounding the MoSe2 layer (Ve =0). Theinterlayer bias voltage VB � Vh � V e and vertical electric field VΔ bothcreate an energy difference in the two layers and tune the effectiveband gap. The gate voltage VG � VTG +VBG tunes the Fermi level andcontrols the net charge ne � nh in the e-h system. We can achievereliable electrical control inmultiple devices and the data shownbelowis highly reproducible (Supplementary Figs. 1, 3 and 4). Unless other-wise specified, the results below are from device D1 at VΔ = 11 V atsample temperature T = 2.5 K.We use optical spectroscopy to quantitatively determine theelectron and hole densities in MoSe2 and WSe2 layers, respectively. InFig. 1c, wefirst showrepresentative reflection spectraR as a function ofthe applied gate voltage VG in the low VB regime where the type-IIband gap is not closed. When the Fermi level is tuned into the gap andboth layers are undoped (middle region of Fig. 1c), we observe strongresonances at 719 nm and 754 nm, which are attributed to the intra-layer excitonic responses in the WSe2 and MoSe2, respectively. WhentheTMD isdopedwith holes or electrons (left and right regions), three-particle bound states commonly known as trions will form, giving riseto an additional absorption peak at longer wavelength31,33. Withincreasing doping, the trion peak becomes stronger, while the excitonpeak blueshifts and loses its oscillator strength due to Pauli blocking32.700 720 740 760 780Wavelength (nm)0.10.20.30.40.50.6Reflectivity0 1 2ne, nh (1012 cm-2)Data in e-h regimeMatched spectrum0 0.5 1VG (V)700720740760780Wavelength(nm)0.250.30.350.40.450.5Reflectivit yMoSe2WSe2hBN10 μm3nm hBN1WSe2 MoSe2Top gateBack gate3nm hBNhBNhBNe–h+hBNregion 0Carrier reservoirregion 1Region of interestlaserVTGVBGVeVhba cMoSe2 excitonWSe2 excitonWSe2 trionMoSe2 trionh+ e–WSe2 contactMoSe2 contact0dMoSe2WSe2Fig. 1 | Electrical control and optical detection of charge states. a Schematiccross-section of the MoSe2/hBN/WSe2 heterostructure device. b Optical micro-scopy imageof thedevicewith layer boundaries outlined. cTypical gate-dependentreflection spectra at a low bias voltage (VB =0:75 V). d Determination of carrierdensities for a typical reflection spectrum in the e-h fluid regime. Black dots,reflection spectrum at VB =0:87 V,VG =0:92 V. Faded colored lines, spectrum-densitymaps forWSe2 (left) andMoSe2 (right) determined from single-layer dopedregime. Solid lines, reflection spectrum from the spectra-density maps thatachieves the best match with the black dots.Article https://doi.org/10.1038/s41467-023-43799-7Nature Communications |         (2023) 14:8264 2Thus, the reflection spectrumprovides a sensitive probe for the carrierdensity in each layer. In thewell-understood single layer doped regime,the undoped TMD layer simply acts as a passive dielectric layer, andthe doping density in the active TMD layer can be calculated from asimple parallel capacitor model (details in Methods). This allows us tocreate a map between the reflection spectra and carrier density foreach layer (Supplementary Fig. 5 and faded colored lines in Fig. 1d).When the bandgap is closed by a large VB, both layers becomedoped, forming a correlated e-h fluid. The black dots in Fig. 1d show arepresentative spectrum, where trion peaks in both layers appear inthis regime. The faded lines in Fig. 1d show the single-layer spectra forelectron and hole densities from 0 − 2 × 1012 cm−2 taken from thespectrum-density maps. We match the measured reflection spectrumwith the single-layer spectrum-densitymaps, allowing us to extract theelectron and hole densities in this less-understood e-h fluid regime.The solid lines display the matched single-layer spectra, which givene = 1:11 × 1012 cm�2 and nh =0:71 × 1012 cm�2. This spectrum-matching method assumes that the intralayer exciton/trion states donot get significantly perturbed by the existence of charges in the otherTMD layer. The intralayer excitons and trions are both tightly boundparticles with a typical size of only ~1 nm34–37, and their optical transi-tions are onlyminimally affectedby chargedparticles in theother TMDlayer that is 3 nm away. In our experiment, the excellent match inwavelength, intensity and line shape for all peaks confirms that ourassumption is valid.Figure 2a, b shows the gate and bias voltage dependence of themeasured electron and hole densities. From the density maps, we canestablish the phase diagram of the carrier doping in the MoSe2/hBN/WSe2 heterostructure (Fig. 2c). At low VB, the type-II band-alignedheterostructure has afinite bandgap, and the gate voltage can tune thesystem from charge neutrality (black) to hole-doped (green) orelectron doped (red). With increasing bias voltage, the band gapdecreases until it closes at a critical VB ≈ 0:82 V. Further increasing VBdopes both layers simultaneously (yellow). This is where interlayerexciton formation is expected. Notably, the carrier density contourlines (dash-dotted lines in Fig. 2a, b) are not straight in this regime.Instead, they show kinks near ne ≈ nh, indicating that the e-h fluid isstrongly correlated.The voltage dependence of ne and nh allows us to determine thecompressibility of interlayer excitons and charge carriers, indepen-dently. Figure 2d shows the exciton compressibility obtained by thepartial derivative of exciton density nx = min ne, nh� �with respect toVB. Figure 2e shows the charge compressibility, defined as the partialderivative of the net charge ne � nh with respect to VG. Interestingly,we find there is a triangle region that extends beyond the excitondoping edge where there is a charge-incompressible, exciton-compressible state. Previous capacitance studies have shown similarresults and have identified this state as an excitonic insulator15.Another important consequence of the formation of interlayerexcitons is the scaling of the carrier density as a function of the biasvoltage at net charge neutrality. In Fig. 2f, we show the bias depen-dence of nx along the ne ≈ nh line (black dashed line in Fig. 2a, b).Interestingly, we observe a sublinear increase of the exciton density.Fitting the experimental data to a power law scaling nx / ΔVηB gives afitted power of η ≈ 0:76±0:02 (Fig. 2f inset). This scaling power isconsistent with previous theoretical studies of strongly correlateddipolar exciton BECs12, where quantum Monte Carlo simulations pre-dict that the ground state energy per exciton scales approximately asn1:36x , i.e., η ≈ 1=1:36 ≈ 0:74.In the non-interacting limit, the electron and hole densities shouldincrease linearly with VB as determined by the geometric capacitance(yellow dashed line). The deviation from the linear bias dependence0.2 0.4 0.6 0.8VG (V)0.750.80.850.9VB(V)-101(ne-nh)/VG(1012cm-2V-1)0.2 0.4 0.6 0.8VG (V)0.750.80.850.9V B(V)a cebd fExcitonic insulatore–h+h+e–0.75 0.8 0.85 0.9 0.95VB (V)00.511.5n x(1012cm-2)ExperimentPower fitGeometriccapacitance10-2 10-1VB(V)10-1100n x(1012cm-2)0.76ne nh0.2 0.4 0.6 0.8VG (V)0.750.80.850.9VB(V)00.511.5n e(1012cm- 2)0.2 0.4 0.6 0.8VG (V)0.750.80.850.9VB(V)00.511.5n h(1012cm- 2)0.2 0.4 0.6 0.8VG (V)0.750.80.850.9VB(V)-10010n x/VB(1012cm-2V-1)Fig. 2 | Carrier density and compressibility maps. a, b False color map ofexperimentally determined electron (a) and hole (b) densities. Contour lines areoverlayed as gray dash-dotted curves. The black dashed line is along ne≈nh. c Phasediagram of the e-h system determined experimentally. Black: no carrier present inthe system. Green: WSe2 is hole doped. Red: MoSe2 is electron doped. Yellow: bothelectrons and holes are present. Four insets schematically show the band alignmentin each phase. d Partial derivative of the exciton density nx = minðne, nhÞ withrespect to bias voltageVB, corresponding to the interlayer exciton compressibility.e Partial derivative of net charge density ne � nh with respect to gate voltage VG,corresponding to the charge compressibility. f Bias dependence of exciton densityalong ne≈nh line. The blue curve is a power function fit to the experimental data(scatters). Yellow dashed line, density determined from the geometric capacitance,showing a linear increase with the bias voltage. Inset: same data but plotted in log-log scale, showing a power-law scaling. Source data are provided as a sourcedata file.Article https://doi.org/10.1038/s41467-023-43799-7Nature Communications |         (2023) 14:8264 3arises from the exciton-exciton interactions. Such interactions can beunderstood in two different physical pictures. In physical picture I, wecan consider the net repulsive dipolar interaction between tightlybound excitons. This repulsion between dipoles causes an energypenalty to generate a high-density exciton fluid. It therefore leads to asublinear scaling where the compressibility at low exciton density ishigher than that at high exciton density. In physical picture II, weconsider the exciton compressibility with respect to the “ideal capa-citor” model, which assumes homogeneous electron-hole distribu-tions. We find that the exciton compressibility approaches the valuepredicted by the geometric capacitance at high density, but has asignificantly enhanced value at low exciton density. It suggests astrongly attractive exciton-exciton interaction at low exciton density ifwe use the homogeneous distribution as the reference point. In thispicture, we can focus on the pure correlation effect by separating outthe electrostatic energy associated with uniform electron and holedistributions. The second picture is widely used in the study of cor-relation effects in electron liquids38–40, and it is theonlypicture that canbe readily used to describe the general electron-hole liquid thatincludes the electron-hole plasma athighdensity aswell as unbalancedelectron and hole densities.To further quantify the thermodynamic behavior of the interlayerexcitons and the exciton-charge complexes in the e-h fluid, we adoptthe second picture to determine the electron, hole, and exciton che-mical potential as a function of the electron and hole densities. Fol-lowing the theory developed in ref. 6, we separate the electrochemicalpotential into the electric potentialϕ, which contains the electrostaticenergy associated with homogeneous charge distributions in an idealcapacitor and is device-geometry dependent, and the chemicalpotential μ, which contains the correlation effects in the e-h fluids. Atequilibrium the chemical potentials (μe, μh) of the electron layer andthe hole layer are related to the electric potentials (ϕe, ϕh) by�eϕe +μe = � eV e � Ec,eϕh +μh = � eVh � Ev,ð1Þwhere e is elementary charge, and Ec and Ev are the conduction andvalence band energies. From the experimentally determined carrierdensities and applied voltages, the electric potentials can be derivedfrom the Poisson equation�eneϵ0ϵBN=ϕe � VTGtt+ϕe � ϕhtm,enhϵ0ϵBN=ϕh � VBGtb+ϕh � ϕetm:ð2ÞHere ϵ0 is vacuum permittivity, ϵBN = 4:2 is the out-of-planedielectric constant for hBN (ref. 41), and tt, tm, tb are the top, middle,andbottomhBN thicknesses, respectively. The chemical potentials canthen be experimentally determined as a function of ne and nh, asshown in Fig. 3a, b.Notably, the electron chemical potential (Fig. 3a) has a dis-continuity across the diagonal line (ne =nh), which is more prominentat low densities, and disappears above a critical density. The holechemical potential also shows a similar jump but with opposite sign(Fig. 3b). In Fig. 3c,we showa typical chemical potential linecut at fixedhole doping nh = 0:15 × 1012 cm�2. By fitting it to a linear backgroundplus a step function, we estimate the chemical potential jump (Δμe,Δμh) at this density to be +13meV and −13 meV for the electrons andholes, respectively. This chemical potential discontinuity effectivelyopens a gap at ne =nh for single charge injection, leading to a charge-incompressible phase at finite density. Nevertheless, the exciton che-mical potential μx =μe +μh is continuous (Fig. 3d), and thereforeexcitons are compressible.The chemical potential jumpacrossnet chargeneutrality hasbeentheoretically predicted as an indication of a finite energy gap in thesuperfluid phase7. It is a definitive signature of interlayer excitons inthe dipolar excitonic insulator. Close to the excitonic insulator phase,the system consists of nx = min ne,nh� �interlayer excitons andne � nh�� �� unpaired charges. When ne<nh, adding onemore electron tothe system creates one interlayer exciton and removes one free hole,gaining an exciton binding energy ϵb. When ne>nh, however, adding-0.5 0 0.5ne–nh (1012cm-2)-50-45-40-35-30x(meV)=–39meV/(1012cm-2)0 0.4 0.8 1.2nh (1012cm-2)00.40.81.2n e(1012cm-2)-60-40-200e(meV) ca b dh0 0.4 0.8 1.2nh (1012cm-2)00.40.81.2n e(1012cm-2)-60-40-200h(meV)0 0.4 0.8 1.2nh (1012cm-2)00.40.81.2n e(1012cm-2)-80-60-40-20x(meV)0 0.2 0.4 0.6ne (1012cm-2)-40-20020e,h(meV)=–66meV/(1012cm-2)– =+31meV/(1012cm-2)eh-40-30-20e(meV)+13meV0 0.1 0.2 0.3ne (1012cm-2)-20-100h(meV)-13meVne=nh0 0.5 1nx (1012cm-2)-20-1001020e,h(meV)ehe f g0 0.2 0.4 0.6 0.8nx (1012cm-2)-80-60-40-20x(meV)g=–70meV/(1012cm-2)Typical errorFig. 3 | Chemical potential of the correlated e-h fluids. a, b Measured chemicalpotential maps for electrons (a) and holes (b). c Line cut of the electron and holechemical potentials at fixed hole densitynh =0:15 × 1012 cm�2. The experimentaldata (empty circles) is fitted to a linear background plus a sigmoid function (solidlines). The error bar represents the estimated typical standard deviation (Methods).dMeasured exciton chemical potentialmap μx =μe +μh. eChemical potential jumpat net chargeneutrality for different excitondensities. f Electron chemical potentialμe ne,nh =0� �and hole chemical potential μh ne,nh =0� �as a function of electrondensity, keeping the hole density zero.g Exciton chemical potential as a function ofexciton density, keeping electron and hole densities equal. h Exciton chemicalpotential as a function of unpaired charge density ne � nh, keeping exciton densitynx =0:05× 1012 cm�2 constant. Dotted lines in f–h are linear expansions of theexperimental data in the low doping region (0:05× 1012 cm�2 to 0:5 × 1012 cm�2).Source data are provided as a source data file.Article https://doi.org/10.1038/s41467-023-43799-7Nature Communications |         (2023) 14:8264 4one electron does not change the number of excitons. Thus, in the low-density limit the chemical potential jump at ne =nh gives the interlayerexciton binding energy.At higher densities, quantumdissociation of the interlayer excitonis expected after the Mott limit is reached42. Figure 3e shows thedensity dependence of the measured chemical potential jump. Thebinding energy in the dilute limit, ϵb ≈ 20meV, is consistent with pre-vious theoretical29 and experimental studies15. The chemical potentialjump decreases with density and vanishes above the Mott density~0.8 × 1012 cm−2, above which the excitons dissociate, and the systemturns into a degenerate electron-hole Fermi gas29.We next examine the general thermodynamic behavior of thecorrelated e-h fluid across the phase diagram. To provide an approx-imate framework to understand the behavior, we can parameterize thetotal energy per unit area of the e-h fluid asE ne,nh� �= � ϵbnx +α2ne � nh� �2 + g2n2x +βnx ne � nh�� ��: ð3ÞHere α, g, and β characterize the charge-charge interactionstrength, exciton-exciton interaction strength, and exciton-chargeinteraction strength, respectively. The electron, hole, and excitonchemical potentials are given by partial derivatives of Ewith respect tone, nh and nx respectively, which yieldμe =∂E∂ne=αðne � nhÞ+βnh, ðne >nhÞαðne � nhÞ+ βnh � ϵb + ðg � 2βÞne, ðne <nhÞ�, ð4Þμh =∂E∂nh=αðnh � neÞ+βne, ðnh >neÞαðnh � neÞ+βne � ϵb + ðg � 2βÞnh, ðnh <neÞ�, ð5Þμx =μe +μh = � ϵb + gnx +β ne � nh�� ��: ð6ÞOur approximate energy expression gives an empirical linearexpansion of the chemical potentials in the low doping regime. InFig. 3f, we show a linecut of μe and μh at zero hole density (corre-sponding to a linecut along the vertical axis in Fig. 3a, b). The electronchemical potential (red circles in Fig. 3f) decreases strongly with ne.This behavior is controlled by the charge-charge interaction α. Fittingthe experimental linecut to a polynomial and comparing it to theansatz electron chemical potential μeðne,nh =0Þ=αne gives an initialslope of α = �66±6 meV=ð1012cm�2Þ: The negative α is a manifesta-tion of the negative compressibility in low-density electron gases andarises from strong exchange interactions38.In contrast, the hole chemical potential μh increases significantlywith the electron density at zero hole density (green circles in Fig. 3f).This unusual behavior arises from a competition between the charge-charge interaction and the charge-exciton interaction, and it highlightsthe strong interlayer correlation effects. With interlayer interactions, theaddition of the hole can be viewed as simultaneously adding an excitonand removing an electron. It results in the hole chemical potentialdescribed by μh ne,nh =0� �= � ϵb + β� αð Þne: The positive slope inFig. 3f is determined by β� α, with β= �35 ± 6 meV=ð1012cm�2Þ.Next, we examine the evolution of the exciton chemical potentialμx. Figure 3g shows that the exciton chemical potential decreases withthe exciton density at ne =nh (corresponding to a diagonal linecut ofFig. 3d). Under this condition, the exciton chemical potentialμx = � ϵb + gnx. Figure 3g shows that the effective interaction betweenexcitons is strongly attractive with g = �70± 2 meV=ð1012cm�2Þ: Anegative g is consistent with the physical picture II described above,where the attractive intralayer exchange interaction, in reference to ahomogeneous charge distribution, dominates over the repulsive inter-layer exchange interaction6,13. At higher doping the chemical potentialdeviates from a linear line, suggesting a positive second order interac-tion 19 ± 2 meV=ð1012cm�2Þ2 as predicted in previous theoreticalstudies6.Figure 3h further displays the evolution of the exciton chemicalpotential with the net charge density ne � nh at fixed exciton densitynx =0:05× 1012 cm�2. μx has a maximum value at the charge neutralstate, and it decreases with either additional electron or hole doping,indicating an attractive interaction between the exciton and charge inthe exciton-charge complex. Using the expressionμx = � ϵb + gnx +β ne � nh�� ��, we can deduce an attractive exciton-charge interaction strength β= �39 ±2 meV=ð1012cm�2Þ. This valueagrees with that value obtained from Fig. 3f.Although we have focused on only a few special line cuts inFig. 3f–h, the parameterized chemical potential (Eqs. 4–6) provides agood description of the complete 2D phase diagram of Fig. 3a, b at lowexciton and charge density regime. Furthermore, our experimentalchemical potential maps can be reproduced semi-quantitatively by ourmean-field theory calculations (Methods and Supplementary Fig. 6).The mean-field calculations clearly predict the chemical potential dis-continuity at ne =nh. They also reproduce the attractive charge-charge,charge-exciton, and exciton-exciton interactions. However, the mean-field theory overestimates the exciton binding energy and cannotcapture the Mott transition due to the lack of screening in thecalculation.Lastly, we investigate the temperature dependence of the chemicalpotential in the e-h fluid. Figure 4a shows the chemical potential jump atneutrality (Δμ= ðΔμe � ΔμhÞ=2) as a function of density and tempera-ture. With increasing temperature, the magnitude of the discontinuitydecreases, suggesting ionization of the interlayer excitons. At the lowestdensity, we estimate the interlayer excitons ionize around 70K. Simi-larly, theMott density decreaseswith increasing temperature, which canbe qualitatively understood as temperature aiding the exciton ioniza-tion. Figure 4b shows the exciton chemical potential change as a func-tion of the exciton density along the ne =nh condition (similar to Fig. 3g)for various temperatures. We find that the exciton-exciton interactionstrength remains constant until ~40K and then becomes weaker withincreasing temperature (Fig. 4b inset). In Fig. 4c, we show linecuts of μxat constant nx (similar to Fig. 3h) for various temperatures. The cusp atcharge neutrality becomes broader with increasing temperature, indi-cating reduced exciton-charge interaction strength. The fitted β alsoremains constant up to ~40K and then weakens (Fig. 4c inset). Thechange in g and β occurs before the disappearance of the chemicalpotential jump, implying a change in the interaction behavior of theexciton-charge complex before the complete thermal ionization of theinterlayer excitons.In summary, we have reported quantitative thermodynamicmeasurements of a strongly coupled electron-hole bilayer system. Thechemical potential discontinuity provides unambiguous evidence ofthe dipolar excitonic insulator state. Attractive exciton-exciton andexciton-charge interactions are observed for the first time. Our workopensmanyexcitingdirections for future studies of interlayer excitonsand the search for the exciton condensate at higher temperatureswithout magnetic fields. The measurements of the chemical potentialand interaction strengths allow direct connection to several theore-tical predictions6,12,13,29,43 and serve as a guide for future studies on theinteractions in the electron-hole fluids and possible exciton con-densates. The technique demonstrated here achieves optical readoutof carrier density and chemical potential and can be easily generalizedto study other correlated 2D electron systems.MethodsDevice fabricationWe use a dry-transfer method based on polyethylene terephthalateglycol (PETG) stamps to fabricate the heterostructures. MonolayerArticle https://doi.org/10.1038/s41467-023-43799-7Nature Communications |         (2023) 14:8264 5MoSe2, monolayer WSe2, few-layer graphene and hBN flakes aremechanically exfoliated frombulk crystals ontoSi substrateswith a 90-nm-thick SiO2 layer. We use 15-25 nm hBN as the gate dielectric, 20-30 nmhBN for the interlayer spacer in region0, and 2-3 nm thinhBNasthe interlayer spacer in region 1. A 0.5mm thick clear PETG stamp isemployed to sequentially pick up the flakes at 65-75 °C. The wholestack is then released onto a 90nm SiO2/Si substrate at 95-100 °C,followed by dissolving the PETG in chloroform at room temperaturefor at least one day. Electrodes (50 nmAuwith 5 nmCr adhesion layer)are defined using photolithography (Durham Magneto Optics, Micro-Writer) and electron beam evaporation (Angstrom Engineering).Supplementary Fig. 1 shows detailed device structures and hBNthickness values for three devices D1 - D3. In addition to the structuredescribed above, D1 has another region (region 2) that is controlled byanother back gate with a thicker dielectric hBN. Data taken in region 2of D1, D2 and D3 are consistent with the main text (SupplementaryFigs. 3 and 4).Electrical contacts and measurementsKeithley 2400 or 2450 source meters are used for applying gate andbias voltages andmonitoring the leakage current. In allmeasurements,the leakage current is kept below 5 pA to make sure the system is inequilibrium with the electrodes. The interlayer leakage current gives alower bound of the exciton recombination lifetime τ. We assume allthe interlayer leakage is fromelectron-hole recombination in the activearea, and the lifetime can be estimated as τ = enxA=I, whereA ≈ 136 μm2 is the region 1 heterostructure area and I is the leakagecurrent. As shown in Supplementary Fig. 2, the average lifetime~320ms is orders ofmagnitude longer than charge injection time fromthe contact, which is on the millisecond time scale.Tomakegood contacts to theTMD layers and achieve equilibriumstates, a thick hBN layer (20-30 nm) is inserted between theTMD layersin the contact region (region 0). The vertical electric field VΔ creates amuch larger voltage difference in region 0 than in region 1, so the bandgap in region 0 will be easily closed. Thus, for the voltage range we areinterested in, region 0 sustains high doping in both layers and formsbetter electrical contact to the graphite electrodes. SupplementaryFig. 2c, d shows the doping phase diagram for region 0 and region 1 atconstant bias volageVB = 1 V. The scan is performed in a “snake”order,i.e., columns have alternating scandirections.We observe hysteresis inregion 0 doping, suggesting that the contact is not effective and has along response time at low doping levels. In region 1, when region 0 isnot heavily doped, we observe hysteresis; but when region 0 hasstrong electron and hole doping (dashed box), the contact resistancereduces significantly, resulting in a clean phase diagram with no hys-teresis. The data shown in the main text is at VΔ = 11 V, where region 0is at a very high doping level. It is also taken as a “snake” scan, and wedo not observe any hysteresis down tomV level, indicating our systemis in equilibrium with the contact.Calibration of hBN thicknessThe dielectric hBN (~20 nm) thickness is determined by optical con-trast calibrated by atomic force microscopy (AFM). We perform AFMthickness measurement and optical microscope imaging for tens ofhBN flakes ranging from 1 to 70 nm. The AFM is operated in contactmode to get accurate thickness values. The opticalmicroscope imagesare taken with a colored camera with fixed color temperature and ISOsettings. The RGB optical contrast of the hBN flake, defined as (hBN –substrate)/substrate for the red, green, andblue channels of the image,is shown in Supplementary Fig. 7a. We fit the RGB contrast to a 4thorder polynomial. Themeasured data points and the fitted curve agreenicely with theoretically calculated reflectivity using Fresnel equations(Supplementary Fig. 7b). Then the thickness of a givenhBNflake canbedetermined bymatching its optical contrast with the fitted curve. For ~20 nm thick hBN flakes, the accuracy is typically within 1 nm. For thethin hBN spacers, the relative error of this method will be worse, so weaccurately determine their thickness from reflectionmeasurements ofthe fabricated heterostructure. We keep the system net charge neutralwhile scanning the VB and VΔ. The exciton gap is closed whenEg � eVB � etmtt + tm + tbVΔ =0, ð7Þwhere Eg is the band gap energy. The slope of the gap-closingboundary gives the ratio of interlayer distance to total hBN thickness.Supplementary Fig. 7c shows a line fit of the gap-closing boundary forD1, which gives tm =3:0±0:2 nm (interpreted as the effectiveinterlayer distance, which is slightly larger than the thin hBN thicknessdue to finite TMD layer thickness), consistent with the valuedetermined from optical contrast (2.7 ± 0.6 nm).Optical measurementsThe optical measurements are performed in an optical cryostat(Quantum Design, OptiCool) with a temperature down to 2 K (nom-inal). We use a supercontinuum laser (Fianium Femtopower 1060Supercontinuum Laser) as the light source for the reflection spectro-scopy. The laser is focused on the sample by a 20×Mitutoyo objectivewith ~1.5 μm beam size. A small beam size provides a local probe thatwe can park in a clean region free of bubbles. We choose a very lowincident laser power (on the order of 10 nW) tominimize photodopingeffects. The spectra are independent of the incident light power up to200 nW. The reflected light is collected by a spectrometer (PrincetonInstruments PIXIS 256e) with 1000ms exposure time. Tominimize theinfluence of laser fluctuations, another laser beam directly reflected0 0.5 1nx (1012cm-2)2.510254570100135T(K)-20-1001020(meV)a b c0 0.2 0.4 0.6 0.8 1nx (1012cm-2)-60-50-40-30-20-100x−x(nx=0)2.510254570100135T (K)101 102T (K)-60-40-20g(me V/(1012cm-2))-0.5 0 0.5ne–nh (1012cm-2)-30-25-20-15-10-505x−x(ne−nh=0)101 102T (K)-40-20(meV/(1012cm-2))Fig. 4 | Temperature dependence. a Chemical potential jump at net charge neu-trality as a function of exciton density and temperature. b Exciton density depen-dence of the exciton chemical potential at various temperatures. The electron andhole densities are kept equal. Inset: Fitted exciton-exciton interaction strength g asa function of temperature. c Exciton chemical potential as a function of unpairedcharge density ne � nh at constant exciton density nx =0:05× 1012 cm�2. Inset:Fitted exciton-charge interaction strength β as a function of temperature. Sourcedata are provided as a source data file.Article https://doi.org/10.1038/s41467-023-43799-7Nature Communications |         (2023) 14:8264 6from a silver mirror is simultaneously collected to normalize thesample reflection spectra.Calibration of carrier densityTo build a map between the reflection spectrum and carrier density,we perform gate voltage scans at low bias voltage and low verticalelectric field. The gap is not closed, so at most one layer is activelydoped and the other layer remains intrinsic. In this regime, the gatedependence of the doping density is well understood and can bedescribed by a parallel capacitor model. Taking the hole layer as anexample, Supplementary Fig. 5a shows such a scan.When the chemicalpotential lies in the bandgap (large positive VG), the hole densityremains zero, so the spectrumdoes not changewithVG. Upon a criticalvoltage, the chemical potential is at the valence band edge, and thespectrum starts to change.We first determine this critical voltage fromgate dependence of k R λð Þ � Rintrinsic λð Þ k, where λ denotes wavelengthand k . . . kmeans vector norm, as shown inSupplementaryFig. 5b. Thisquantity is a measure of the spectrum change relative to the intrinsicspectrum.Wefit the initial increase to a polynomial. The crossing pointof this polynomial and the constant baseline gives the critical voltage.When the gate voltage is further decreased, holes start to fill thevalence band. The hole density is then solved fromμhðnhÞ+ eϕhðnhÞ= eVh � Ev, as shown in Supplementary Fig. 5b.Hartree-Fock theory gives the relation between the chemical potentialand density38:dμhdnh=π_2m*h� e24πϵ0ϵhffiffiffiffiffiffiffiffiffi2πnhs, ð8Þwhere _ is the reduced Planck constant, and m*h is the hole effectivemass. Thefirst term is the non-interacting kinetic energy, and the secondterm describes the Coulomb interactions. ϵh = 5:7 is the effectivedielectric constant for the hole layer, which is given by the followingprocedure: We build a layered dielectric environment according to ourdevice geometry, where the anisotropic dielectric constant of hBN,WSe2 and MoSe2 are all considered44. The screening from the graphitegates is included by applying Dirichlet boundary condition for theelectrostatic potential. A point charge is placed inside the WSe2 layer,and we numerically solve the Poisson’s equation to get the electricpotential as a function of the in-plane distance r to the point charge. Thispotential is fitted to a 1=r decay to get the effective dielectric constant.In fact, the quantum capacitance is much larger than the geo-metrical capacitance in our device geometry. Even if we ignore thisintralayer correlation effect, the carrier density calibration only chan-ges by ~10%. Supplementary Fig. 5c displays the WSe2 reflectionspectrum as a function of hole density. The electron density in theMoSe2 layer is determined in the same way (Supplementary Fig. 5d).With such density calibration maps, the spectrum-matching gives theelectron and hole densities in the e-h regime, which is then used todetermine the chemical potentials. We check the self-consistency bycomparing the determined μhðnh,ne =0Þ to Eq. (8) and estimate thetypical error (standard deviation) of the experimentally determinedchemical potential to be 4meV.Mean-field theoryWeself-consistently solve themean-fieldHamiltonianHMF =H0 +HX inwhich H0 =Pkð_2k22m* +Eg2 Þðayckack � ayvkavkÞ, the single-particle con-tribution, contains parabolically dispersing conduction and valencebands with equal effective masses m* = 0:5me, and the exchange self-energy HX = � 1APl0 ,l,k0 ,kVl0l k0 � k� �δρl0l k0� �ayl0kalk where ay,a are thecreation and annihilation operators, and l0,l = c,v are the band (layer)indices. The density-matrix δρl0 l k0� �is the difference between thedensity matrix of the negative energy band and the density matrix of afull valence and empty conduction band. The gap Eg is a tuningparameter that controls the exciton density. The interaction Vl0l qð Þtakes the dual-gate-screened Coulomb form that distinguishes intra-layer and interlayer interactions:Vl0lðqÞ=2πe2ϵqðeqd�e�qDÞðe�qD�e�qDÞ1�e�2qD , ðl0 = lÞ,2πe2ϵqeqd ðe�qd�e�qDÞ21�e�2qD , ðl0 ≠ lÞ,8<: ð9Þwhere d is the distance between the electron and hole layers and D isthedistancebetween the top andback gates. Thedielectric constant ofhBN has in-plane component ϵ? = 7 and out-of-plane componentϵzz =4:2. The quantities ϵ,d, and D that appear in the above equationare replaced by their effective values ϵ=ffiffiffiffiffiffiffiffiffiffiffiffiϵ?ϵzzp, d =d0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiϵ?=ϵzzp, andD=D0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiϵ?=ϵzzpwhere d0 and D0 are the physical distances. In ourcalculations, we take d0 = 3 nm and D0 = 40 nm. The density matrix ρis defined relative to the state inwhich the valenceband isfilled and theconduction band empty: ρl0l kð Þ= haylkal0ki � δl0 lδlv.The electron and hole chemical potentials are defined relative tothe conduction band bottom and valence band top respectively:μe =μ� Eg2 ,μh = � μ� Eg2 where μ is the chemical potential in themean-field calculations. The electrostatic (Hartree) potential is not includedin the definition of chemical potentials because it has already beenaccounted for in the gate-geometry-dependent electric potential cal-culation. This separation of the chemical potential into a gate-geometry-dependent part and a part that depends only on the inter-acting electrons and holes is necessitated by the long-range of theCoulomb interaction.Data availabilityThe main data that support the findings of this study are availablewithin the article and its Supplementary Information files. More sup-porting data are available from the corresponding authors uponrequest. Source data are provided with this paper.References1. Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P. Experimental obser-vation of the quantum Hall effect and Berry’s phase in graphene.Nature 438, 201–204 (2005).2. Klitzing, K. V., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based onquantized hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).3. Andrei, E. Y. et al. Observation of a magnetically induced wignersolid. Phys. Rev. Lett. 60, 2765–2768 (1988).4. Smoleński, T. et al. Signatures of Wigner crystal of electrons in amonolayer semiconductor. Nature 595, 53–57 (2021).5. Zhou, Y. et al. Bilayer Wigner crystals in a transition metal dichal-cogenide heterostructure. Nature 595, 48–52 (2021).6. Zeng, Y. &MacDonald, A. H. Electrically controlled two-dimensionalelectron-hole fluids. Phys. Rev. B 102, 085154 (2020).7. Pieri, P., Neilson, D. & Strinati, G. C. Effects of density imbalance onthe BCS-BEC crossover in semiconductor electron-hole bilayers.Phys. Rev. B 75, 113301 (2007).8. Jauregui, L. A. et al. Electrical control of interlayer exciton dynamicsin atomically thin heterostructures. Science 366, 870–875 (2019).9. Paik, E. Y. et al. Interlayer exciton laser of extended spatial coher-ence in atomically thin heterostructures. Nature 576, 80–84 (2019).10. Rivera, P. et al. Observation of long-lived interlayer excitons inmonolayer MoSe2–WSe2 heterostructures. Nat. Commun. 6,6242 (2015).11. Conti, S. et al. Chester supersolid of spatially indirect excitons indouble-layer semiconductor heterostructures. Phys. Rev. Lett. 130,057001 (2023).12. Lozovik, Y. E., Kurbakov, I. L., Astrakharchik, G. E., Boronat, J. &Willander, M. Strong correlation effects in 2D Bose–Einstein con-densed dipolar excitons. Solid State Commun. 144,399–404 (2007).Article https://doi.org/10.1038/s41467-023-43799-7Nature Communications |         (2023) 14:8264 713. Wu, F.-C., Xue, F. & MacDonald, A. H. Theory of two-dimensionalspatially indirect equilibrium exciton condensates. Phys. Rev. B 92,165121 (2015).14. Perali, A., Neilson, D. & Hamilton, A. R. High-temperature super-fluidity in double-bilayer graphene. Phys. Rev. Lett. 110,146803 (2013).15. Ma, L. et al. Strongly correlated excitonic insulator in atomic doublelayers. Nature 598, 585–589 (2021).16. Zhang, Z. et al. Correlated interlayer exciton insulator in hetero-structures of monolayer WSe2 and moiré WS2/WSe2. Nat. Phys. 18,1214–1220 (2022).17. Astrakharchik, G. E., Boronat, J., Kurbakov, I. L. & Lozovik, Yu. E.Quantum phase transition in a two-dimensional system of dipoles.Phys. Rev. Lett. 98, 060405 (2007).18. Kogar, A. et al. Signatures of exciton condensation in a transitionmetal dichalcogenide. Science 358, 1314–1317 (2017).19. Lu, Y. F. et al. Zero-gap semiconductor to excitonic insulator tran-sition in Ta2NiSe5. Nat. Commun. 8, 14408 (2017).20. Butov, L. V., Zrenner, A., Abstreiter, G., Böhm, G. & Weimann, G.Condensation of indirect excitons in coupled AlAs/GaAs quantumwells. Phys. Rev. Lett. 73, 304–307 (1994).21. Nandi, D., Finck, A. D. K., Eisenstein, J. P., Pfeiffer, L. N. &West, K. W.Exciton condensation and perfect Coulomb drag. Nature 488,481–484 (2012).22. Du, L. et al. Evidence for a topological excitonic insulator in InAs/GaSb bilayers. Nat. Commun. 8, 1971 (2017).23. Liu, X.,Watanabe, K., Taniguchi, T., Halperin, B. I. & Kim, P. QuantumHall drag of exciton condensate in graphene. Nat. Phys. 13,746–750 (2017).24. Li, J. I. A., Taniguchi, T., Watanabe, K., Hone, J. & Dean, C. R. Exci-tonic superfluid phase in double bilayer graphene. Nat. Phys. 13,751–755 (2017).25. Wang, G. et al. Excitons in atomically thin transition metal dichal-cogenides. Rev. Mod. Phys. 90, 021001 (2018).26. Regan, E. C. et al. Emerging exciton physics in transition metaldichalcogenide heterobilayers. Nat. Rev. Mater. 7, 778–795(2022).27. Chernikov, A. et al. Exciton binding energy and nonhydrogenicRydberg series in monolayer WS2. Phys. Rev. Lett. 113,076802 (2014).28. Calman, E. V. et al. Indirect excitons in van der Waals hetero-structures at room temperature. Nat. Commun. 9, 1895 (2018).29. Fogler, M. M., Butov, L. V. & Novoselov, K. S. High-temperaturesuperfluidity with indirect excitons in van der Waals hetero-structures. Nat. Commun. 5, 4555 (2014).30. Debnath, B., Barlas, Y.,Wickramaratne, D., Neupane,M. R. & Lake, R.K. Exciton condensate in bilayer transition metal dichalcogenides:Strong coupling regime. Phys. Rev. B 96, 174504 (2017).31. Mak, K. F. et al. Tightly bound trions in monolayer MoS2. Nat. Mater.12, 207–211 (2012).32. Scuri, G. et al. Large excitonic reflectivity of monolayer MoSe2encapsulated in hexagonal boron nitride. Phys. Rev. Lett. 120,037402 (2018).33. Ross, J. S. et al. Electrical control of neutral and charged excitons ina monolayer semiconductor. Nat. Commun. 4, 1474 (2013).34. Christopher, J.W., Goldberg, B. B. & Swan, A. K. Long tailed trions inmonolayerMoS2: temperature dependent asymmetry and resultingred-shift of trion photoluminescence spectra. Sci. Rep. 7,14062 (2017).35. Zhang, C., Wang, H., Chan, W., Manolatou, C. & Rana, F. Absorptionof light by excitons and trions in monolayers of metal dichalco-genide MoS2: Experiments and theory. Phys. Rev. B 89,205436 (2014).36. Liu, E. et al. Exciton-polaron Rydberg states in monolayer MoSe2and WSe2. Nat. Commun. 12, 6131 (2021).37. Goryca,M. et al. Revealing excitonmasses anddielectric propertiesof monolayer semiconductors with high magnetic fields. Nat.Commun. 10, 4172 (2019).38. Nagano, S., Singwi, K. S. & Ohnishi, S. Correlations in a two-dimensional quantum electron gas: The ladder approximation.Phys. Rev. B 29, 1209–1213 (1984).39. Li, L. et al. Very large capacitance enhancement in a two-dimensional electron system. Science 332, 825–828 (2011).40. Kravchenko, S. V., Rinberg, D. A., Semenchinsky, S. G. & Pudalov, V.M. Evidence for the influence of electron-electron interaction onthe chemical potential of the two-dimensional electron gas. Phys.Rev. B 42, 3741–3744 (1990).41. Regan, E. C. et al. Mott and generalized Wigner crystal states inWSe2/WS2 moiré superlattices. Nature 579, 359–363 (2020).42. Kappei, L., Szczytko, J., Morier-Genoud, F. & Deveaud, B. Directobservation of the mott transition in an optically excited semi-conductor quantum well. Phys. Rev. Lett. 94, 147403 (2005).43. López Ríos, P., Perali, A., Needs, R. J. & Neilson, D. Evidence fromquantum monte carlo simulations of large-gap superfluidity andBCS-BEC crossover in double electron-hole layers. Phys. Rev. Lett.120, 177701 (2018).44. Laturia, A., Van de Put, M. L. & Vandenberghe, W. G. Dielectricproperties of hexagonal boron nitride and transition metal dichal-cogenides: from monolayer to bulk. Npj 2D Mater. Appl. 2,1–7 (2018).AcknowledgementsThis work was supported primarily by the U.S. Department of Energy,Office of Science, Office of Basic Energy Sciences, Materials Sciencesand Engineering Division under contract no. DE-AC02-05-CH11231 (vanderWaals heterostructures programme, KCWF16). The data analysiswasalso supported by the AFOSR award FA9550-23-1-0246. S.T. acknowl-edges support from DOE-SC0020653, NSF CMMI 1933214, NSF mid-scale 1935994, NSF 1904716, NSF DMR 1552220, and DMR 1955889.K.W. and T.T. acknowledge support from JSPS KAKENHI (Grant Numbers19H05790, 20H00354).Author contributionsF.W. conceived the research. R.Q. and A.Y.J. fabricated the devices.A.Y.J. andR.Q. performed the opticalmeasurementswith help fromZ.Z.,E.R., J.X, and Z.L. R.Q., A.Y.J., and F.W. analyzed the data. Y.Z. andA.H.M.did the mean-field calculations. T.Z., Q.F., and M.F.C. contributed to thedevice fabrication. S.T. grew WSe2 and MoSe2 crystals. K.W. and T.T.grew hBN crystals. All authors discussed the results and wrote themanuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-43799-7.Correspondence and requests for materials should be addressed toAndrew Y. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2023Article https://doi.org/10.1038/s41467-023-43799-7Nature Communications |         (2023) 14:8264 9http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Thermodynamic behavior of correlated electron-hole fluids in van der Waals heterostructures Results Methods Device fabrication Electrical contacts and measurements Calibration of hBN thickness Optical measurements Calibration of carrier density Mean-field�theory Data availability References Acknowledgements Author contributions Competing interests Additional information