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Yanhao Tang, Jie Gu, Song Liu, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), James C. Hone, Kin Fai Mak, Jie Shan

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[Dielectric catastrophe at the Wigner-Mott transition in a moiré superlattice](https://mdr.nims.go.jp/datasets/fb05d0fd-f225-4fac-a936-6b6b204582e5)

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Dielectric catastrophe at the Wigner-Mott transition in a moiré superlatticenature communicationsArticle https://doi.org/10.1038/s41467-022-32037-1Dielectric catastrophe at the Wigner-Motttransition in a moiré superlatticeYanhao Tang1,2,7 , Jie Gu 1,7, Song Liu3, Kenji Watanabe 4,Takashi Taniguchi 4, James C. Hone 3, Kin Fai Mak1,5,6 & Jie Shan 1,5,6The bandwidth-tuned Wigner-Mott transition is an interaction-driven phasetransition from a generalized Wigner crystal to a Fermi liquid. Because thetransition is generally accompanied by both magnetic and charge-orderinstabilities, it remains unclear if a continuous Wigner-Mott transition exists.Here, we demonstrate bandwidth-tuned metal-insulator transitions at fixedfractional fillings of a MoSe2/WS2 moiré superlattice. The bandwidth is con-trolled by an out-of-plane electric field. The dielectric response is probedoptically with the 2s exciton in a remote WSe2 sensor layer. The excitonspectral weight is negligible for the metallic state with a large negativedielectric constant. It continuously vanisheswhen the transition is approachedfrom the insulating side, corresponding to a diverging dielectric constant or a‘dielectric catastrophe’ driven by the critical charge dynamics near the tran-sition. Our results support the scenario of continuousWigner-Mott transitionsin two-dimensional triangular lattices and stimulate future explorations ofexotic quantum phases in their vicinities.Metal-insulator transitions (MITs) accompanied by large electricalconductivity change are widely observed in condensed-mattersystems1–5. One particularly intriguing origin of charge localizationand insulating phases is Coulomb repulsion between electrons6. Theprototypemodel for interacting electrons in a lattice is the single-bandHubbard model with electronic bandwidth, W , and on-site Coulombrepulsion,U. The ground state of the electronic system for half bandorfull lattice filling is a Mott insulator in the strong interaction limit(U ≫W ), and a Fermi liquid in the weak interaction limit (U ≪W ). AMIT, the Mott transition, is expected near U ~W 6. Similarly, if theelectrons are localized by the extended Coulomb repulsion, V , insteadof U, a Wigner-Mott insulator (or a generalized Wigner crystal) thatspontaneously breaks the underlying lattice space-group symmetriesis formed at fractional lattice fillings7–10; a Wigner–Mott transition isexpected near V ~W 11–15. The evolution of a Mott or Wigner–Mottinsulator into a metal as a function of the interaction strength remainsa challenging theoretical problem4,11–15; it also raises an excitingopportunity for realizing exotic quantum phases near the transition ifit is continuous11,12,14–17.Experimentally, the interaction strength, or equivalently, thebandwidth can be tuned by applying isostatic or chemical pressure1. Inalmost all knownmaterials, bandwidth-tunedMITs (the caseof interestfrom now on) are driven first order1 because the MITs are oftenaccompanied by magnetic and structural phase transitions; it is diffi-cult for the various kinds of transitions involving different orderparameters to occur simultaneously without fine-tuning1,14,16. Therecent experimental breakthroughs in semiconducting transitionmetal dichalcogenide (TMD)moirématerials7,8,18–20 have opened a newavenue to realize and study continuous bandwidth-tuned MITs15,21–26.These materials form a two-dimensional (2D) triangular lattice thatsuppresses magnetic ordering due to geometric frustration. Theelectrons are trapped by the periodic moiré potential; they can tunnelReceived: 5 April 2022Accepted: 12 July 2022Check for updates1School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA. 2Interdisciplinary Center for Quantum Information, Zhejiang Province KeyLaboratory of Quantum Technology, and Department of Physics, Zhejiang University, Hangzhou 310027, China. 3Department of Mechanical Engineering,ColumbiaUniversity,NewYork,NY, USA. 4National Institute forMaterials Science, 1-1Namiki, 305-0044Tsukuba, Japan. 5Laboratory of AtomicandSolidStatePhysics, Cornell University, Ithaca, NY, USA. 6Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, USA. 7These authors contributed equally: YanhaoTang, Jie Gu. e-mail: yanhaotc@zju.edu.cn; kinfai.mak@cornell.edu; jie.shan@cornell.eduNature Communications |         (2022) 13:4271 11234567890():,;1234567890():,;http://orcid.org/0000-0003-1510-2379http://orcid.org/0000-0003-1510-2379http://orcid.org/0000-0003-1510-2379http://orcid.org/0000-0003-1510-2379http://orcid.org/0000-0003-1510-2379http://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/0000-0002-1467-3105http://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/0000-0002-8084-3301http://orcid.org/0000-0002-8084-3301http://orcid.org/0000-0002-8084-3301http://orcid.org/0000-0002-8084-3301http://orcid.org/0000-0002-8084-3301http://orcid.org/0000-0003-1270-9386http://orcid.org/0000-0003-1270-9386http://orcid.org/0000-0003-1270-9386http://orcid.org/0000-0003-1270-9386http://orcid.org/0000-0003-1270-9386http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-32037-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-32037-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-32037-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-32037-1&domain=pdfmailto:yanhaotc@zju.edu.cnmailto:kinfai.mak@cornell.edumailto:jie.shan@cornell.edubetween the moiré sites and interact with each other via both the on-site and extended Coulomb repulsions. This realizes the triangular-lattice extended Hubbard model13,21,27,28. A variety of correlated insu-lating states are reported, including the Mott insulator7,18–20 at oddinteger filling and Wigner-Mott insulators7–10,29 at fractional fillings ofthemoiré superlattice. A continuous bandwidth-tunedMott transitionis also observed by electrical measurements22,23. However, the Wigner-Mott transition at fractional fillings remains elusive. In particular, itremains unclear if a continuousWigner-Mott transition exists given theadditional charge-ordering transition (besides possible magnetictransitions) near the MIT11,14.Here we report the observation of continuous bandwidth-tunedWigner-Mott transitions in an angle-aligned MoSe2/WS2 heterobilayerby the exciton sensing technique8,30. The heterobilayer forms a trian-gular moiré lattice with a lattice density of ≈1:9× 1012 cm−2. It isencapsulated in hexagonal boron nitride (hBN) and gated by a top andbottom few-layer graphite gate (Fig. 1a). The dual-gate structureenables independent tuning of the electron density in themoiré latticeν (in units of the moiré density), and the out-of-plane electric field, E(>0 for field pointing from the MoSe2 to WS2 layer). The MoSe2/WS2heterobilayer has a type-I band alignment with both the conductionand valence band edges located in the MoSe2 layer (Fig. 1b). The bandoffsets areΔc ~ 100meV for the conductionbands andΔv ~ 320meV forthe valencebands fromoptical spectroscopymeasurement (Methods).We tune the electronic bandwidth at a fixed doping density by theelectric-field effect22,23. The out-of-plane electric field varies the moirépotential depth by controlling the band offset and, correspondingly,the resonance interlayer hopping amplitude21,22. Because Δc <Δv, theelectric-field effect on the conduction bandwidth is much larger; wefocus on the case of electron doping.The dielectric response of themoiré system, which can reflect thecritical charge dynamics near continuous MITs31–33, is probed by theexciton sensing technique. Recent studies show that the technique ishighly sensitive to the insulating states8,30. These states perturb theelectric field between the optically excited electrons and holes (exci-tons) in a charge-neutral WSe2 monolayer that is separated from themoiré superlattice by a bilayer hBN. The spacer thickness is smallerthan the Bohr radius of the 2s and higher-energy exciton states. Weprobe the effective dielectric constant, ε, of themoiré heterobilayer bymeasuring the 2s exciton that has the largest spectral weight. Both the2s exciton resonance energy and spectral weight, S, depend on ε 34. Weanalyze the spectral weight near the MITs (Methods). It is difficult todetermine the exciton binding energy accurately since the band-to-band transitions are also renormalized and cannot be easily measured.Unless otherwise specified, all measurements are performed at 3.6 K;the corresponding thermal excitation energy is substantially lowerthan the characteristic energy scales (U, V and W ) of the electronicsystem. Details on the device fabrication and optical measurementsare provided in Methods.Results and discussionsCorrelated insulating states in the strong interaction limitFigure 1c–e shows the reflectance contrast spectrum of device 1 as afunction of electrondensity ν in the strong interaction limit (U,V >W ),corresponding to the moiré exciton of MoSe2, the 2s exciton of thesensor layer, and the moiré exciton of WS2, respectively. The moiréexciton spectra are consistentwith a previous study35; the fundamentalexciton in WS2 remains robust for the entire doping range, indicatingthat the electrons are doped only into the MoSe2 layer in the type-Iheterostructure. A series of incompressible states emerge at integermultiples and specific fractions of the moiré density. They modulatethe moiré exciton features, but more significantly, the sensor 2s exci-ton. At each incompressible state, the 2s exciton shows enhancedreflectance contrast or spectral weight (as well as spectral blueshift); itis consistent with small ε. The 2s exciton cannot be identified for thecompressible states; it is merged into the band-to-band transitions30.The quenching of the 2s exciton is consistent with large negativedielectric constant for a metallic phase.The insulating states at even integers (ν = 2 and 4) are the single-particle moiré band insulators. The odd integer states (ν = 1 and 3) arethe Mott or charge-transfer insulators27. The fractional states (e.g.ν = 4/3, 3/2, 5/3 etc.) are theWigner–Mott insulators. Similar results arereported for a related WSe2/WS2 moiré superlattice7–9,19. Generally,with increasing doping density the 2s exciton reflectance contrastdecreases, and less fractional states can be identified. It reflects thedecreasing importance of the Coulomb interaction at large dopingFig. 1 | Correlated insulating states in MoSe2/WS2 moiré heterobilayers.a Schematic illustration of a dual-gated MoSe2/WS2 moiré heterobilayer with anintegratedWSe2 monolayer sensor separated by bilayer hBN. Voltage Vt and Vb areapplied to the top and bottom hBN-graphite gates, respectively. Both the moiréheterobilayer and the sensor are grounded.bMoSe2/WS2moiré heterobilayer has atype-I band alignment with Δc ≈0.1 eV and Δv ≈0.3 eV. c–e Reflectance contrastspectrumofdevice 1 as a functionof electrondopingdensity ν (in unitsof themoirédensity), corresponding to the moiré exciton of MoSe2 (c), the sensor 2 s exciton(d), and the moiré exciton of WS2 (e), respectively. Full lattice fillings are denotedby dashed lines. The incompressible states are identified by the enhanced reflec-tance contrast of the 2 s exciton.Article https://doi.org/10.1038/s41467-022-32037-1Nature Communications |         (2022) 13:4271 2densities. The secondmoiré band has a larger bandwidth compared tothe first moiré band (Methods). We study the second moiré band withdoping density ν = 2–4, for which the field-tuned MITs are more easilyachieved. For the same field range, a weak electric-field effect isobserved for the first moiré band with ν =0–2 (Supplementary Fig. 1).Bandwidth-tuned metal-insulator transitionsFigure 2 illustrates the evolution of the incompressible states probedby the sensor 2s exciton for ν = 2−4 under increasing electric fields. Toenhance the optical contrast of these states, we show the energyderivative of the reflectance contrast spectrum dR=dϵ. As electric fieldincreases, the insulating states gradually disappear, first the bandinsulating state at ν = 4, followed by the Wigner–Mott state at ν = 7/3and 8/3, and the Mott state at ν = 3. The band insulating state at ν = 2remains robust. The order of disappearance of these states can differslightly in different devices. The result for device 2 (SupplementaryFig. 2) shows that the fractional states disappear first, followed by theν = 4 and 3 states; the latter two disappear at similar electric fields.Fig. 2 | Electric-field-tuned MITs. a–e The energy-derivative of the reflectancecontrast spectrum (dR=dϵ) of the sensor 2s exciton as a function of electron fillingfactor (ν = 2–4) under out-of-plane electric field of −0.02V/nm (a), 0.02V/nm (b),0.06 V/nm (c), 0.12 V/nm (d) and 0.2 V/nm (e). The incompressible states atE = −0.02V/nm manifest enhanced reflectance contrast and are labeled by thedashed black lines. They gradually disappear as electric field increases.Fig. 3 | Dielectric catastrophe near the Mott and Wigner–Mott transitions.a, b Reflectance contrast spectrum of the sensor 2s exciton under varying electricfields atfixed electron filling factor of ν = 3 (a) and 7/3 (b). The spectra are verticallydisplaced by a constant 0.03. The dashed lines represent the reflectance contrastspectra at high electric fields (0.28V/nm and 0.24V/nm for ν = 3 and 7/3, respec-tively), in which the broad humps correspond to the band-to-band transitions. Thevertical dotted linesdenote the 2s excitonpeak. c, Reflectance contrast spectrumatν = 3, normalized by that at 0.28 V/nm (in the metallic phase), at representativeelectric fields. The spectral weight S is extracted by integrating the shaded area.d The extracted 2s spectral weight S (symbols) as a function of electric field (bot-tom axis) and 4c (top axis) for ν = 3 (upper panel) and ν = 7/3 (lower panel). Thebandoffset4c is calculatedusing the applied electricfield as described inMethods.The dashed lines are the smoothed data using the Savitzky-Golay algorithm with awindow of 80mV/nm. The solid lines are the dielectric constant �ε of the moiréheterobilayer (right axis) that is normalized to unity at 0.02V/nm and −0.08V/nm(upper and lower panel, deep in the insulating phase). It is obtained from thedashed lines using the empirical relation, S / ε�0:7 (Methods).Article https://doi.org/10.1038/s41467-022-32037-1Nature Communications |         (2022) 13:4271 3The above observation is consistent with the bandwidth-tunedMITs. As electric field increases (inducing larger Δc and shallowermoiré potential), the moiré bandwidth W increases; this is the pre-dominant effect since W is exponentially dependent on the moirépotential depth21. Thedisappearanceof the ν = 4 state indicates closingof the band gap between the second and third moiré band as Wincreases. The vanishing ν = 3 and the fractionalfilling states reflect theclosing of the Mott charge gap and the Wigner-Mott charge gap whenW becomes comparable to U and V , respectively. Because U >V , thefractional states disappear at smaller critical fields than the ν = 3 statesin all devices examined in this study. On the other hand, the relativeimportance of the Mott gap and the band gap is sample dependent. Aplausible origin is the twist angle and moiré density variations sincethese gaps are generally charge-density dependent due to the strongcorrelation effects.Next we investigate the bandwidth-tuned Mott and Wigner-Mott transitions systematically at fixed electron density of ν = 3 and7/3, respectively. Figure 3a,b illustrate the sensor 2s exciton spec-trum as a function of electric field. The 2s exciton resonance van-ishes above a critical field, Ec. To determine the exciton spectralweight, we first normalize the reflectance contrast spectrumby thatat a large field (e.g. 0.28 V/nm, above the critical field for ν = 3). Inthe metallic phase, the 2s exciton resonance is quenched (Fig. 2);the reflectance contrast spectrum is dominated by a broad humpcorresponding to the band-to-band transitions30; the spectrum isnearly identical to that at incommensurate fillings. Figure 3c showsthe normalized spectrum at representative fields for ν = 3 (theresult for ν = 7/3 is included in Supplementary Fig. 3). The inte-grated spectral weight (corresponding to the shaded area) is shownin Fig. 3d for ν = 3 and 7/3 as a function of electric field. As electricfield or bandwidth increases, the spectral weight continuouslydecreases to zero. In addition, we do not observe any electric-fieldhysteresis within the experimental uncertainty. (The ν = 3 spectralweight decreases for negative electric fields because the MoSe2moiré bands are approaching the WS2 bands.).We infer the dielectric constant of the moiré heterobilayer fromthe measured sensor exciton spectral weight by modeling excitons inthe 2D sensor layer using realistic device geometry (SupplementaryFig. 4). We numerically solve the electron-hole Schrodinger equationwith screened Coulomb potential by the heterobilayer and the hBNsubstrate with dielectric constant ε and εBN , respectively. For ε≫ εBN ,which holds for the insulting side near the transition, we obtain anempirical relation, S / ε�0:7 (Methods). The field-dependence of εinferred using the relation is included in Fig. 3d (black lines). Here ε isnormalized by its value deep in the insulating phase, for which the 2sspectral weight plateaus.We also limit the electric-field range such thatthe signal-to-noise ratio of S stays above 1. We find that ε increasessharply towards the critical point for both ν = 3 and 7/3; the electric-field dependence of ε is compatible with a power-law dependence,ε / ∣E � Ec∣�γ, with exponent γ = 1.6–2.2 for ν = 3 and 1.2–1.8 for ν = 7/3(Supplementary Fig. 5). The dependence of ε on the metallic sidecannot be extracted because of the vanished 2s exciton.Fig. 4 | Temperature dependence. a, b Contour plot of the sensor 2s spectralweight as a function of temperature and electricfield for ν = 3 (a) and ν = 7/3 (b).MI,WI, and M represent, respectively, the Mott insulator, Wigner-Mott insulator, andmetal. Regions with enhanced spectral weight (red) correspond to the incom-pressible states (MI and WI); regions with negligible spectral weight (blue)correspond to the compressible states (metal). c,d Electric-field dependenceof the2s spectral weight at selected temperatures for ν = 3 (c) and ν = 7/3 (d). The spectralweight maximum is normalized to unity. The metal-insulator crossover is broa-dened at high temperatures.Article https://doi.org/10.1038/s41467-022-32037-1Nature Communications |         (2022) 13:4271 4The diverging dielectric constant is consistent with the expecta-tions for a continuous Mott/Wigner–Mott transition32,36. As the MITcritical point is approached from the insulating side, polarizationfluctuations and the holon/doublon density proliferate, giving rise to a‘dielectric catastrophe’ ε ! 1 (Ref. 36); the charge gap Δ / ε�1=2 van-ishes continuously32. The result reflects the critical charge dynamicsnear a continuous MIT and is not expected for a first order MIT. ThecontinuousMott transition at ν =3 agreeswith recent transport studiesof other TMD moiré materials22,23. The continuous Wigner–Mott tran-sition at ν = 7=3 is in qualitative agreement with theoretical analyses onthe continuous bandwidth-tuned MIT at fractional lattice filling11,12,14,15.This transition can proceed through either a two-step process thatinvolves an intermediate charge-density-wave metal11,14 or a one-stepprocess that involves a deconfined quantum critical point12,14,15. Ourresults are, however, unable to resolve the two scenarios. AdditionalHall effect and optical conductivitymeasurements on themetallic sideare required in future studies.Crossover at elevated temperaturesFinally, we examine the temperature dependence of these transi-tions. Figures 4a and 4b show the 2s exciton spectral weight as afunction of electric field and temperature at ν = 3 and 7/3, respec-tively. The spectral weight is always negligible on the metallic side;it gradually decreases on the insulating side with increasing tem-perature because the thermally excited free carriers in the moiréheterobilayer screen the excitonic interaction in the sensor. Themelting temperature is estimated to be ~65 K and 25 K, respectively,for the ν = 3 and 7/3 states. We compare the electric-field depen-dence of S at 3.6 K and an elevated temperature for the two states inFigs. 4c, d. The spectral weight deep in the insulating phase isnormalized to unity. At 3.6 K, the thermal excitation energy is smallcompared to the charge gap of both the Mott and Mott-Wignerinsulators. Compared to the low-temperature behavior at 3.6 K, theMIT becomes a broadened metal-insulator crossover at high tem-peratures, a manifestation of critical point at lower temperatures.Our result thus suggests either a continuous Mott andWigner–Motttransition with quantum critical point at zero temperature or weaklyfirst-order transitions with critical point substantially below 3.6 K.The reduced dimensionality, the geometrically frustrated triangularlattice and the presence of disorders are known to favor continuousor weakly first-order transitions14–16. Above the critical point, thesetwo scenarios are almost identical37; future experiments down tolower temperatures are required to distinguish them.In conclusion, we have demonstrated bandwidth-tuned Wigner-Mott transitions at fixed band fillings of a Hubbard system based onsemiconducting moiré materials. The transitions manifest a dielectriccatastrophe when the critical point is approached from the insulatingside. Our results present new opportunities to simulate Hubbardphysics in the interesting regime of comparable Coulomb repulsionðU,V Þ and bandwidth (W ), and to search for quantum spin liquids nearthe transitions12,14–17.MethodsDevice fabricationWe fabricate dual-gate devices of a MoSe2/WS2 moiré heterobilayerwith an integrated WSe2 monolayer sensor using the reported drytransfer method38. Briefly, atomically thin flakes are first exfoliatedfrombulk crystals onto Si substrates and then stacked using a polymerstamp to form the desired heterostructure. MonolayerMoSe2 andWS2flakes are angle alignedwith aprecisionof about0.5°C. Theorientationand relative alignment of these crystals are determined from the angle-resolved optical second harmonic measurement19. The WSe2 sensor isseparated from the moiré heterobilayer by a bilayer hBN. The TMDmoiré heterobilayer and the sensor are grounded through few-layergraphite electrodes. The entire heterostructure is gated by hBN gatedielectrics (≈25 nm) and few-layer graphite gates on both sides.Optical reflectance contrast measurements and analysisDetails of the reflectance contrast measurement are reported in theliterature8,19. Briefly, broadband white light from a tungsten-halogenlamp is focused under normal incidence to a diffraction-limited spoton the device by a high-numerical-aperture objective. The device ismounted in a closed-cycle cryostat with base temperature of 3.6 K(attoDry 1000). The reflected light is collected by the same objectiveand detected by a spectrometer with a liquid-nitrogen cooled charge-coupled device (CCD). The reflectance contrast spectrumR � ðI 0 � IÞ=Iis obtained by comparing the reflected light spectrum from the sample(I 0) with a featureless background spectrum (I).To analyze the 2s exciton spectral weight, we integrate thenormalized reflectance contrast over a spectral window of 5 nmcentered at the resonance. Specifically, we define a straight line thatconnects the end points of the integrationwindow as a baseline, andintegrate the area above it (Fig. 3c). To verify the reliability of theprocedure, we have varied the size of the integration window; theprocess does not affect the electric field dependence of the 2sspectral weight.Band alignment of the MoSe2/WS2 heterobilayerWe determine the band alignment of the MoSe2/WS2 heterobilayer byexamining a sample with large twist angle to avoid themoiré effect forsimplicity. The reflectancecontrast spectrum ismeasured as a functionof doping density under zero applied electric field. SupplementaryFig. 6a and b show the fundamental exciton resonance in monolayerMoSe2 andWS2, respectively. Theneutral exciton feature turns into thecharged exciton (or polaron) feature in MoSe2 with both electron andhole doping; the optical response of the WS2 layer remains largelyunperturbed. Charges are therefore introduced only into the MoSe2layer upon both electron and hole doping; the MoSe2/WS2 hetero-bilayer has a type-I band alignment.We determine the band offsets by measuring the electric-field (E)dependence of the reflectance contrast spectrum. We choose a fixedelectron doping density (≈3.7 × 1012 cm−2); the chemical potential isslightly above the conduction band edge of MoSe2. As E increases inthe WS2 to MoSe2 direction (E < 0), the charged exciton feature inMoSe2 changes to the neutral exciton above E0 ≈�0:33 V/nm (Sup-plementary Fig. 6c); at the same time, the neutral exciton feature inWS2 turns into the charged exciton (Supplementary Fig. 6d). Thespectral changes correspond to the onset of charge transfer fromMoSe2 to WS2 when the two conduction bands become nearlydegenerate. Using the reported interlayer dipole moment in theMoSe2/WS2 heterobilayer35, d ≈0:3 e �nm,we estimate the conductionband offset to be 4C =d � E0 ≈ 0.1 eV. The valence band offset can beevaluated as 4V ≈ EWg � EMog �4C ≈ 0.32 eV, where EWg ≈ 2:04 eV andEMog ≈ 1:62 eV are the optical gaps of monolayer MoSe2 and WS2,respectively. Supplementary Fig. 6e illustrates the inferred bandalignment.Estimate of the bandwidthWe estimate the first moiré conduction bandwidth, W0 ~_2ma2M~5–10meV (corresponding to a temperature scale ~60–120 K), fromthe moiré period aM . Here _ and m denote the Planck’s constant andthe conduction bandmass ofmonolayerMoSe2, respectively (m ≈0.56m0 in termsof the free electronmassm0)39. The combinedwidthof thefirst two moiré bands is 4W0; the second moiré bandwidth is about3W0 (~180–360K). These values are substantially smaller than 4C ≈0.1 eV (~1200K). We focus on the second moiré conduction band withlarger bandwidth so that the sample temperature is small compared tothe bandwidth and the bandwidth-tuned MITs are easier to achieve.Article https://doi.org/10.1038/s41467-022-32037-1Nature Communications |         (2022) 13:4271 5The electrons reside in theMoSe2 layer for all electricfields anddopingdensities in this study.Modeling the 2s exciton of the sensor layerQuantitative estimate of the effective dielectric constant of the moiréheterobilayer is obtained by modeling the sensor 2s exciton (Supple-mentary Fig. 4). We solve the Schrödinger equation,HΨns ρð Þ= EnsΨns ρð Þ, for the energy (Ens) and wavefunction (Ψns ρð Þ) ofthe ns (n = 1, 2, …) exciton state using the finite difference method40.For radially symmetric ns excitons confined in the 2D sensor plane, theHamiltonian is given by H = � _22mR∂2ρ +1ρ ∂ρ� �+V ρð Þ, where ρ is thedistance between the electron and hole,mR (≈0.2m0)41 is the reducedmass of the exciton, and V ðρÞ is the electrostatic potential between theelectron and hole. Wemodel V ρð Þ using the device geometry shown inSupplementary Fig. 4a. The thickness of the sensor layer, which issubstantially smaller than the 2s exciton Bohr radius (~5 nm), isignored; the thickness of the hBN spacer and the moiré heterobilayerare d1 and d2, respectively; the hBN gate dielectric is assumed to beinfinitely thick. The latter is a good approximation when the excitonBohr radius does not exceed substantially the hBN thickness andscreening by the graphite gates is negligible. We express the potentialas follows42V ρð Þ= � e24πε0εBN� 1ρ+ ∑1j =0β2j + 1 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρ2 + 4 d1 + jd2� �2q � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρ2 + 4 d1 + j + 1ð Þd2� �2q0B@1CA264375:ð1ÞHere ε0 is the vacuum permittivity; β= ðε� εBN Þ=ðε+ εBNÞ is given bythe dielectric constant of hBN (εBN = 4.5)8 and the heterobilayer (ε); wetake d1 = 0.9 nm for a 2L-hBN spacer and d2 = 0.6 nm for electronsresiding in theMoSe2 layer of themoiré.We assume ε to be a real valuefor the insulating states for simplicity. For ρ larger than the 1 s excitonBohr radius (<2 nm), the potential V ρð Þ in Eq. (1) is a goodapproximation for the more accurate Rytova-Keldysh potential for2D excitons40,43,44 (Supplementary Fig. 7). Since we focus on 2s sensingin this study, Eq. (1) is sufficient; we can ignore the finite sensorthickness.Supplementary Fig. 4b, c illustrate the spatial distribution ofpotential V ρð Þ and wavefunction Ψ2s ρð Þ of the 2s exciton for severalvalues of ε=εBN . The 2s exciton radius r2s (=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΨ2s ∣ρ2 ∣Ψ2sh ip) and bindingenergy E2s as a functionof ε=εBN are shown in Supplementary Fig. 4d, e,respectively. For ε=εBN = 2, screening by the heterobilayer is negligible;we have V ρð Þ≈� e24πε0εBρand r2s≈ 4.7 nm. The latter agrees well with thereported valueof r2s ≈ 6.6 nm formonolayerWSe2 embedded inhBN40.As ε=εBN increases, V ρð Þ is suppressed and Ψ2s ρð Þ is flattened; r2sincreases and E2s decreases. For ε=εBN = 104, r2s exceeds 50 nm. Welimit the range of ε=εBN to <104 (so that the correction from the gatescreening effect remains small) and perform a power-law analysis ofE2s. The binding energy is well described by ~ð εεBNÞ�0:7 for εεBN> 10 (solidline in Supplementary Fig. 4e). The exciton spectral weight is expectedto follow the same scaling lawon the dielectric constant since both theexciton spectral weight and binding energy scale quadratically withinverse of the exciton radius, which represents the total number ofelectronic states constituting the exciton45 (Supplementary Fig. 4fshows E2s ~ ðr2sÞ�2 for the entire range of dielectric constant). Weexamine the result for several hBN space thicknesses. The power-lawexponent of�0:7 remains a good approximation as long as the spacerthickness is much smaller than the 2s exciton radius. In the main text,we use the power-law dependence to extract the evolution of thedielectric constant on the out-of-plane electric field from the experi-mental spectral weight.Data availabilitySource data are provided. Additional data that support this workare available upon reasonable request to the correspondingauthors. 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The research was sup-ported by the National Science Foundation (NSF) under DMR-2114535(development of the sensing technique) and the US Army ResearchOffice under grant number W911NF-17-1-0605 (device fabrication).Growth of the MoSe2 and WSe2 crystals was supported by the U.S.Department of Energy (DOE), Office of Science, Basic Energy Sciences(BES), under Award # DE-SC0019481 and growth of the hBN crystals bythe Elemental Strategy Initiative of MEXT, Japan and CREST(JPMJCR15F3), JST. This work made use of the Cornell NanoScaleFacility, an NNCI member supported by NSF Grant NNCI-2025233.K.F.M. acknowledges support from the David and Lucille Packard Fel-lowship. Y.T. acknowledges support from a startup funding by ZhejiangUniversity.Author contributionsY.T. and J.G. fabricated the devices and performed the optical mea-surements. S.L. and J.H. grew the bulk TMD crystals. K.W. and T.T. grewthe bulk hBN crystals. Y.T., K.F.M., and J.S. designed the study, per-formed the analysis, and co-wrote themanuscript. All authors discussedthe results and commented on the manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary informationTheonline version contains supplementarymaterial available athttps://doi.org/10.1038/s41467-022-32037-1.Correspondence and requests for materials should be addressed toYanhao Tang, Kin Fai Mak or Jie Shan.Peer review information Nature Communications thanks the anon-ymous reviewers for their contribution to the peer review of thiswork. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2022Article https://doi.org/10.1038/s41467-022-32037-1Nature Communications |         (2022) 13:4271 7https://doi.org/10.1038/s41467-022-32037-1http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Dielectric catastrophe at the Wigner-Mott transition in a moiré superlattice Results and discussions Correlated insulating states in the strong interaction limit Bandwidth-tuned metal-insulator transitions Crossover at elevated temperatures Methods Device fabrication Optical reflectance contrast measurements and analysis Band alignment of the MoSe2/WS2 heterobilayer Estimate of the bandwidth Modeling the 2s exciton of the sensor layer Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information