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Kaining Yang, Xiang Gao, Yaning Wang, Tongyao Zhang, Yuchen Gao, Xin Lu, Shihao Zhang, Jianpeng Liu, Pingfan Gu, Zhaoping Luo, Runjie Zheng, Shimin Cao, Hanwen Wang, Xingdan Sun, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Xiuyan Li, Jing Zhang, Xi Dai, Jian-Hao Chen, Yu Ye, Zheng Han

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[Unconventional correlated insulator in CrOCl-interfaced Bernal bilayer graphene](https://mdr.nims.go.jp/datasets/95831a7f-ac35-46dd-a96b-f8e8bcc762c9)

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Unconventional correlated insulator in CrOCl-interfaced Bernal bilayer grapheneArticle https://doi.org/10.1038/s41467-023-37769-2Unconventional correlated insulator inCrOCl-interfaced Bernal bilayer grapheneKaining Yang1,2,18, Xiang Gao1,2,18, Yaning Wang3,4,18, Tongyao Zhang 1,2,18,YuchenGao5,6, Xin Lu 7 , Shihao Zhang 7, Jianpeng Liu 7,8, PingfanGu 5,6,Zhaoping Luo3, Runjie Zheng9, Shimin Cao 9,10, Hanwen Wang 3,Xingdan Sun 3, Kenji Watanabe 11, Takashi Taniguchi 12, Xiuyan Li3,Jing Zhang1,2, Xi Dai 13,14 , Jian-Hao Chen 9,10,15,16 , Yu Ye 5,6 &Zheng Han 1,2,17The realization of graphene gapped states with large on/off ratios over widedoping ranges remains challenging. Here, we investigate heterostructuresbased on Bernal-stacked bilayer graphene (BLG) atop few-layered CrOCl,exhibiting an over-1-GΩ-resistance insulating state in a widely accessible gatevoltage range. The insulating state could be switched into ametallic state withan on/off ratio up to 107 by applying an in-plane electric field, heating, orgating. We tentatively associate the observed behavior to the formation of asurface state in CrOCl under vertical electric fields, promotingelectron–electron (e–e) interactions in BLG via long-range Coulomb coupling.Consequently, at the charge neutrality point, a crossover from single particleinsulating behavior to an unconventional correlated insulator is enabled,below an onset temperature.We demonstrate the application of the insulatingstate for the realization of a logic inverter operating at low temperatures. Ourfindings pave the way for future engineering of quantum electronic statesbased on interfacial charge coupling.AB-stacked BLG hosts fascinating emerging physics and can be abuilding block for intriguing nanoelectronics1–7. In the single-particlepicture, when subjected to vertical electric fields, Bernal-stacked BLGyields a layer-polarized gap at charge neutrality, which is tunable andreaches about 250meV in an experimentally applicable maximumdisplacement field of about 3 V/nm8. However, the correspondingresistances areusually peaked in a very small doping range5,9–12,makingit limited for further explorations in such gapped states.On the other hand, charge neutral BLG is strongly susceptible toCoulomb interactions and is predicted to exhibit ground states withReceived: 8 November 2022Accepted: 30 March 2023Check for updates1State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan, PR China. 2CollaborativeInnovation Center of ExtremeOptics, Shanxi University, Taiyuan, PR China. 3Shenyang National Laboratory for Materials Science, Institute of Metal Research,Chinese Academy of Sciences, Shenyang, China. 4School ofMaterial Science and Engineering, University of Science and Technology of China, Anhui, China.5Collaborative Innovation Center of Quantum Matter, Beijing, China. 6State Key Lab for Mesoscopic Physics and Frontiers Science Center for Nano-Optoe-lectronics, School of Physics, Peking University, Beijing, China. 7School of Physical Science and Technology, ShanghaiTech University, Shanghai, China.8ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai, China. 9International Center for Quantum Materials, School ofPhysics, Peking University, Beijing, China. 10Beijing Academy of Quantum Information Sciences, Beijing, China. 11Research Center for Functional Materials,National Institute for Materials Science, Tsukuba, Japan. 12International Center for Materials Nanoarchitectonics, National Institute for Materials Science,Tsukuba, Japan. 13Materials Department, University of California, Santa Barbara, CA, USA. 14Department of Physics, The Hongkong University of Science andTechnology,HongKong,China. 15Key Laboratory for the Physics andChemistry of Nanodevices, PekingUniversity, Beijing, China. 16Hefei National Laboratory,Hefei,China. 17LiaoningAcademyofMaterials, Shenyang,China. 18These authorscontributedequally: KainingYang,XiangGao,YaningWang, TongyaoZhang.e-mail: lvxin@shanghaitech.edu.cn; daix@ust.hk; chenjianhao@pku.edu.cn; ye_yu@pku.edu.cn; vitto.han@gmail.comNature Communications |         (2023) 14:2136 11234567890():,;1234567890():,;http://orcid.org/0000-0002-7947-1635http://orcid.org/0000-0002-7947-1635http://orcid.org/0000-0002-7947-1635http://orcid.org/0000-0002-7947-1635http://orcid.org/0000-0002-7947-1635http://orcid.org/0000-0002-6228-1480http://orcid.org/0000-0002-6228-1480http://orcid.org/0000-0002-6228-1480http://orcid.org/0000-0002-6228-1480http://orcid.org/0000-0002-6228-1480http://orcid.org/0000-0002-5787-5022http://orcid.org/0000-0002-5787-5022http://orcid.org/0000-0002-5787-5022http://orcid.org/0000-0002-5787-5022http://orcid.org/0000-0002-5787-5022http://orcid.org/0000-0002-8564-0415http://orcid.org/0000-0002-8564-0415http://orcid.org/0000-0002-8564-0415http://orcid.org/0000-0002-8564-0415http://orcid.org/0000-0002-8564-0415http://orcid.org/0000-0002-4203-794Xhttp://orcid.org/0000-0002-4203-794Xhttp://orcid.org/0000-0002-4203-794Xhttp://orcid.org/0000-0002-4203-794Xhttp://orcid.org/0000-0002-4203-794Xhttp://orcid.org/0000-0001-9245-586Xhttp://orcid.org/0000-0001-9245-586Xhttp://orcid.org/0000-0001-9245-586Xhttp://orcid.org/0000-0001-9245-586Xhttp://orcid.org/0000-0001-9245-586Xhttp://orcid.org/0000-0003-2212-6205http://orcid.org/0000-0003-2212-6205http://orcid.org/0000-0003-2212-6205http://orcid.org/0000-0003-2212-6205http://orcid.org/0000-0003-2212-6205http://orcid.org/0000-0003-3811-2326http://orcid.org/0000-0003-3811-2326http://orcid.org/0000-0003-3811-2326http://orcid.org/0000-0003-3811-2326http://orcid.org/0000-0003-3811-2326http://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/0000-0002-2396-0966http://orcid.org/0000-0002-2396-0966http://orcid.org/0000-0002-2396-0966http://orcid.org/0000-0002-2396-0966http://orcid.org/0000-0002-2396-0966http://orcid.org/0000-0002-9485-1759http://orcid.org/0000-0002-9485-1759http://orcid.org/0000-0002-9485-1759http://orcid.org/0000-0002-9485-1759http://orcid.org/0000-0002-9485-1759http://orcid.org/0000-0001-6046-063Xhttp://orcid.org/0000-0001-6046-063Xhttp://orcid.org/0000-0001-6046-063Xhttp://orcid.org/0000-0001-6046-063Xhttp://orcid.org/0000-0001-6046-063Xhttp://orcid.org/0000-0001-5721-6206http://orcid.org/0000-0001-5721-6206http://orcid.org/0000-0001-5721-6206http://orcid.org/0000-0001-5721-6206http://orcid.org/0000-0001-5721-6206http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-37769-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-37769-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-37769-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-37769-2&domain=pdfmailto:lvxin@shanghaitech.edu.cnmailto:daix@ust.hkmailto:chenjianhao@pku.edu.cnmailto:ye_yu@pku.edu.cnmailto:vitto.han@gmail.comspontaneous symmetry breaking6,13,14. Ultra-clean BLG samples undervertical electric fields are often demanded to observe such uncon-ventional insulating states, yet within narrowly distributed parameterspaces6. To favor e–e interactions, recent attempt was also devoted tosuch as moiré double bilayer graphene, where excitonic insulatingbehaviors are seen at charge neutrality due to the overlapping electronand hole pockets at different wave vector k15.Here, we attempt to devise a different route to identify anunconventional correlated insulating phase in BLG atop few layeredCrOCl. It is known that interaction effects are alreadymanifested in thescenario of monolayer graphene/CrOCl heterostructure, where areconstruction of the Dirac dispersion induced by e–e interaction isthe most prominent effect16–19. In the case of BLG, the quadratic bandsaround the charge neutrality endow larger density of states at theFermi level, which are expected to have more dramatic correlationeffects, as such evidence can be found in free-standing ultra-cleanBLG6. Here, in a hBN-BLG-CrOCl device, the interaction effects in BLGare further enhanced by the interfacial coupling to a presumably long-wavelength charge order at the surface of CrOCl, leading to gate-tunable correlated gap in neutral BLG with a sheet resistance largerthan GΩ within a much expanded effective gate range. This observa-tion is markedly different compared to conventional dual-gated BLGsystems. Nevertheless, the displacement field D in BLG in the currentsystem is estimated to be at the order of ~ 1 V/nm, similar to thosefound in conventional BLG samples. Systematic transport measure-ments together with theoretical modeling self-consistently suggestthat this insulating ground state can neither be explained by localiza-tion nor be trivially categorized into a band insulator. As a result, thecharge coupling from the surface charge order triggered in CrOCl isthe key ingredient to drive the crossover from single-particle insulat-ing phase to a correlated insulator in neutral BLG, with a maximumonset temperature Tinsulator of a full insulating state (with the zerobiased conductance reaching the noise level) seen at about 40 K. Thewide gate range of the insulating phase further allows the demon-stration of a logic inverter using BLG/CrOCl devices. Our findings pavethe way for the engineering of interfacial coupling between 2D elec-tron gases in van der Waals heterostructure, which may be expandedto a broader library of materials.ResultsInsulating behaviors in BLG/CrOCl hetero-systemBernal-stacked BLG, thin CrOCl flakes, and encapsulating hexagonalboron nitride (h-BN) flakes were exfoliated from high-quality bulkcrystals and stacked in ambient conditions using the dry transfermethod20, with more detail of the fabrication process and samplemorphologies described in Supplementary Figs. 1–2. We cooled downthe samples to a base temperature of 1.5 K, and measured the long-itudinal channel resistanceRxx as a functionof topgate Vtg and bottomgate Vbg, as shown in Fig. 1a. Two key observations are to be under-stood in Fig. 1a, i.e., the extremely wide gate range of an insulatingregion that reaches 109Ω, and the largely bent phase boundaries of thegapped state, which is markedly different from the charge neutralitypoint (CNP) resistive peaks found in conventional ultra-clean BLGsamples5,6,8–12. A side-by-side comparison of the gapped states in h-BN/BLG/CrOCl and h-BN/BLG/h-BN hetero-systems can be seen in Sup-plementary Fig. 3 and Supplementary Table 1.In the following, we define total carrier densityntot = (CtgVtg +CbgVbg)/e − n0, and the effective displacement fieldDeff = (CtgVtg−CbgVbg)/2ϵ0 −D0, where Ctg and Cbg are the top andbottom gate capacitances per area, and n0 and D0 are residual dopingand residual displacement field, respectively. Figure 1b plots a lineprofile (along the black dashed line in Fig. 1a, with a fixed Deff = 0.4 V/nm). It shows that the channel resistance can be tuned from a fewhundred Ω into an OFF state by gating, with ON-OFF ratios reaching107. By examining multiple devices, we exclude the possibility of gateleakages (meaning that the bulk CrOCl itself is always insulating anddoes not contribute to transport throughout the measurements) orimpurity-dominated parasitic effects for this observed unconventionalgapped sates, shown in Supplementary Figs. 4–5. Atomic resolution ofthe cross-section of a typical heterostructure can be seen in the high-angle annular dark-field scanning transmission electron microscopy(HAADF-STEM) image in the inset in Fig. 1b, showing a clean interfacebetween the layered compounds.For a typical Vbg and Vtg which correspond to the black starredpoint in Fig. 1a, we performed the dI/dV (differential conductanceobtained by differentiating the DC I–V curves) as a function of biasvoltage Vbias at different temperatures, as shown in Fig. 1c. It displaysthat the low-biased insulating phase with negligible conductance canbe killed at both high temperatures and high bias voltages. Line pro-files along dashed lines in Fig. 1c are shown in Fig. 1d, which illustratedI/dV versus temperature in the insulating state (Vbias = 0 V) and themetallic state with conductance at the order of mS (Vbias = 300mV),respectively. Interestingly, a drastic drop in zero-biased dI/dV with theonset temperature Tinsulator of a full insulating state reaching the noiselevel is seen at about 35 K, indicated by the solid arrow. Similar dI/dVcurves were also seen in spontaneous symmetry breaking states insuspended ultra-clean BLG owning to electronic correlation6. The lat-ter, however, was at a much lower Tinsulator (below 10 K) and lack of afull study in the parameter space of ntot-Deff.Extracting the single particle gap at the CNP in BLG/CrOClTo understand the observed insulating state in our system, we firstdetermine the nature of doping in the insulating region by measuringthe device in the quantum Hall limit, so that the exact doping in thebilayer graphene nBLG can be deduced by the filling fractionsν = nBLGh/eBof each Landau levels (LLs), whereh is Planck constant, e iselementary charge, andB is the perpendicularmagneticfield. Figure 2ashows longitudinal conductivity σxx =RxxðR2xx +R2xyÞatB = 12T andT = 1.5 Kofthe same sample as in Fig. 1. It shows that, compared to that seen inconventional BLG cases, the BLG/CrOCl system exhibits distinct fea-tures of LLs, with a crossover from straight stripes to cascades-likebent stripes as ntot is varied from negative to positive in general. Fig-ure 2b shows line profiles along the dashed line in Fig. 2a (more datacan be found in Supplementary Figs. 6–7), indicating that nBLG in theinsulating states corresponds to a filling fraction of ν =0 (yellow sha-dowed area in Fig. 2b), i. e., the charge neutrality. Meanwhile, fulldegeneracy lifting can be seen at each integer filling fractions fromν = −1 to −10. This speaks the high quality of BLG itself.To further clarify the observed bent CNP, we set up a simplifiedelectrostatic model (see also Supplementary Note 1) to introduce anextra capacitance induced by the interfacial band with density ofstates n2 that is very close to the surface of CrOCl (with a distance d2).While top and bottom gates are located at distances of d1 and d3,respectively, illustrated in the cartoon image and capacitance modelin Fig. 2c, d, respectively. The potential on the interfacial state isdefined as V2, and dielectric constants εi (i = 1, 2, 3) are assigned toeach of the capacitor. By evaluating the electrostatic model usingGauss’s law, we found that the above-mentioned two major experi-mental observations can be well reproduced, as shown in the phasediagram in Fig. 2e, where the iso-doping lines of nBLG are highlightedin thentot-Deff space. A clear phase boundary is indicated by thewhitedashed line, separating the Phase-i (conventional phase with n2 = 0,and nBLG isDeff-independent) and Phase-ii (interfacial coupling phasewith n2 > 0, i. e., interfacial band is filled with electrons via tunnelingfrom the BLG, where nBLG depends on both Deff and ntot). Accordingto the above analysis, a diagram showing typical transition process inour system from Phase-i to Phase-ii is illustrated in Supplemen-tary Fig. 8.Moreover, as discussed in Supplementary Note 1, by quantify-ing the average vertical electric field in the BLG, iso-DBLG lines canArticle https://doi.org/10.1038/s41467-023-37769-2Nature Communications |         (2023) 14:2136 2also be plotted in Fig. 2f. In general, due to the existence of n2,electric fields in BLG is bent toward lower Deff in Phase-ii as com-pared to that in Phase-i. As a result, the calculated DBLG at chargeneutrality in Phase-ii is in the range of 0.5 to 1.3 V/nm, similar to thatestimated in conventional h-BN/BLG/h-BN cases, which in thesingle-particle picture corresponds to the gap size of about50–130meV8.Temperature dependence (Supplementary Fig. 9) shows that theinsulating phase weakens upon heating, and the resistance remains atthe order of MΩ at 80K. As plotted in Fig. 2g, we tracked 6 typicalpoints at the resistance maxima in the dual-gate map at B = 0 andT = 80K, as shown in Supplementary Fig. 10. The Ids-T−1 curves for these6 points are obtained using DC 2-probe measurement with a fixedVbias = 5mV. The thermal activation gaps (defined asIds / expð�Δ=2kBTÞ) of each curve are then extracted to be from17.41meV to 70.04meV, with each corresponding ntot calculated fromtheir gate voltages (Supplemental Fig. 10). These measured thermalgap sizes at the temperature range of 40 to 100 K are in good agree-ment with those expected from the layer-polarized gaps induced bythe displacement fields at charge neutrality shown in Fig. 2f. However,such a single-particle gap picture contradicts the experimentallyobserved unconventional insulating behaviors such as the peculiar I–Va121086420Top Gate (V)-8 -6 -4 -2 0 2 4 6 8Bottom Gate (V)Insulating State(> 1 GΩ)1086421010101010R   (Ω)xxD   =0.4 V/nmbh-BN/BLG/CrOCl108642Resistance (Ω)1.00.50-0.51 GΩ1010101010On-Off Ratio > 10 7n    (X10   cm   )-213toteffT = 1.5 Kd10-1010-910-810-710-610-510-410-3Temperature (K)dI/dV (S)Zero BiasHigh Bias80604020T  insulator‘Normal’ Metallic StateInsulating State10010-1010-910-810-710-610-510-4-300 0 100c 140 130 120 110 100 90 80 70 60 50 40 30 20 10 1.5T (K)V     (mV)dI/dV (S)ZeroBiasNoise Level h-BNBLGCrOClHighBias-200 -100 200 300Switching VoltagebiasFig. 1 | Unconventional insulating phase in neutral bilayer graphene atop few-layered CrOCl. a Color map of a dual gate scan of channel resistance in a typicalsample, measured using DC Ohm meter at T= 1.5 K and B =0 T. b Line profile oflongitudinal resistance Rxx at D =0.4 V/nm, along the dashed line in a in the rangeof −0.8 × 1013 cm−2 <ntot < 1.5 × 1013 cm−2, herentot = (CtgVtg +CbgVbg)/e−n0,whereCtgand Cbg are the top and bottom gate capacitances per area, Vtg and Vbg are the topand bottom gate voltage, and n0 is residual doping. A contact resistance of about5800Ω is subtracted. Inset shows a high-angle annular dark-field scanning trans-mission electron microscopy (HAADF-STEM) image of the cross-section of a typicalsample (Device-S1), and the scale bar is 2 nm. c Differential conductance dI/dV as afunction of bias voltage, measured at the black starred point in a at different tem-peratures.d dI/dV as a function of temperature along fixed bias voltages as indicatedin dashed lines in c.Article https://doi.org/10.1038/s41467-023-37769-2Nature Communications |         (2023) 14:2136 3curves and the onset temperature Tinsulator characteristics as shown inFig. 1c, d.Ruling out the possibility of a band insulatorWe can rule out the observed insulating behavior from the possibilityof being a trivial band insulator. First, as discussed in Fig. 1c, in I–Vcurves, the gapped state exhibits below a critical source-drain voltageVC (indicated as switching voltage in Fig. 1c), beyond which the insu-lating state breaks down and turns into a normal metallic state withresistances of a few kΩ. Moreover, critical behavior is also observableas a function of temperature, as a drastic drop of zero-bias con-ductance occurs at Tinsulator, indicated by the solid arrow in Fig. 1d.Meanwhile, at high bias voltage, the system is showing only metallicstate. Such behavior could be a hallmark that differs from trivial bandinsulators, with more evidence discussed in the coming text.In the following, the Vbias is converted into an L-independent in-plane electric field E!k =Vds=L, since VC is proportional to thedistance between electrodes L (Supplementary Fig. 11), as illustratedin Fig. 3a. Figure 3b displays the dual-gate maps of channel resis-tance at different Vbias. It shows that, with increasing E!k from bot-tom to top, a significant portion of the insulating region is switchedtometallic states. This suggests that increasing E!k has an effect thatis similar to that of either gating or temperature on the insulatingstate, indicating a continuously tunable phase transition. The loweredges of the insulating phase at CNP are plotted in Fig. 3c, with theI–V spectra at typical points (indicated by colored solid circles alongthe dashed line of ntot = 0.5 × 1013 cm−2) measured in a trace-retracemanner, shown in Fig. 3d.Importantly, as plotted in Supplementary Fig. 11c, the insulatorbreakdownelectricalfield E!C (defined as the E!k atVC) as a function of1/L clearly shows a trend that does not extrapolate to zero as 1/L→0,which is clearly different from Zener-type breakdown ( E!C ! 0 as 1/L→0) for a band insulator. This is likely a characteristic behavior ori-ginated from the pair-breaking mechanism for a Coulomb induceddce-CrOClAuh-BNAue- e-BLGdVtgVbgf 1/T (K   )-110-1210-1110-1010-9I   (A)0.0350.0300.0250.0200.015 Point I Point II Point III Point IV Point V Point VIV   = 5 mV dsds17.41 meV 34.87 meV 47.95 meV 61.22 meV 67.53 meV 70.04 meV gTop GateBLGCrOClBottom GateE1E2E3ntnbn BLGn  2VtgV2Vbg0ε  d  11ε  d  22ε  d  33a b0xy e( 2 )h/T=1.5 KB=12 T0 0.4-0.4-0.40.60.81.210.40.2-0.200.02.0xx (e /h2 )InsulatingStateν= 0ChargeNeutral-0.6n    (X10   cm   )-213totD   (V/nm)eff-5-10ν = -12-1-2-3-4-5-6-7-8-10+40 0.4-0.4n    (X10   cm   )-213totPhase BoundaryCalculated-2X10   cm  -212-4X10   cm  -212-6X10   cm  -212-8X10   cm  -212+2X10   cm  -2120 0.4-0.4n    (X10   cm   )-213totCalculatede-0.40.60.81.210.40.2-0.20-0.6D   (V/nm)eff1.2 V/nm0.9 V/nm0.6 V/nm0.3 V/nm0 V/nm-0.3 V/nmDBLGnBLGPhase-iiPhase-in    = 0BLG n    = 0BLGFig. 2 | Phase diagram of CrOCl-interfaced BLG in the space of ntot and Deff.a σxx =Rxx=ðR2xx +R2xyÞ of Device-S8 plotted in the ntot-Deff space, measured at B = 12T and T = 1.5 K. Here Rxx and Rxy are longitudinal and transverse resistance,respectively. b Line profile of Hall conductance σxy at ntot = 0.2 × 1013 cm−2 along thedashed line in a, in the range of -0.55 V/nm<Deff < 1.12 V/nm. Here, the effectivedisplacement field is defined as Deff = (CtgVtg −CbgVbg)/2ϵ0 −D0, where Ctg and Cbgare the top and bottom gate capacitances per area, Vtg and Vbg are the top andbottomgate voltage, andD0 is the residual displacement field, respectively. Shadedarea denotes the region of the charge neutrality. c Schematic image showing theinterfacial states near the interface between CrOCl and BLG, with an average dis-tance between these states and BLG defined as d (more details in Methods andSupplementary Note 1). d Simplified electrostatic model with three capacitors witheachof their parameters indicated.Here,di (i = 1, 2, 3), εi (i = 1, 2, 3), and Ei (i = 1, 2, 3),are the distance, dielectric constant, and electrical field of the top (between the topgate andgraphene),middle (betweengraphene and the surface state ofCrOCl), andbottom (between bottom gate and the surface state of CrOCl) capacitor, respec-tively. nt, n2, and nb, denote the carrier concentration induced in the top gate,surface state in CrOCl, and the bottom gate, respectively. Ids and Vds denote thesource-drain current and voltage, respectively. e Calculated carrier density nBLG inBLG (step size between each iso-doping line is 2 × 1012 cm−2) at B =0, using themodel described in Supplementary Note 1. f Calculated displacement field DBLG inBLG, with each iso-DBLG line stepped by 0.3 V/nm. Phase-boundary of the two dis-tinct phases (i.e., conventional Phase-i, and interfacial coupling Phase-ii) is indi-cated by the black dashed line. Shaded areas in e, f denote the charge neutrality ofthe system. g Ids as a function of 1/T in a semi-log plot. Thermal activation gapsextracted from each curve are labeled, with the solid line being fitted linear slopes.Article https://doi.org/10.1038/s41467-023-37769-2Nature Communications |         (2023) 14:2136 4excitonic insulator described in a recent theoretical model21. It is alsoworth noting that themesoscopic samples studied in our work exhibitinsulating breakdown at E!C ∼ 105 V/m, orders of magnitudes smallerthan the values of Zenerbreakdown for band insulators ( ~ 107 V/mfor agap of about 0.1meV,whichhas also a size-independent characteristic,largely distinct from the L-dependent small E!C observed in thiswork)22, in agreement with the theoretical model in the limit of smallL21. We have to emphasize that other experimental factors and/orsample details, including charge disorder concentration, metal-to-graphene contact, and fringe electric fields, are not taken into accountin the theoretical model21.We now consider the Tinsulator of the insulating state at the phaseboundary with a finite E!k=100 kV/m applied to the ground state,shown in the inset in Fig. 3e. At several typical points (Points A-F) alongthe phase edge, zero-biased differential conductances dI/dV wererecorded as a function of temperature, shown in Fig. 3e. We see trendsof drastic drop of differential conductance as the temperature islowered, which can be explained as a crossover of the gap nature fromthe single-particle picture at high temperatures to correlation domi-nant type at low temperatures. Moreover, the Tinsulator (from ~ 10 K atPoint-A to ~ 40 K at Point-F) is readily tunable by gating.In addition, it is notice that such switching-like (sometimesasymmetric) I-V characteristic found in the current system is onlyreported in a few systems suchasBCS superconductors23,24, someMottinsulator systems25,26, and, perhaps most pertinently, the 2D quantumelectron crystals27. Indeed, one can further see clear hysteresis in the I-V curves (indicated by solid arrows in Fig. 3d), especially at relativelylow values of Deff. To this point, we can see that in the observed I-Vcurves, gate-tunable Tinsulator below which a zero-biased conductancereaches the zero limit (while the system exhibits a metallic behavior athigh bias above E!C), relatively small E!C, together with a plausiblepair-breakingmechanism for the in-plane electrical field breakdownofthe insulator, all point to a correlated insulator behavior. At last, wewould like to mention that although magnetic phase transition occursin CrOCl at around 27 K (also structural phase transition at Néel tem-perature of ~ 14 K)28–30, we found no connection between the observedinsulating behavior and the magnetism of CrOCl since insulating fea-tures prevail up to 80K.Coulomb interaction augmented correlated insulatorWe have understood that the actual electric field in the BLG/CrOClhetero-system is in the same order as those reported in conventionalBLG samples. Yet, the neutral BLG in the interfacial coupling phase(CNP in Phase-ii) has a much larger gap under such magnitude ofelectric fields, and correlation behaviors of the insulating state at theCNP are revealed in transport measurements. To further elucidate thephysical origin of the correlated insulator,we now consider theoreticalmodelings of the current system. At certain vertical electric fields, theinterfacial band from CrOCl (mainly from the 3d orbital of top Cratoms) starts to overlap with the Fermi level of BLG16,17, this triggerscharge transfer (via tunneling) from BLG to the interfacial band, asindicated in the band alignment diagram in Fig. 4a. Calculations16c100.5-0.5D  (V/nm)0 0.5 1 1.5-40040I ds(μA)-0.8 -0.4 0.0 0.4 0.8 Vds (V) Point 1 Point 2 Point 3 Point 4 Point 5Insulating Phase Edge120406080100 E //(kV/m)n    (X10   cm   )-213toteff12345CrOClVbias E//Distance Led1.510-1010-910-810-710-6Zero Bias Conductance (S)80604020T (K) E // ~ 0 V/mPoint -A Point -BPoint -DPoint -CPoint -EPoint -F-8 0 812840V   (V) E //V   (V)bgtg10 26log Rxx1.00.00.5D (V/nm)0 0.5 1n   (x10  cm  )13 -2tot1026log Rxx E // = 100 kV/meffA B C D EFabFig. 3 | Correlated insulatorbehaviors inBLG/CrOCl hetero-system. a Schematicpictureof in-plane electricfield applied to the BLG/CrOCl heterostructure.Vbias andE!k denote the bias voltage between source and drain, and the in-plane electricalfield, respectively. b Dual-gate maps of longitudinal channel resistance Rxx at dif-ferent in-plane electric field E!k of 20, 40, 60, 80, and 100kV/m, in increasing orderfrom bottom to top, respectively. c The lower edges of the insulating phaseextracted from b in the ntot-Deff space. d I–V curves for Points 1–5 indicated inc, with traces and retraces recorded. Data in b–dwere obtained at 1.5 K. e Zero-biasdifferential conductance as a function of temperature measured in the coloredpoints indicated in the inset, which is the same data as the top map in b, i. e., Rxxmeasured at E!k = 100kV/m, plotted in ntot-Deff space.Article https://doi.org/10.1038/s41467-023-37769-2Nature Communications |         (2023) 14:2136 5suggest that long-wavelength charge order should appear in theinterfacial states in the top layer of CrOCl due to e–e interaction anddoes not contribute to transport, illustrated in the cartoon in Fig. 4b.The length scale of such charge order Ls (~5 nm, inversely pro-portional to the square root of the surface electronic density in CrOClsurface bands) ismuch larger than the carbon-carbon bond distance inbilayer graphene. Thus, if we are only interested in the low-energyphysics in BLG, the charge degrees of freedom in the long-wavelengthcharge order on the CrOCl side can be integrated out to provide aneffective background superlattice potential (arising from the Coulombpotentials of the charge order in the CrOCl substrate) for electrons inBLG, as revealed in the scenario of monolayer graphene-CrOClheterostructure16,17.Based on the effective model, we can then investigate the e–einteraction effects in the bilayer graphene-CrOCl heterostructureusing the renormalization group (RG) assisted Hartree-Fock (HF)approximations16, asdescribed in theMethods section. Clearly, the gapat Dirac point at certain vertical displacement field and a fixed Ls, issignificantly magnified by interactions compared to the non-interacting case 2Δe. Taking D =0.6 V/nm (corresponding to a2Δe ~ 50meV) and Ls = 5 nm for example, Fig. 4c juxtaposes the bandstructures of BLG obtained from the RG-HF calculations (solid lines)and that in the non-interacting case (dashed lines). Moreover, we plotin Fig. 4d the interaction gap at the Dirac point ΔHF as a function of thedoping of interfacial state nCrOCl at D =0.6 V/nm, with a clearenhancement of ΔHF with respect to 2Δe upon increasing nCrOCl.To check whether the gap enhancement results indeed from thesuperlattice-promoted e–e interaction effects, a straightforwardmeans is to lift the height of BLG above the underneathCrOCl surfacepotential, which will weaken the long-ranged interlayer Coulombcoupling between BLG and nCrOCl in an exponentially decayingmanner. Indeed, by intercalating amonolayer of h-BN in the interfaceof BLG/CrOCl, it appears a significant weakening of the insulatingstate at CNP, as shown in Fig. 4e, f. Further increase of the thicknessof intercalation h-BN, the insulating phase gradually fades away, andthe bent CNP is recovered in the dual gatemap, while the behavior ofchannel resistance for conventional h-BN/BLG/h-BN is fully restoredwith an intercalation of above 4.99 nm, as shown in Fig. 4g–i. Thisis a strong support to the previously discussed picture for theorigin of the correlated insulator ground state at CNP in BLG/CrOCle-e-e-e-e- e-e-e-e-eeeee-eeeeeeeeee-ee-fb eBottom Gate (V)g h iTop Gate (V)840-4840-40-10 10481201650-5840-450-5840-440-4840-41026log RxxBernal BLG0.68 nm 4.99 nm 6.16 nm0.34 nmNo IntercalationCrOClIntercalation LayerjdεΔCrOCl BLGCNPInterfacialBanda cEnergy (eV)0.60.40.20.0-0.2-0.4Γ X S Y Γs s s s sInteraction Gap Δ     (eV)0.140.120.100.080.06n       (X10   cm   )-212CrOCl1 2 3 4 5 6D = 0.6 V/nmHFInteraction Strength η (arb.units.)0.0Lifted Distance of BLG Above CrOCl (nm)1.5 3.0 4.5 6.0Bare BLGBLG/CrOClSingle ParticleInteractionDevice-S4 Device-S9 Device-S15 Device-S17 Device-S180.2350.2250.2150.205Fig. 4 | Interaction enhanced gap in neutral BLG coupled to interfacial chargeorder in CrOCl. a Schematics of the band diagram of the system. The relativeenergy differencebetween thebottomenergy of the interface states and the chargeneutrality point (CNP) energy of BLG can be tuned by the gate voltages. Δ denotesthe single particle gap. b Schematic picture of the BLG/CrOCl hetero-system withthe long wavelength charge order (red circles labeled with e−) in the surface ofCrOCl. A thin h-BN may be intercalated for examining the gap as a function ofdistance d between BLG and the interfacial states. c Band structures of BLGobtained from the renormalization group Hartree-Fock calculations (solid lines)and that in the non-interacting case (dashed lines).dThe calculated interaction gapΔHF as a function of the interfacial charge density nCrOCl in CrOCl. e–i Dual-gatemaps of longitudinal channel resistance Rxx at different intercalation distance d of0,0.34, 0.68, 4.99, and 6.16 nm, respectively. (j) The calculated interaction strengthη as a function of d for BLG/CrOCl (green solid circle and line) and bare BLG (greendashed line), respectively.Article https://doi.org/10.1038/s41467-023-37769-2Nature Communications |         (2023) 14:2136 6hetero-system. The origin of gap change by electric field can be ruledout. Since one can see that, in Fig. 4e–g, the channel resistance in theinsulating phase has reduced by more than 5 orders of magnitude.Roughly estimated by the thermal gap Δ∼ � kBT � logðIdsÞ, the gapchange is then about 3–4 times, way more than the value of a fewpercent induced from the change of displacement field due to the0.68 nm intercalation to the original thickness of more than 30 nmdielectric. Furthermore, by adding an h-BN layer in the effectivemodel, at D = 0.6 V/nm and Ls = 7 nm, the interaction strength η(defined as ratio between the e–e interaction strength and thebandwidth in BLG) is suppressed by increasing the lifted distance ofBLG above CrOCl, which is approaching to the same value as the h-BNencapsulated BLG at above 4.5 nm (Fig. 4j). Weaker e–e interactionsthus lead to smaller region of correlated insulating phase, in goodagreement with the tendency observed in our experiments. Weemphasize that, although themodel of long-wavelength chargeorderat the surface state in CrOCl can self-consistently address theexperimental observations, it is so far a theoretical hypothesis thatneeds further direct experimental evidence.CMOS-like graphene inverter based on tunable correlatedinsulatorBased on the gate tunable phase transition from metal to correlatedinsulator, one can obtain both P- and N-like metal oxide semi-conductor field effect transistor (MOS-FET) behaviors in the BLG/CrOCl systems in a specific gate range. Taking samples Device-7 andDevice-8 for example, as shown in Supplementary Fig. 12, the ON andOFF state can be out-of-phase when scanned along the dashed lines inthe two different samples. More specifically, by setting Vbg at -2.9 V inDevice-S26 (setting Vtg at +12.673 V in Device-S22), and scan Vtg (Vbg)in the range of 0 to 10 V, a PMOS-like (NMOS-like)field-effect curve canbe realized, as shown in Fig. 5a, b, respectively, with Vds set to be 5mV.Log scale plot of each curve is shown in their insets.One then can design a logic inverter out of the correlated insu-lator state, using two BLG/CrOCl devices, as illustrated in Fig. 5c (seemore details in Supplementary Figs. 12–13). The diagram of the BLG/CrOCl logic inverter is similar to a standard Si CMOS inverter, but twoextra setting voltages (VPset and VNset) are needed to maintain the shapeof the desired field-effect curves. Vdd denotes the supply voltage (i.e.,a V    (mV) I    (nA)bc de V  V VGND Vbg Vtg Vbg VtgDevice 1Device 2PMOS-likeNMOS-like V VoutinsetpsetNddoutdd120804001.20.80.40.0Gain800V   (V)I    (μA)dstg V    = -2.9 V bgPMOS-likeDevice S26I    (A)dsV   (V)tg V    = 5 mV ds2.01.51.00.50.010864202.510-1310-1210-1110-1010-910-810-710-6108642010-14I    (μA)dsV   (V)bg V    = 12.673 V tgNMOS-likeDevice S22V   (V)bgI    (A)ds V    = 5 mV ds10864200.80.60.40.20.01.010-1310-1210-1110-1010-910-810-710-61086420160 V    (V)in V    (V)in876543 V    = 150 mVdd2001501005001086420V    (V)inGain1.20.01000.6 T  = 1.5 K V    = 200 mVdd V    (V)in V    (V)in 1.5 10 20 30 40 50 60 70 80T (K) V    (mV)outfFig. 5 | CMOS-like inverter based on gate tunable correlated insulator.a, b PMOS- and NMOS-like field effect curves in the same gate range, swept alongdash lines in Supplementary Fig. 12a, b. Insetof each shows the log scale of the samedata. c Schematic pictureof the BLG/CrOClCMOS logic inverter.Here,VPset andVNsetdenote the setting voltages to maintain the shape of the desired P- or N-type field-effect curves. d The performance of a typical graphene inverter. e Source-draincurrent Idd flowing in the graphene inverter in d as a function of input voltageduring working. fOutput voltage Vout as a function of input voltage Vin at differenttemperatures, with the supply voltage Vdd fixed at 200 meV. Inset shows the gainfor each curves in f.Article https://doi.org/10.1038/s41467-023-37769-2Nature Communications |         (2023) 14:2136 7Vds in the previous conventions), and Vin is the input voltage of theinverter, which is sent to Vtg and Vbg for each device, as shown inFig. 5c. The performanceof such a typical BLGCMOS inverter at 1.5 K isshown in Fig. 5d, with a Vdd= 150mV maintained in the measurement.The output voltage Vout is identical to Vdd and flipped to zero at athreshold voltage of about 5.2 V, yielding a gain of about 1.1. During theworking process of the BLG CMOS-like inverter in Fig. 5d, a maximumIdd of about 120 nA was seen (Fig. 5e), corresponding to a power con-sumption of 18 nW. Figure 5f shows, in a typical sample at fixed Vdd,Vout as a function of Vin at different temperatures up to 80 K, with thevalues of gain for each curves indicated in the inset of Fig. 5f. Morecharacterizations of temperature dependence and temporal dynamicsof such logic devices can be found in Supplementary Figs. 14–15.In conclusion, we have designed a hybrid system with AB-stackedBLG interfaced with an antiferromagnetic insulator CrOCl. An insulat-ing phase (with sheet resistanceR□ > 1GΩ) at a largely distorted CNP isfound in the dual gate mapping of channel resistance, which is mark-edly distinct from the known picture in conventional BLG. The sim-plified electro-static model suggests that the vertical electric field inBLG is inferior to those found in conventional BLG, which howeverdoes not explain the enhanced insulating behavior within the single-particle picture. Systematic transport measurements suggest that thisheterostructure enables the coupling between the interfacial states inCrOCl and the BLG, which further allows the enhancement of elec-tronic interaction in BLG. It hence triggers a crossover from conven-tional single particle insulating behavior to the e–e interactionenhanced quantum insulator at charge neutrality, with a gate-tunableTinsulator, as further confirmed by theoretical modelings. Such corre-lated insulating ground state can be switched into ametallic state withan on/off ratio up to 107, by applying an in-plane electric field, heating,or by electrostatic gating,which is unusual in all known carbon species.Demonstration of a logic circuit using such quantum insulating statesis also shown. Our results shed lights on a tuning knob, i.e., interfacialcharge order coupling, for engineering future quantum electronicstate in 2D electron gases in vdW heterostructures.MethodsSample fabrications and characterizationsThe CrOCl/bilayer-graphene/h-BN heterostructures were fabricated inambient conditions using the dry-transfer method, with the flakesexfoliated from high-quality bulk crystals. CrOCl bulk crystals weregrown via a chemical vapor transport method. Thin CrOCl layers werepatterned using an ionmilling with Ar plasma, and dual-gated samplesare fabricated using standard e-beam lithography (Zeiss Sigma300 +Raith ELPHY Quantum). A Bruker Dimension Icon atomic forcemicroscope was used for thicknesses and morphology tests. Theelectrical performances of the devices were measured using a OxfordTeslaTron with a base temperature of 1.5 K and a superconductingmagnet of 12 T maximum. A probe station (Cascade Microtech Inc.EPS150) is used for room temperature electrical tests. For AC mea-surements, StandfordSR830 lock-in amplifierswereused at 17.77 Hz toobtain 4-wire resistances, in constant-current configuration with a100MΩ AC bias resistor. For DC measurements, we used Keithley2636B multimeters for high precision current measurements, andKeithley 2400 source meters for providing gate voltages. The STEMand EDS investigations were conducted using a double aberrationcorrected FEI ThemisG260–300electronmicroscope equippedwith aSuperX-EDS detector and operated at 300 kV.Effective Hamiltonian formalismA simplified effective Hamiltonian of BLG coupled with a superlatticepotential, capturing the low-energyphysicsof our system, is proposed.Since Ls is much larger than the lattice constant of graphene, we couldthus safely omit the intervalley coupling and model graphene as twoseparate continua of Dirac fermions from two valleys. Explicitly, theHamiltonian in a given valley τ readsHτeff =Hτ0 +UdðrÞ ð1Þwhere τ = ± indicates the valley K=K 0, respectively, and Hτ0 is the non-interacting k ⋅p Hamiltonian for AB-stacked bilayer grapheneexpanded around the valley τ31. In the layer-sublattice basisð∣1,Ai,∣1,Bi,∣2,Ai,∣2,BiÞ, Hτ0 reads (ℏ = 1)Hτ0ðqÞ=Δe vFQ� �v?Q� t?vFQ + Δe �v?Q+ �v?Q��v?Q+ �v?Q� �Δe vFQ�t? �v?Q + vFQ� �Δe0BBB@1CCCA ð2Þwhere Q− = τqx − iqy, Q+ = τqx + iqy, and the sublattice A of Layer 1 is onthe top of the sublattice B of Layer 2, Δe is the potential differencebetween two layers of graphene in the presence of out-of-planeelectric field and all other parameters are given by the Slater-Kostertransfer integral32,33.The background superlattice potential Ud(r) has a periodUd(r) =Ud(r +Rs). The superlattice vector Rs is assumed to be com-mensurate with the atomic rectangular lattice of CrOCl, but is sig-nificantly enlarged due to the low carrier density. Technically, thespacing between Cr atoms and graphene is found to be d = 7Å from aDFT lattice relaxation study16. We assume that the coupling is only viathe long-ranged Coulomb interactions, i. e., neglecting the couplingfromorbital overlaps suchas interlayer hoppings, which is screenedbydielectric constants εd = 4. The superlattice constant is set tobe aroundLs = 5 nm corresponding to doping of 6.7 × 1012 cm−2 in the interfacialband. Suppose that Layer 1 is closer to theCrOCl substrate thanLayer 2due to the interlayer distance between two layers is d0 = 3.35 Å. Then,the magnitude of the superlattice potential affects stronger Layer 1than Layer 2. In particular, the homogeneous contributions in Ud(r),i.e., the Fourier component ~UdðG =0Þ, give rise to an electrostaticpotential differenceexactly ashomogeneousout-of-plane electricfieldso that it is absorbed in Δe. Furthermore, our electrostatic modelexplicitly takes into account this part and encodes it into the effectivedisplacement field Deff. Therefore, the Fourier component ~UdðG =0Þ isincluded in Δe in our formalism.As a result, the underlying superlattice would fold Dirac conesinto its small Brillouin zone forming subbands. The degeneracy pointsdue to the folding are gapped out by~UdðG ≠0Þ= e2ϵ0ϵdΩ0� e�GdiGð3Þwhere Ω0 = LxLy is the area of the primitive cell of the superlattice, thedistance between CrOCl and graphene sheets d1 = d for Layer 1 andd2 = d + d0 for Layer 2.When Deff = 0.6 V/nm, i. e., Δe = 25 meV for εd = 4, layer polariza-tion due to an out-of-plane electric field opens a gap31Δ=2Δet?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4Δ2e + t2?q ≈ 2Δe ð4Þat finite momentumq=ΔevFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4Δ2e +2t2?4Δ2e + t2?s≈ 0 ð5Þknowing that t⊥≫Δe.Hartree-Fock calculationsAdouble-gate screenedCoulomb interactionwith a dielectric constantεr = 4 and the thickness of the device ds = 400 Å are used in the model.Article https://doi.org/10.1038/s41467-023-37769-2Nature Communications |         (2023) 14:2136 8The Coulomb interactions are written in the subband eigenfunctionbasis. As interaction effects are most prominent around the CNP, weproject the Coulomb interactions onto only a low-energy windowincluding three valence and three conduction subbands that are clo-sest to the Dirac point for each valley and spin. We use a mesh of18 × 18 k-points to sample the mini Brillouin zone of the superlattice.To incorporate the influences of Coulomb interactions from the high-energy remote bands, we rescale the Fermi velocity vF and the inter-layer hopping t⊥ within the low-energy window of the effectiveHamiltonian using the formula derived from the RG approach16,34,35.Note that the ratio vF/t⊥ remains unchanged after the RG procedure.We keep other parameters of the non-interacting effective Hamilto-nian unchanged since their RG correction is of higher order, thus canbe neglected. Feeding with the initial conditions in the form of orderparameters, we self-consistently obtain the gap at the CNP and thesingle-particle excitation spectrum.Data availabilityThedata that support thefindings of this study areavailable atZenodo,https://doi.org/10.5281/zenodo.6569307.Code availabilityThe codes used in theoretical simulations and calculations are avail-able from the corresponding authors upon request.References1. E. Feldman, B. E., Martin, J. & Yacoby, A. Broken-symmetry statesand divergent resistance in suspended bilayer graphene.Nat. Phys.5, 889–893 (2009).2. Maher, P. et al. Tunable fractional quantum hall phases in bilayergraphene. Science 345, 61–64 (2014).3. Zhou,H.et al. Isospinmagnetismandspin-polarizedsuperconductivityin Bernal bilayer graphene. Science 375, 774–778 (2022).4. 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).5. Weitz, R. T., Allen, M. 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Quantum valley hall effect, orbitalmagnetism, and anomalous hall effect in twisted multilayer gra-phene systems. Phys. Rev. X 9, 031021 (2019).34. Gonzalez, J., Guinea, F. & Vozmediano, M. A. H. Non-Fermi liquidbehavior of electrons in the half-filled honeycomb lattice (a renor-malization group approach). Nucl. Phys. B 424, 595–618 (1994).35. Vafek, O. & Kang, J. Renormalization group study of hidden sym-metry in twisted bilayer graphene with coulomb interactions. Phys.Rev. Lett. 125, 257602 (2020).AcknowledgementsThis work is supported by the National Key R&D Program of China withGrants. 2019YFA0307800, 2017YFA0206301, 2018YFA0306900,2019YFA0308402, and 2018YFA0305604. The authors acknowledgesupport from the National Natural Science Foundation of China (NSFC)with Grants 92265203, 11974357, U1932151, 11934001, 11774010,92265106, and 11921005. Thegrowthof hexagonal boronnitride crystalswas supported by the Elemental Strategy Initiative conducted by theMEXT, Japan, Grant Number JPMXP0112101001, JSPS KAKENHI GrantNumber JP20H00354 and A3 Foresight by JSPS. Jian-Hao Chenacknowledges support from Beijing Municipal Natural ScienceArticle https://doi.org/10.1038/s41467-023-37769-2Nature Communications |         (2023) 14:2136 9https://doi.org/10.5281/zenodo.6569307http://arxiv.org/abs/2206.05659http://arxiv.org/abs/2206.05659http://arxiv.org/abs/2302.07543Foundation (Grant No. JQ20002) and the technical support from PekingNanofab.Author contributionsZ.H. and Y.Y. conceived the experiment and supervised the overallproject. K.Y., X.G., and Y.W. carried out device fabrications; X.G., K.Y.,Y.W., T.Z., P.G., X.S., R.Z., S.C., J.-H.C., Y.Y., andZ.H. carriedout electricaltransport measurements; P.G. and Y.Y. performed synthesis of bulkCrOCl crystals; Z.L., H.W., and X.L. (Xiuyan Li) carried out TEM char-acterizations; K.W. and T.T. provided high-quality h-BN bulk crystals.Z.H., Y.Y., J.Z., J.L., andX.D. analyzed the experimental data. X.L. (Xin Lu),S.Z., and J.L. performed effective Hamiltonian and RG+HF calculations.X.D. and Y.G. carried out the electrostatic modelings. The manuscriptwas written by Z.H., J.L., and X.L. (Xin Lu) with discussion and input fromall authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-37769-2.Correspondence and requests formaterials should be addressed to XinLu, Xi Dai, Jian-Hao Chen, Yu Ye or Zheng Han.Peer review information Nature Communications thanks the anon-ymous reviewer(s) 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) 2023Article https://doi.org/10.1038/s41467-023-37769-2Nature Communications |         (2023) 14:2136 10https://doi.org/10.1038/s41467-023-37769-2http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Unconventional correlated insulator in CrOCl-interfaced Bernal bilayer graphene Results Insulating behaviors in BLG/CrOCl hetero-system Extracting the single particle gap at the CNP in BLG/CrOCl Ruling out the possibility of a band insulator Coulomb interaction augmented correlated insulator CMOS-like graphene inverter based on tunable correlated insulator Methods Sample fabrications and characterizations Effective Hamiltonian formalism Hartree-Fock calculations Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information