Abstract
The realization of graphene gapped states with large on/off ratios over wide doping ranges remains challenging. Here, we investigate heterostructures based on Bernalstacked bilayer graphene (BLG) atop fewlayered CrOCl, exhibiting an over1GΩresistance insulating state in a widely accessible gate voltage range. The insulating state could be switched into a metallic state with an on/off ratio up to 10^{7} by applying an inplane electric field, heating, or gating. We tentatively associate the observed behavior to the formation of a surface state in CrOCl under vertical electric fields, promoting electron–electron (e–e) interactions in BLG via longrange Coulomb coupling. Consequently, at the charge neutrality point, a crossover from single particle insulating behavior to an unconventional correlated insulator is enabled, below an onset temperature. We demonstrate the application of the insulating state for the realization of a logic inverter operating at low temperatures. Our findings pave the way for future engineering of quantum electronic states based on interfacial charge coupling.
Introduction
ABstacked BLG hosts fascinating emerging physics and can be a building block for intriguing nanoelectronics^{1,2,3,4,5,6,7}. In the singleparticle picture, when subjected to vertical electric fields, Bernalstacked BLG yields a layerpolarized gap at charge neutrality, which is tunable and reaches about 250 meV in an experimentally applicable maximum displacement field of about 3 V/nm^{8}. However, the corresponding resistances are usually peaked in a very small doping range^{5,9,10,11,12}, making it limited for further explorations in such gapped states.
On the other hand, charge neutral BLG is strongly susceptible to Coulomb interactions and is predicted to exhibit ground states with spontaneous symmetry breaking^{6,13,14}. Ultraclean BLG samples under vertical electric fields are often demanded to observe such unconventional insulating states, yet within narrowly distributed parameter spaces^{6}. To favor e–e interactions, recent attempt was also devoted to such as moiré double bilayer graphene, where excitonic insulating behaviors are seen at charge neutrality due to the overlapping electron and hole pockets at different wave vector k^{15}.
Here, we attempt to devise a different route to identify an unconventional correlated insulating phase in BLG atop few layered CrOCl. It is known that interaction effects are already manifested in the scenario of monolayer graphene/CrOCl heterostructure, where a reconstruction of the Dirac dispersion induced by e–e interaction is the most prominent effect^{16,17,18,19}. In the case of BLG, the quadratic bands around the charge neutrality endow larger density of states at the Fermi level, which are expected to have more dramatic correlation effects, as such evidence can be found in freestanding ultraclean BLG^{6}. Here, in a hBNBLGCrOCl device, the interaction effects in BLG are further enhanced by the interfacial coupling to a presumably longwavelength charge order at the surface of CrOCl, leading to gatetunable correlated gap in neutral BLG with a sheet resistance larger than GΩ within a much expanded effective gate range. This observation is markedly different compared to conventional dualgated BLG systems. Nevertheless, the displacement field D in BLG in the current system is estimated to be at the order of ~ 1 V/nm, similar to those found in conventional BLG samples. Systematic transport measurements together with theoretical modeling selfconsistently suggest that this insulating ground state can neither be explained by localization nor be trivially categorized into a band insulator. As a result, the charge coupling from the surface charge order triggered in CrOCl is the key ingredient to drive the crossover from singleparticle insulating phase to a correlated insulator in neutral BLG, with a maximum onset temperature T_{insulator} of a full insulating state (with the zero biased conductance reaching the noise level) seen at about 40 K. The wide gate range of the insulating phase further allows the demonstration of a logic inverter using BLG/CrOCl devices. Our findings pave the way for the engineering of interfacial coupling between 2D electron gases in van der Waals heterostructure, which may be expanded to a broader library of materials.
Results
Insulating behaviors in BLG/CrOCl heterosystem
Bernalstacked BLG, thin CrOCl flakes, and encapsulating hexagonal boron nitride (hBN) flakes were exfoliated from highquality bulk crystals and stacked in ambient conditions using the dry transfer method^{20}, with more detail of the fabrication process and sample morphologies described in Supplementary Figs. 1–2. We cooled down the samples to a base temperature of 1.5 K, and measured the longitudinal channel resistance R_{xx} as a function of top gate V_{tg} and bottom gate V_{bg}, as shown in Fig. 1a. Two key observations are to be understood in Fig. 1a, i.e., the extremely wide gate range of an insulating region that reaches 10^{9}Ω, and the largely bent phase boundaries of the gapped state, which is markedly different from the charge neutrality point (CNP) resistive peaks found in conventional ultraclean BLG samples^{5,6,8,9,10,11,12}. A sidebyside comparison of the gapped states in hBN/BLG/CrOCl and hBN/BLG/hBN heterosystems can be seen in Supplementary Fig. 3 and Supplementary Table 1.
In the following, we define total carrier density n_{tot} = (C_{tg}V_{tg} + C_{bg}V_{bg})/e − n_{0}, and the effective displacement field D_{eff} = (C_{tg}V_{tg} − C_{bg}V_{bg})/2ϵ_{0} − D_{0}, where C_{tg} and C_{bg} are the top and bottom gate capacitances per area, and n_{0} and D_{0} are residual doping and residual displacement field, respectively. Figure 1b plots a line profile (along the black dashed line in Fig. 1a, with a fixed D_{eff} = 0.4 V/nm). It shows that the channel resistance can be tuned from a few hundred Ω into an OFF state by gating, with ONOFF ratios reaching 10^{7}. By examining multiple devices, we exclude the possibility of gate leakages (meaning that the bulk CrOCl itself is always insulating and does not contribute to transport throughout the measurements) or impuritydominated parasitic effects for this observed unconventional gapped sates, shown in Supplementary Figs. 4–5. Atomic resolution of the crosssection of a typical heterostructure can be seen in the highangle annular darkfield scanning transmission electron microscopy (HAADFSTEM) image in the inset in Fig. 1b, showing a clean interface between the layered compounds.
For a typical V_{bg} and V_{tg} which correspond to the black starred point in Fig. 1a, we performed the dI/dV (differential conductance obtained by differentiating the DC I–V curves) as a function of bias voltage V_{bias} at different temperatures, as shown in Fig. 1c. It displays that the lowbiased insulating phase with negligible conductance can be killed at both high temperatures and high bias voltages. Line profiles along dashed lines in Fig. 1c are shown in Fig. 1d, which illustrate dI/dV versus temperature in the insulating state (V_{bias} = 0 V) and the metallic state with conductance at the order of mS (V_{bias} = 300 mV), respectively. Interestingly, a drastic drop in zerobiased dI/dV with the onset temperature T_{insulator} of a full insulating state reaching the noise level is seen at about 35 K, indicated by the solid arrow. Similar dI/dV curves were also seen in spontaneous symmetry breaking states in suspended ultraclean BLG owning to electronic correlation^{6}. The latter, however, was at a much lower T_{insulator} (below 10 K) and lack of a full study in the parameter space of n_{tot}D_{eff}.
Extracting the single particle gap at the CNP in BLG/CrOCl
To understand the observed insulating state in our system, we first determine the nature of doping in the insulating region by measuring the device in the quantum Hall limit, so that the exact doping in the bilayer graphene n_{BLG} can be deduced by the filling fractions ν = n_{BLG}h/eB of each Landau levels (LLs), where h is Planck constant, e is elementary charge, and B is the perpendicular magnetic field. Figure 2a shows longitudinal conductivity \({\sigma }_{{{{{{{{\rm{xx}}}}}}}}}=\frac{{R}_{{{{{{{{\rm{xx}}}}}}}}}}{({R}_{{{{{{{{\rm{xx}}}}}}}}}^{2}+{R}_{{{{{{{{\rm{xy}}}}}}}}}^{2})}\) at B = 12 T and T = 1.5 K of the same sample as in Fig. 1. It shows that, compared to that seen in conventional BLG cases, the BLG/CrOCl system exhibits distinct features of LLs, with a crossover from straight stripes to cascadeslike bent stripes as n_{tot} is varied from negative to positive in general. Figure 2b shows line profiles along the dashed line in Fig. 2a (more data can be found in Supplementary Figs. 6–7), indicating that n_{BLG} in the insulating states corresponds to a filling fraction of ν = 0 (yellow shadowed area in Fig. 2b), i. e., the charge neutrality. Meanwhile, full degeneracy lifting can be seen at each integer filling fractions from ν = −1 to −10. This speaks the high quality of BLG itself.
To further clarify the observed bent CNP, we set up a simplified electrostatic model (see also Supplementary Note 1) to introduce an extra capacitance induced by the interfacial band with density of states n_{2} that is very close to the surface of CrOCl (with a distance d_{2}). While top and bottom gates are located at distances of d_{1} and d_{3}, respectively, illustrated in the cartoon image and capacitance model in Fig. 2c, d, respectively. The potential on the interfacial state is defined as V_{2}, and dielectric constants ε_{i} (i = 1, 2, 3) are assigned to each of the capacitor. By evaluating the electrostatic model using Gauss’s law, we found that the abovementioned two major experimental observations can be well reproduced, as shown in the phase diagram in Fig. 2e, where the isodoping lines of n_{BLG} are highlighted in the n_{tot}D_{eff} space. A clear phase boundary is indicated by the white dashed line, separating the Phasei (conventional phase with n_{2} = 0, and n_{BLG} is D_{eff}independent) and Phaseii (interfacial coupling phase with n_{2} > 0, i. e., interfacial band is filled with electrons via tunneling from the BLG, where n_{BLG} depends on both D_{eff} and n_{tot}). According to the above analysis, a diagram showing typical transition process in our system from Phasei to Phaseii is illustrated in Supplementary Fig. 8.
Moreover, as discussed in Supplementary Note 1, by quantifying the average vertical electric field in the BLG, isoD_{BLG} lines can also be plotted in Fig. 2f. In general, due to the existence of n_{2}, electric fields in BLG is bent toward lower D_{eff} in Phaseii as compared to that in Phasei. As a result, the calculated D_{BLG} at charge neutrality in Phaseii is in the range of 0.5 to 1.3 V/nm, similar to that estimated in conventional hBN/BLG/hBN cases, which in the singleparticle picture corresponds to the gap size of about 50–130 meV^{8}.
Temperature dependence (Supplementary Fig. 9) shows that the insulating phase weakens upon heating, and the resistance remains at the order of MΩ at 80 K. As plotted in Fig. 2g, we tracked 6 typical points at the resistance maxima in the dualgate map at B = 0 and T = 80 K, as shown in Supplementary Fig. 10. The I_{ds}T^{−1} curves for these 6 points are obtained using DC 2probe measurement with a fixed V_{bias} = 5 mV. The thermal activation gaps (defined as \({I}_{{{{{{{{\rm{ds}}}}}}}}}\propto \exp (\Delta /2{k}_{{{{{{{{\rm{B}}}}}}}}}T)\)) of each curve are then extracted to be from 17.41 meV to 70.04 meV, with each corresponding n_{tot} calculated from their gate voltages (Supplemental Fig. 10). These measured thermal gap sizes at the temperature range of 40 to 100 K are in good agreement with those expected from the layerpolarized gaps induced by the displacement fields at charge neutrality shown in Fig. 2f. However, such a singleparticle gap picture contradicts the experimentally observed unconventional insulating behaviors such as the peculiar I–V curves and the onset temperature T_{insulator} characteristics as shown in Fig. 1c, d.
Ruling out the possibility of a band insulator
We can rule out the observed insulating behavior from the possibility of being a trivial band insulator. First, as discussed in Fig. 1c, in I–V curves, the gapped state exhibits below a critical sourcedrain voltage V_{C} (indicated as switching voltage in Fig. 1c), beyond which the insulating state breaks down and turns into a normal metallic state with resistances of a few kΩ. Moreover, critical behavior is also observable as a function of temperature, as a drastic drop of zerobias conductance occurs at T_{insulator}, indicated by the solid arrow in Fig. 1d. Meanwhile, at high bias voltage, the system is showing only metallic state. Such behavior could be a hallmark that differs from trivial band insulators, with more evidence discussed in the coming text.
In the following, the V_{bias} is converted into an Lindependent inplane electric field \({\overrightarrow{E}}_{\parallel }={V}_{{{{{{{{\rm{ds}}}}}}}}}/L\), since V_{C} is proportional to the distance between electrodes L (Supplementary Fig. 11), as illustrated in Fig. 3a. Figure 3b displays the dualgate maps of channel resistance at different V_{bias}. It shows that, with increasing \({\overrightarrow{E}}_{\parallel }\) from bottom to top, a significant portion of the insulating region is switched to metallic states. This suggests that increasing \({\overrightarrow{E}}_{\parallel }\) has an effect that is similar to that of either gating or temperature on the insulating state, indicating a continuously tunable phase transition. The lower edges of the insulating phase at CNP are plotted in Fig. 3c, with the I–V spectra at typical points (indicated by colored solid circles along the dashed line of n_{tot} = 0.5 × 10^{13} cm^{−2}) measured in a traceretrace manner, shown in Fig. 3d.
Importantly, as plotted in Supplementary Fig. 11c, the insulator breakdown electrical field \({\overrightarrow{E}}_{{{{{{{{\rm{C}}}}}}}}}\) (defined as the \({\overrightarrow{E}}_{\parallel }\) at V_{C}) as a function of 1/L clearly shows a trend that does not extrapolate to zero as 1/L → 0, which is clearly different from Zenertype breakdown (\({\overrightarrow{E}}_{{{{{{{{\rm{C}}}}}}}}}\to\) 0 as 1/L → 0) for a band insulator. This is likely a characteristic behavior originated from the pairbreaking mechanism for a Coulomb induced excitonic insulator described in a recent theoretical model^{21}. It is also worth noting that the mesoscopic samples studied in our work exhibit insulating breakdown at \({\overrightarrow{E}}_{{{{{{{{\rm{C}}}}}}}}} \sim 1{0}^{5}\) V/m, orders of magnitudes smaller than the values of Zener breakdown for band insulators ( ~ 10^{7} V/m for a gap of about 0.1 meV, which has also a sizeindependent characteristic, largely distinct from the Ldependent small \({\overrightarrow{E}}_{{{{{{{{\rm{C}}}}}}}}}\) observed in this work)^{22}, in agreement with the theoretical model in the limit of small L^{21}. We have to emphasize that other experimental factors and/or sample details, including charge disorder concentration, metaltographene contact, and fringe electric fields, are not taken into account in the theoretical model^{21}.
We now consider the T_{insulator} of the insulating state at the phase boundary with a finite \({\overrightarrow{E}}_{\parallel }\)=100 kV/m applied to the ground state, shown in the inset in Fig. 3e. At several typical points (Points AF) along the phase edge, zerobiased differential conductances dI/dV were recorded as a function of temperature, shown in Fig. 3e. We see trends of drastic drop of differential conductance as the temperature is lowered, which can be explained as a crossover of the gap nature from the singleparticle picture at high temperatures to correlation dominant type at low temperatures. Moreover, the T_{insulator} (from ~ 10 K at PointA to ~ 40 K at PointF) is readily tunable by gating.
In addition, it is notice that such switchinglike (sometimes asymmetric) IV characteristic found in the current system is only reported in a few systems such as BCS superconductors^{23,24}, some Mott insulator systems^{25,26}, and, perhaps most pertinently, the 2D quantum electron crystals^{27}. Indeed, one can further see clear hysteresis in the IV curves (indicated by solid arrows in Fig. 3d), especially at relatively low values of D_{eff}. To this point, we can see that in the observed IV curves, gatetunable T_{insulator} below which a zerobiased conductance reaches the zero limit (while the system exhibits a metallic behavior at high bias above \({\overrightarrow{E}}_{{{{{{{{\rm{C}}}}}}}}}\)), relatively small \({\overrightarrow{E}}_{{{{{{{{\rm{C}}}}}}}}}\), together with a plausible pairbreaking mechanism for the inplane electrical field breakdown of the insulator, all point to a correlated insulator behavior. At last, we would like to mention that although magnetic phase transition occurs in CrOCl at around 27 K (also structural phase transition at Néel temperature of ~ 14 K)^{28,29,30}, we found no connection between the observed insulating behavior and the magnetism of CrOCl since insulating features prevail up to 80 K.
Coulomb interaction augmented correlated insulator
We have understood that the actual electric field in the BLG/CrOCl heterosystem is in the same order as those reported in conventional BLG samples. Yet, the neutral BLG in the interfacial coupling phase (CNP in Phaseii) has a much larger gap under such magnitude of electric fields, and correlation behaviors of the insulating state at the CNP are revealed in transport measurements. To further elucidate the physical origin of the correlated insulator, we now consider theoretical modelings of the current system. At certain vertical electric fields, the interfacial band from CrOCl (mainly from the 3d orbital of top Cr atoms) starts to overlap with the Fermi level of BLG^{16,17}, this triggers charge transfer (via tunneling) from BLG to the interfacial band, as indicated in the band alignment diagram in Fig. 4a. Calculations^{16} suggest that longwavelength charge order should appear in the interfacial states in the top layer of CrOCl due to e–e interaction and does not contribute to transport, illustrated in the cartoon in Fig. 4b.
The length scale of such charge order L_{s} (~5 nm, inversely proportional to the square root of the surface electronic density in CrOCl surface bands) is much larger than the carboncarbon bond distance in bilayer graphene. Thus, if we are only interested in the lowenergy physics in BLG, the charge degrees of freedom in the longwavelength charge order on the CrOCl side can be integrated out to provide an effective background superlattice potential (arising from the Coulomb potentials of the charge order in the CrOCl substrate) for electrons in BLG, as revealed in the scenario of monolayer grapheneCrOCl heterostructure^{16,17}.
Based on the effective model, we can then investigate the e–e interaction effects in the bilayer grapheneCrOCl heterostructure using the renormalization group (RG) assisted HartreeFock (HF) approximations^{16}, as described in the Methods section. Clearly, the gap at Dirac point at certain vertical displacement field and a fixed L_{s}, is significantly magnified by interactions compared to the noninteracting case 2Δ_{e}. Taking D = 0.6 V/nm (corresponding to a 2Δ_{e} ~ 50 meV) and L_{s} = 5 nm for example, Fig. 4c juxtaposes the band structures of BLG obtained from the RGHF calculations (solid lines) and that in the noninteracting case (dashed lines). Moreover, we plot in Fig. 4d the interaction gap at the Dirac point Δ_{HF} as a function of the doping of interfacial state n_{CrOCl} at D = 0.6 V/nm, with a clear enhancement of Δ_{HF} with respect to 2Δ_{e} upon increasing n_{CrOCl}.
To check whether the gap enhancement results indeed from the superlatticepromoted e–e interaction effects, a straightforward means is to lift the height of BLG above the underneath CrOCl surface potential, which will weaken the longranged interlayer Coulomb coupling between BLG and n_{CrOCl} in an exponentially decaying manner. Indeed, by intercalating a monolayer of hBN in the interface of BLG/CrOCl, it appears a significant weakening of the insulating state at CNP, as shown in Fig. 4e, f. Further increase of the thickness of intercalation hBN, the insulating phase gradually fades away, and the bent CNP is recovered in the dual gate map, while the behavior of channel resistance for conventional hBN/BLG/hBN is fully restored with an intercalation of above 4.99 nm, as shown in Fig. 4g–i. This is a strong support to the previously discussed picture for the origin of the correlated insulator ground state at CNP in BLG/CrOCl heterosystem. The origin of gap change by electric field can be ruled out. Since one can see that, in Fig. 4e–g, the channel resistance in the insulating phase has reduced by more than 5 orders of magnitude. Roughly estimated by the thermal gap \(\Delta \sim {k}_{{{{{{{{\rm{B}}}}}}}}}T\cdot \log ({I}_{{{{{{{{\rm{ds}}}}}}}}})\), the gap change is then about 3–4 times, way more than the value of a few percent induced from the change of displacement field due to the 0.68 nm intercalation to the original thickness of more than 30 nm dielectric. Furthermore, by adding an hBN layer in the effective model, at D = 0.6 V/nm and L_{s} = 7 nm, the interaction strength η (defined as ratio between the e–e interaction strength and the bandwidth in BLG) is suppressed by increasing the lifted distance of BLG above CrOCl, which is approaching to the same value as the hBN encapsulated BLG at above 4.5 nm (Fig. 4j). Weaker e–e interactions thus lead to smaller region of correlated insulating phase, in good agreement with the tendency observed in our experiments. We emphasize that, although the model of longwavelength charge order at the surface state in CrOCl can selfconsistently address the experimental observations, it is so far a theoretical hypothesis that needs further direct experimental evidence.
CMOSlike graphene inverter based on tunable correlated insulator
Based on the gate tunable phase transition from metal to correlated insulator, one can obtain both P and Nlike metal oxide semiconductor field effect transistor (MOSFET) behaviors in the BLG/CrOCl systems in a specific gate range. Taking samples Device7 and Device8 for example, as shown in Supplementary Fig. 12, the ON and OFF state can be outofphase when scanned along the dashed lines in the two different samples. More specifically, by setting V_{bg} at 2.9 V in DeviceS26 (setting V_{tg} at +12.673 V in DeviceS22), and scan V_{tg} (V_{bg}) in the range of 0 to 10 V, a PMOSlike (NMOSlike) fieldeffect curve can be realized, as shown in Fig. 5a, b, respectively, with V_{ds} set to be 5 mV. Log scale plot of each curve is shown in their insets.
One then can design a logic inverter out of the correlated insulator state, using two BLG/CrOCl devices, as illustrated in Fig. 5c (see more details in Supplementary Figs. 12–13). The diagram of the BLG/CrOCl logic inverter is similar to a standard Si CMOS inverter, but two extra setting voltages (\({V}_{{{{{{{{\rm{set}}}}}}}}}^{{{{{{{{\rm{P}}}}}}}}}\) and \({V}_{{{{{{{{\rm{set}}}}}}}}}^{{{{{{{{\rm{N}}}}}}}}}\)) are needed to maintain the shape of the desired fieldeffect curves. V_{dd} denotes the supply voltage (i.e., V_{ds} in the previous conventions), and V_{in} is the input voltage of the inverter, which is sent to V_{tg} and V_{bg} for each device, as shown in Fig. 5c. The performance of such a typical BLG CMOS inverter at 1.5 K is shown in Fig. 5d, with a V_{dd }= 150 mV maintained in the measurement. The output voltage V_{out} is identical to V_{dd} and flipped to zero at a threshold voltage of about 5.2 V, yielding a gain of about 1.1. During the working process of the BLG CMOSlike inverter in Fig. 5d, a maximum I_{dd} of about 120 nA was seen (Fig. 5e), corresponding to a power consumption of 18 nW. Figure 5f shows, in a typical sample at fixed V_{dd}, V_{out} as a function of V_{in} at different temperatures up to 80 K, with the values of gain for each curves indicated in the inset of Fig. 5f. More characterizations of temperature dependence and temporal dynamics of such logic devices can be found in Supplementary Figs. 14–15.
In conclusion, we have designed a hybrid system with ABstacked BLG interfaced with an antiferromagnetic insulator CrOCl. An insulating phase (with sheet resistance R_{□} > 1 GΩ) at a largely distorted CNP is found in the dual gate mapping of channel resistance, which is markedly distinct from the known picture in conventional BLG. The simplified electrostatic model suggests that the vertical electric field in BLG is inferior to those found in conventional BLG, which however does not explain the enhanced insulating behavior within the singleparticle picture. Systematic transport measurements suggest that this heterostructure enables the coupling between the interfacial states in CrOCl and the BLG, which further allows the enhancement of electronic interaction in BLG. It hence triggers a crossover from conventional single particle insulating behavior to the e–e interaction enhanced quantum insulator at charge neutrality, with a gatetunable T_{insulator}, as further confirmed by theoretical modelings. Such correlated insulating ground state can be switched into a metallic state with an on/off ratio up to 10^{7}, by applying an inplane electric field, heating, or by electrostatic gating, which is unusual in all known carbon species. Demonstration of a logic circuit using such quantum insulating states is also shown. Our results shed lights on a tuning knob, i.e., interfacial charge order coupling, for engineering future quantum electronic state in 2D electron gases in vdW heterostructures.
Methods
Sample fabrications and characterizations
The CrOCl/bilayergraphene/hBN heterostructures were fabricated in ambient conditions using the drytransfer method, with the flakes exfoliated from highquality bulk crystals. CrOCl bulk crystals were grown via a chemical vapor transport method. Thin CrOCl layers were patterned using an ion milling with Ar plasma, and dualgated samples are fabricated using standard ebeam lithography (Zeiss Sigma300 + Raith ELPHY Quantum). A Bruker Dimension Icon atomic force microscope was used for thicknesses and morphology tests. The electrical performances of the devices were measured using a Oxford TeslaTron with a base temperature of 1.5 K and a superconducting magnet of 12 T maximum. A probe station (Cascade Microtech Inc. EPS150) is used for room temperature electrical tests. For AC measurements, Standford SR830 lockin amplifiers were used at 17.77 Hz to obtain 4wire resistances, in constantcurrent configuration with a 100 MΩ AC bias resistor. For DC measurements, we used Keithley 2636B multimeters for high precision current measurements, and Keithley 2400 source meters for providing gate voltages. The STEM and EDS investigations were conducted using a double aberration corrected FEI Themis G2 60–300 electron microscope equipped with a SuperXEDS detector and operated at 300 kV.
Effective Hamiltonian formalism
A simplified effective Hamiltonian of BLG coupled with a superlattice potential, capturing the lowenergy physics of our system, is proposed. Since L_{s} is much larger than the lattice constant of graphene, we could thus safely omit the intervalley coupling and model graphene as two separate continua of Dirac fermions from two valleys. Explicitly, the Hamiltonian in a given valley τ reads
where τ = ± indicates the valley \(K/{K}^{{\prime} }\), respectively, and \({H}_{0}^{\tau }\) is the noninteracting k ⋅ p Hamiltonian for ABstacked bilayer graphene expanded around the valley τ^{31}. In the layersublattice basis \((\left1,A\right\rangle,\left1,B\right\rangle,\left2,A\right\rangle,\left2,B\right\rangle )\), \({H}_{0}^{\tau }\) reads (ℏ = 1)
where Q_{−} = τq_{x} − iq_{y}, Q_{+} = τq_{x} + iq_{y}, and the sublattice A of Layer 1 is on the top of the sublattice B of Layer 2, Δ_{e} is the potential difference between two layers of graphene in the presence of outofplane electric field and all other parameters are given by the SlaterKoster transfer integral^{32,33}.
The background superlattice potential U_{d}(r) has a period U_{d}(r) = U_{d}(r + R_{s}). The superlattice vector R_{s} is assumed to be commensurate with the atomic rectangular lattice of CrOCl, but is significantly enlarged due to the low carrier density. Technically, the spacing between Cr atoms and graphene is found to be d = 7 Å from a DFT lattice relaxation study^{16}. We assume that the coupling is only via the longranged Coulomb interactions, i. e., neglecting the coupling from orbital overlaps such as interlayer hoppings, which is screened by dielectric constants ε_{d} = 4. The superlattice constant is set to be around L_{s} = 5 nm corresponding to doping of 6.7 × 10^{12} cm^{−2} in the interfacial band. Suppose that Layer 1 is closer to the CrOCl substrate than Layer 2 due to the interlayer distance between two layers is d_{0} = 3.35 Å. Then, the magnitude of the superlattice potential affects stronger Layer 1 than Layer 2. In particular, the homogeneous contributions in U_{d}(r), i.e., the Fourier component \({\tilde{U}}_{d}({{{{{{{\bf{G}}}}}}}}=0)\), give rise to an electrostatic potential difference exactly as homogeneous outofplane electric field so that it is absorbed in Δ_{e}. Furthermore, our electrostatic model explicitly takes into account this part and encodes it into the effective displacement field D_{eff}. Therefore, the Fourier component \({\tilde{U}}_{d}({{{{{{{\bf{G}}}}}}}}=0)\) is included in Δ_{e} in our formalism.
As a result, the underlying superlattice would fold Dirac cones into its small Brillouin zone forming subbands. The degeneracy points due to the folding are gapped out by
where Ω_{0} = L_{x}L_{y} is the area of the primitive cell of the superlattice, the distance between CrOCl and graphene sheets d_{1} = d for Layer 1 and d_{2} = d + d_{0} for Layer 2.
When D_{eff} = 0.6 V/nm, i. e., Δ_{e} = 25 meV for ε_{d} = 4, layer polarization due to an outofplane electric field opens a gap^{31}
at finite momentum
knowing that t_{⊥} ≫ Δ_{e}.
HartreeFock calculations
A doublegate screened Coulomb interaction with a dielectric constant ε_{r} = 4 and the thickness of the device d_{s} = 400 Å are used in the model. The Coulomb interactions are written in the subband eigenfunction basis. As interaction effects are most prominent around the CNP, we project the Coulomb interactions onto only a lowenergy window including three valence and three conduction subbands that are closest to the Dirac point for each valley and spin. We use a mesh of 18 × 18 kpoints to sample the mini Brillouin zone of the superlattice. To incorporate the influences of Coulomb interactions from the highenergy remote bands, we rescale the Fermi velocity v_{F} and the interlayer hopping t_{⊥} within the lowenergy window of the effective Hamiltonian using the formula derived from the RG approach^{16,34,35}. Note that the ratio v_{F}/t_{⊥} remains unchanged after the RG procedure. We keep other parameters of the noninteracting effective Hamiltonian unchanged since their RG correction is of higher order, thus can be neglected. Feeding with the initial conditions in the form of order parameters, we selfconsistently obtain the gap at the CNP and the singleparticle excitation spectrum.
Data availability
The data that support the findings of this study are available at Zenodo, https://doi.org/10.5281/zenodo.6569307.
Code availability
The codes used in theoretical simulations and calculations are available from the corresponding authors upon request.
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Acknowledgements
This work is supported by the National Key R&D Program of China with Grants. 2019YFA0307800, 2017YFA0206301, 2018YFA0306900, 2019YFA0308402, and 2018YFA0305604. The authors acknowledge support from the National Natural Science Foundation of China (NSFC) with Grants 92265203, 11974357, U1932151, 11934001, 11774010, 92265106, and 11921005. The growth of hexagonal boron nitride crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan, Grant Number JPMXP0112101001, JSPS KAKENHI Grant Number JP20H00354 and A3 Foresight by JSPS. JianHao Chen acknowledges support from Beijing Municipal Natural Science Foundation (Grant No. JQ20002) and the technical support from Peking Nanofab.
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Z.H. and Y.Y. conceived the experiment and supervised the overall project. 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., and Z.H. carried out electrical transport measurements; P.G. and Y.Y. performed synthesis of bulk CrOCl crystals; Z.L., H.W., and X.L. (Xiuyan Li) carried out TEM characterizations; K.W. and T.T. provided highquality hBN bulk crystals. Z.H., Y.Y., J.Z., J.L., and X.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 manuscript was written by Z.H., J.L., and X.L. (Xin Lu) with discussion and input from all authors.
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Yang, K., Gao, X., Wang, Y. et al. Unconventional correlated insulator in CrOClinterfaced Bernal bilayer graphene. Nat Commun 14, 2136 (2023). https://doi.org/10.1038/s41467023377692
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DOI: https://doi.org/10.1038/s41467023377692
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