Abstract
Interfacial moiré superlattices in van der Waals vertical assemblies effectively reconstruct the crystal symmetry, leading to opportunities for investigating exotic quantum states. Notably, a twodimensional nanosheet has top and bottom open surfaces, allowing the specific case of doubly aligned supermoiré lattice to serve as a toy model for studying the tunable lattice symmetry and the complexity of related electronic structures. Here, we show that by doubly aligning a graphene monolayer to both top and bottom encapsulating hexagonal boron nitride (hBN), multiple conductivity minima are observed away from the main Dirac point, which are sensitively tunable with respect to the small twist angles. Moreover, our experimental evidences together with theoretical calculations suggest correlated insulating states at integer fillings of −5, −6, −7 electrons per moiré unit cell, possibly due to intervalley coherence. Our results provide a way to construct intriguing correlations in 2D electronic systems in the weak interaction regime.
Introduction
Lattice symmetry of a crystal often defines its fundamental physical properties, yet the measures to manipulate in real space the crystallographic symmetries are so far rather limited in solidstate systems. Owing to the feasibility of controllable twisting of van der Waals (vdW) twodimensional (2D) building blocks, a direct reconstruction of the interfacial moiré superlattice can be realized^{1,2}, leading to the modifications of electronic band structure of such hybrid systems. As a result, emerging quantum phenomena, such as strongcorrelation superconductivity,^{3,4,5} orbital ferromagnetism,^{6,7,8} unusual excitonic properties,^{9,10} and nematic electronic orderings,^{11,12,13,14} have been intensively investigated in twistangled 2D heterostructures that are fundamentally influenced by the moiré superpotentials.
Although moirédominated electronic and optical behaviours have been largely investigated, it is noticed that a 2D vdW electronic layer with duplicated moiré superlattices on both of its surfaces is of particular interests. However, restricted by a sophisticated stacking and twisting operation of doubly aligned moiré superlattice, very few reports have been focused on such a model.^{15,16,17,18} Indeed, doubly aligned hBN/graphene/hBN interfaces give rise to an extra moiré wavelength (defined as composite supermoiré) that is possible to arrange the spectrum reconstruction at arbitrary low energies.^{17} More recently, scanning probe tipassisted dynamic rotating of a top hBN reveals a subtle distinction of symmetry for 0^{∘}/0^{∘} and 0^{∘}/60^{∘} configurations of the hBN/graphene/hBN doubly aligned sandwich.^{15} Nevertheless, experimental demonstrations of such model systems in a broader gate range and more detailed understandings of the unusual band modification, such as possible flat bands and correlated electronic states, remain unexplored.
In this work, we investigate the case of doubly aligned hBN/graphene/hBN devices, in which closetozero angle alignments were achieved on both top and bottom graphene/hBN interfaces, allowing one to fully map the detailed fillings of each emerging minibands thanks to the composite supermoiré lattices. We first demonstrate theoretically that in the scenario when encapsulating hBNs are in perfect alignment (i.e., 0^{∘}–0^{∘}–0^{∘} aligned), there is only one single moiré superlattice, which yields expected flat minibands. Experimentally, the minibands of doubly aligned composite moiré in hBN/graphene/hBN heterostructure can be sensitively tuned by the twisting angle. In the bestaligned double moiré sample S7, we observed resistive peaks at fillings of −4, −8, −12 electrons per moiré unit cell. In other samples S12 and S4 with slightly different, yet rather close to zero, dual twists from top and bottom hBN/graphene interfaces, evidence of possible correlated insulating (CI) states was observed in −5, −6, −7 electrons per moiré unit cell. Our results suggest that lattice reconstruction of graphene using dually aligned moiré superpotential can be an effective way to realize correlated states.
Results
Characterizations of doubly aligned heterostructures
Monolayered graphene and thin encapsulating hBN flakes were exfoliated from bulk crystals and stacked with the drytransfer method.^{19} The graphene/hBN vertical heterostructure was patterned into Hall bars with electrodes edgecontacted. The device was equipped with Au top and graphite bottom gates, as sketched in Fig. 1a (See also Supplementary Fig. 1). A zoomedin art view is illustrated in Fig. 1b, with the small rotation angle between the top and bottom graphene/hBN interfaces denoted as θ_{t} and θ_{b}, respectively. Experimentally, the alignment between graphene and the two encapsulating hBN flakes was realized by carefully aligning the exfoliated straight edges of each flake using the optical micrograph images with a graphic analysis tool software. An example of optically wellaligned sample is given in Fig. 1c, with a typical dualgated device using this doubly aligning method shown in Fig. 1d.
It is rather evident that when the three layers are in the alignment regime, the first holeside satellite resistive peak \({R}_{{n}_{s}}\) (n_{s} corresponding to a carrier filling of 4\({n}_{0}=\frac{8}{\sqrt{3}{L}^{2}}\), with n_{0} defined as one electron filling per moiré unit cell area \(A=\frac{\sqrt{3}{L}^{2}}{2}\). L is the moiré wavelength^{15}) will be about 1.5 times higher than the central major Dirac peak (with a resistance of \({R}_{D}^{{{{{{{{\rm{neutral}}}}}}}}}\)) at the charge neutrality at room temperature.^{15} Such a “wellaligned regime” has a tolerance of less than 1^{∘}.^{15} Therefore, the above feature of fieldeffect curves can be a criterion to distinguish devices with different alignments. Indeed, as shown in Fig. 1e, typical samples with different θ_{t} /θ_{b} values obtained by measuring their straight edges in the optical images exhibit significant variations in their room temperature field effect curves. As a result, sample S8 has a best alignment of θ_{t}/θ_{b}, yielding a field effect curve with \({R}_{{n}_{s}}\) higher than \({R}_{D}^{{{{{{{{\rm{neutral}}}}}}}}}\). Meanwhile, other samples with larger misalignment showing lower \({R}_{{n}_{s}}\) than \({R}_{D}^{{{{{{{{\rm{neutral}}}}}}}}}\), with the satellite peak disappearing in, for example, sample S20. Raman spectroscopy is another means to check the twist angles of the doubly aligned hBN/graphene/hBN. As indicated in Fig. 1f, the corresponding devices in Fig. 1e show distinct Raman spectra. The bestaligned sample yields a graphene 2D Raman peak with slightly wider fullwidth at the half maximum (FWHM) and lowest peak height compared to other lessaligned samples, in agreement with previously reported results.^{15,17}
Band structure of doubly aligned hBN/graphene/hBN
Before further experimental measurements, we study the electronic band structures of the doubly aligned hBN/graphene/hBN with a composite moiré superlattice (tightbinding simulations and continuum models are both used, and found to be consistent. See “Methods” and Supplementary Note 1). In the case of monolayer graphene/hBN single moiré superlattice, the original band structure of graphene will be folded and give rise to, away from the major Dirac cone, duplicates of a set of mini Dirac cones at electron filling of ±n_{s}.^{1,2} The composite moiré superlattice with perfect alignment 0^{∘}–0^{∘}–0^{∘} (illustrated in Fig. 2a) however splits the holesided bands at higher energy into minibands with fully developed gaps (Δ_{i}, i = 1, 2, 3) at electron fillings of −4 n_{0}, −8 n_{0} and −12 n_{0}, as illustrated in Fig. 2b. Compared to the single interfaced hBN/graphene heterostructure, the consequence of a doubly aligned hBN/graphene/hBN moiré superlattice is the strong band anticrossing originated from an extra moiré potential in the Hamiltonian (Supplementary Note 1). For the specific scenario of perfect alignment, we calculated via continuum model the bandwidth of the first three bands below the charge neutrality, shown in Fig. 2c. It is seen that band I and II increase monotonously with twist angle θ, while band III has a minimum at about 0.65^{∘} with a bandwidth of about 35 meV. More details can be found in the Supplementary Information.
Transport measurements of doubly aligned hBN/graphene/hBN
Bearing in mind with a band structure of 0^{∘}–0^{∘}–0^{∘} illustrated in Fig. 2b, we now come to the experimental examination of the theoretical models. A typical sample S7 was characterized by recording fieldeffect curves as a function of magnetic field at 45 mK. It develops into a Landau fan with a fractal structure known as the Hofstadter butterfly, which can be renormalised into a diagram defined by the Diophantine equation^{20,21,22}:
where topological integers ν denotes the Hall conductivity in units of a conductance quantum e^{2}/h, and s is the index of band filling. ϕ = B ⋅ A is flux per moiré unit area at magnetic field B, and ϕ_{0} = h/e is a flux quantum with h being the Planck’s constant. It is seen in Fig. 2d that pseudogaps at each filling of −4 n_{0}, −8 n_{0}, −12 n_{0} can be extrapolated from the Landau fans at B = 0, agreeing well with the expected minibands filling for perfect doubly aligned hBN/graphene/hBN. Typical subsets of the quantum Hall states (QHS) that satisfy Eq. (1) are identified as black solid lines in Fig. 2e, with topological index ν to be ±2, ±6, ±10, etc. More data on other samples with less well alignment can be found in Supplementary Figs. 2–9.
Except for those resistive maxima at full band fillings, extra resistive peaks can be observed. These peaks may come from the slightly nonperfect alignment. To quantitatively analyse them, a phenomenological model was given, which, for simplicity, assumes that the −4 n_{0} filling corresponds a fixed hBN/graphene moiré with a smaller twist (wavelength L_{α}), and all extra peaks are from another moiré with a larger twist (wavelength L_{β}). Thus, one can then extract and fix L_{α}, and solve L_{β} using the formula^{17,23}
where a is graphene’s lattice constant, l_{β} is the reciprocal lattice vector of L_{β}, and δ is the lattice mismatch between graphene and hBN^{24}. Experimentally, L_{α} can be larger than the theoretical maximum of 13.8 nm in a graphene/hBN superlattice, due to possibly existing strains in the heterostructures^{17,25}. The abovementioned analytical formula yields six solutions at each side of the charge neutrality, because of the six vectors formed by the two of moiré Brillouin zone from top and bottom surfaces of graphene^{17,23}. With this method, one can have access to the relatively precise twist angles in measured samples via their resistive peaks in the fieldeffect spectra. For example, twist angles in sample S7 is found to be of 0^{∘}–0^{∘}–0.45^{∘}. Besides the above discussed geometrical model, we also performed atomic force microscopy characterizations of the doubly aligned moiré superlattice in real space, as shown in Supplementary Figs. 10, 11.
Discussion
With sample S7, we obtained a filling sequence that resembles perfect aligned 0^{∘}–0^{∘}–0^{∘} moiré superlattice. The minibands are actually super sensitive to twisting angles of each constituent layer. When at slightly different finite dual twists, the resulted composite moiré superlattices will lead to tunable resistive peaks in their spectra of field effect. Calculated density of states (DOS) for the band structure in Fig. 2b is plotted in Fig. 2f, while Fig. 2g summarizes typical samples with different twist angles. By quantifying their exact alignment angles using the same method for sample S7 (more details can be found in Supplementary Figs. 6, 9), we can thus overlay their fieldeffect curves at B = 0 in the renormalized axis of n/n_{0}, in Fig. 2g. It is seen that, even with a tiny difference from 0^{∘}–0^{∘}–0.45^{∘} (sample S7) to 0^{∘}–0^{∘}–0.5^{∘} (sample S12 and S4), the resistive peaks are largely tuned. For sample S2 with relatively larger twist angle (0.5^{∘}–0^{∘}–1.5^{∘}), only n = 4 n_{0} band filling was observed. Notice that samples S12 and S4 are almost identical in their fieldeffect curves. Interestingly, resistive peaks in the second miniband are seen in them, as indicated by the black solid arrows in Fig. 2g. It is seen that those resistive peaks correspond to integer fillings of −5 n_{0}, −6 n_{0} and −7 n_{0}, where the host miniband has calculated bandwidth to be of a few tens of meV, marked in purple. It thus suggests possible CI behaviour as observed in other flat band samples previously reported, depicted by the correlationdriven gap opening picture at integer fillings of the moiré unit cell.^{4,26,27,28} Notably, flat band with bandwidth of similar energy was reported theoretically,^{29} and were found experimentally possible for the construction of correlated superconductivity in Bernalstacking bilayer graphene/hBN superlattice^{30}. Among the over 60 samples fabricated, we have 5 dualmoiré samples showing the correlated states at integer fillings from −5 to −7 n_{0}. However, none of them exhibit superconductivity in a reachable doping range and to the lowest temperature we could obtain. Possible explanations for the absence of superconductivity in our samples, even though correlated states signatures are observed, could be that the microscopic mechanism of interactioninduced superconductivity in such as magic angle twisted bilayer graphene are different from that in our case.
To further characterize the CI states in such as sample S12, we performed temperature dependence of its fieldeffect curves, shown as the colour map in Fig. 3a. It is observed that at temperatures at the order of 100 K, multiple peaks start to develop, with the line cuts of fieldeffect curves at different temperatures shown in Fig. 3b. Some of the resistive peak at each side of the charge neutrality develops into fine structures, that can be identified as either band fillings (marked by black solid triangles), or composite moiré peaks (marked by black solid circles)(more details of other samples can be seen in Supplementary Figs. 12–16). Notably, the broad satellite peak on the holeside gradually deconvolutes into individual resistive peaks at −4, −5, −6, −7 and −8 n_{0} at the ground state. These insulating pseudogaps are located in band II, with the ones at fillings of −5, −6 and −7 n_{0} attributed to electron correlation, as in the noninteracting picture, the doped system is metallic at these three fillings. We thus extract the thermal activation gaps for representative peaks marked by black dashed lines in Fig. 3b, and fitted with the Arrhenius law in Fig. 3c. The thermal gaps of the CIs are fitted to be 0.06 meV, 0.47 meV and 0.22 meV, for fillings of −5, −6 and −7 n_{0}, respectively. This energy scale is relatively smaller than the theoretically calculated bandwidth as shown in Fig. 2bc. It is noticed that the thermal gaps extracted (less than 1 meV) are also small as compared to the energy scale of the temperature range used for extracting them. As in our experiment, the thermal activation gaps are obtained from the temperature range of 10–50 K. This is mainly because those “gaps” are not as robust as band insulators, and are often defined as incipient gaps due to relatively weakinteraction strength. In transport experiment, they turn out to be weak resistive peaks that cannot develop into more insulating ground state, and become saturating at temperatures below a few Kelvin. In other recently reported studies on twisted graphene systems, similar temperature ranges are often used for the fitting of thermal activation gaps for such weak resistive peaks^{15,27}. Moreover, among four samples (S4, S12, S60 and S63) that exhibit similar resistive peaks at integer fillings from −5 to −7 n_{0}, the thermal activation gaps are extracted to be rather consistent, as shown in Supplementary Fig. 17, which speaks the reproducibility of the observed phenomenon.
We further performed, in a typical sample (S60), the zoomedin scan of these CI states in the parameter space of carrier density and magnetic field, with the longitudinal (R_{xx}) and transverse resistance (R_{xy}) shown in Fig. 4a, b, respectively. Line cuts of R_{xx} and R_{xy} at different magnetic fields are plotted in Fig. 4c, d. It is noticed that in the band from −4 n_{0} to −8 n_{0}, insulating states are seen at each integer fillings, with positive longitudinal magnetoresistances at finite magnetic fields. Direct comparisons of samples showing similar CI behaviours are shown in Supplementary Figs. 18, 19. We attribute the pseudogaps to strong Coulomb interaction, which lifts the degeneracy and induce the correlating gaps, at fillings of −5, −6 and −7 n_{0}, as will be discussed in the coming text.
To address the origin of this correlating physical phenomenon, we performed theoretical calculations for those correlated gaps using Hartree–Fock methods (see “Methods”). In our calculations, one target band per valley per spin is taken into consideration, thus total four energy bands are involved in the Hartree–Fock analysis. Depending on whether valley U(1) symmetry is broken or not, a gapped state driven by Coulomb interactions can be divided into two types: in the first type the gap is induced by valley and/or spin polarization, while in other type originates from an intervalley coherent (IVC) order that couples the two valleys and breaks valley charge conservation (valley U(1) symmetry) spontaneously. First, we consider the characteristic screened Coulomb interaction in this moiré system U_{M} = e^{2}/(4πεε_{0}L_{M}) (see more details in “Methods”), with L_{M} being the moiré lattice constant in real space. It is noticed that, using the above relation, at a twist angle θ_{G} = 0.5^{∘}, the amplitude of U_{M} is only ~28 meV, which is roughly the energy scale of the valley and spin splittings if a valley and/or spinpolarized state were the ground state at partial integer fillings. However, this value is still far lower than the bandwidth of band II (~80 meV) shown in Fig. 2c. Thus, we conjecture that an IVC order is needed in order to open a global gap for such relatively weak Coulomb interaction strength (notice that other hypothesis of gap openings from such as Mott insulator on triangular lattice, or from strong Coulomb interactioninduced valley splitting, can be ruled out, as we are in a weakinteraction regime). As shown in Supplementary Fig. 20, it is seen that the IVC orders are existing at all fillings, with each average amplitude to be 1.42, 1.90 and 1.35 meV for fillings of −5, −6 and −7 n_{0}, respectively. However, it is worthwhile to note that at all of the three fillings, the IVC order amplitudes are highly dependent on the moiré wavevector, which can reach a maximal value of 50–90 meV at some small regions in the moiré Brillouin zone (see Supplementary Fig. 20), which is comparable to the bandwidth shown in Fig. 2c. Hence, thanks to these peaks of IVC orders, correlated states with notable energy gaps can occur at low temperatures in spite of the relatively weak Coulomb interaction strength and large bandwidth.
As shown in the calculated energy bands, when taking IVC order into account at θ_{G} = 0. 5^{∘}, in Fig. 4e–g, global gaps openings can be seen as indicated by the lightblue regions, with the value of 11.8, 9.7 and 1.8 meV, for ν = −5, −6 and −7 n_{0}, respectively. The calculated correlated gaps are larger than the experimentally extracted thermal activation ones, which may come from the underestimated quantum fluctuations (thus overestimated gaps) in the Hartree–Forck simulations. It is known that the gap itself also strongly depends on experimental details, such as errors in the twist angle, and the exact dielectric constant in the materials. Nevertheless, the general trend of the three correlated gaps are in qualitative agreement with the ones obtained in Supplementary Fig. 17. We further obtain the global correlated gap sizes as a function of twist angle θ_{G}, shown in Supplementary Fig. 21. It is seen that the global gap can reach up to the order of 10–20 meV for the perfectly aligned situation. However, considering the error of twist angle in the dual moiré sample fabrications, the global gap sizes can be much reduced. In the mean time, it is also found that the global gap sizes drastically diminish with an increased dielectric constant, as shown in Supplementary Fig. 22. On the other hand, to observe the correlated states at fillings of −5, −6 and −7 n_{0} in the dualmoiré hBN/graphene/hBN devices, one has also to minimize as much as possible the inhomogeneities, as distortion of the superlattice can be seen in the AFM scans in Supplementary Fig. 11. Finally, we have to emphasize that the doubly aligned moiré superlattice in the studied system plays a crucial role to reduce the bandwidth into an appropriate scale (a few tens of meV of band II, as calculated in Fig. 2c, while band I is still too large with a width exceeding 100 meV, and band III may be too much influenced by the hybridization of lower energy bands). Overall, the concurrence of a doubly aligned moiré superlattice, and the hypothesis of the existence of IVC orders, together with the observed correlated states at fillings of −5, −6 and −7 n_{0} in multiple dualmoiré samples, are selfconsistent in this work.
We have demonstrated a series of monolayered graphene samples with a dual moiré constructed from both top and bottom hBN/graphene interfaces. Thanks to the lattice symmetry tuned by the composite double moiré super potential, the system exhibits tunable band structure, with minibands at the order of a few tens meV for perfect alignment analysed using tightbinding and continuum models. By experimentally controlling the small twist angles of the doubly aligned superlattice, the signatures of the accompanied high order resistive peaks can indeed be sensitively tuned in multiple samples. In a bestaligned sample, the experimental observations agree with the picture of a holeside filling sequence of each moiré minibands with −4, −8 and −12 n_{0}, in resemblance to the perfect align scenario. We also found correlated insulating states at integer fillings at −5 n_{0} to −7 n_{0} in typical doubly aligned single layer graphene samples S12, S4, S60 and S63. This correlated behaviour is further attributed to a concurrence of the doubly aligned moiré superlattice and the existence of intervalley coherent orders in the weak coupling regime. Our results suggest that composite supermoiré lattices with a dual twist from both graphene/hBN interfaces can be a platform for the effective tuning of moiré minibands, and may provide insights for future correlated states engineering in such systems.
Methods
Device fabrication and measurements
The BNencapsulated graphene heterostructures were fabricated in ambient condition, using the drytransfer method, with the crystals aligned to each of their exfoliated straight edges by a mechanical rotator. A Bruker Dimension Icon AFM was used for thicknesses and morphology tests. Raman measurements were performed by an HR 800 JobinYvon Horiba Raman spectroscopy. The electrical performances of the devices were measured using a BlueFors LD250 at mK temperature, a Cryogenics 4K system for temperature scannings, and a probe station (Cascade Microtech Inc. EPS150) for room temperature tests. Standard 4probe measurements were applied with the low frequency lockin technique.
Band structure calculations
For band structure calculations of the moiré superlattice, we employed a tightbinding (TB) model. Only p_{z} orbitals were considered since only the states near the Fermi level are of interest. The TB Hamiltonian has the form^{31}
where \(\lefti\right\rangle\) is the p_{z} orbital of the atom located at \({\overrightarrow{{{{{{{{\bf{r}}}}}}}}}}_{i}\), ϵ_{i} is the onsite energy, and t_{ij} is the coupling parameter. For the hBN/graphene/hBN heterostructure, both ppπ and ppσ interactions are important. According to the SlaterKoster formula^{32}, t_{ij} is given by:
where n is the direction cosine along the zdirection of the vector r_{j} − r_{i} = r_{ij}, and r_{ij} = ∣r_{ij}∣. Following previous studies^{33,34}, the functions V_{ppπ} and V_{ppσ} are assumed to have the following forms:
Here, \({V}_{pp\pi }^{0}\) = 0.48 eV and \({V}_{pp\sigma }^{0}\) = − 2.7 eV are the hopping integrals for the ppπ and ppσ interactions, respectively. q = 2.218 is chosen so that the nextnearest intralayer hopping is 0.1\({V}_{pp\sigma }^{0}\). a_{0} = 1.418 Å is the CC bond length in graphene, and d_{0} = 3.35 Å is the spacing between the graphene and BN layers. The intra and interlayer hoppings are truncated with a 6 Å cutoff. The onsite energies for C, B and N are 0 eV, 3.34 eV and −1.40 eV, respectively. These parameters have been widely used to study bilayer graphene and graphene/BN systems^{23,29,33}.
We consider the intersite Coulomb interaction H_{C} as:
for Hartree–Fock meanfield calculations in the moiré system. Here α refers to layer and sublattice indices. k and q are atomic wavevectors expanded around the Dirac points, and \(\sigma ,{\sigma }^{\prime}\) are the spin indices. We choose the doublegate screened Coulomb interaction, which is expressed as \(V({{{{{{{\bf{q}}}}}}}})={e}^{2}\tanh ({{{{{{{\bf{q}}}}}}}}{d}_{s})/(2{{{\Omega }}}_{{{{{{{{\rm{M}}}}}}}}}\varepsilon {\varepsilon }_{0}{{{{{{{\bf{q}}}}}}}})\). Here ε ~ 4.0 is background dielectric constant, and will be further changed as a variable in our calculations. d_{s} = 200 Å is the screening length. Ω_{M} is the area of moiré Brillouin zone. The screened Coulomb interaction can be rewritten as \(V({{{{{{{\bf{q}}}}}}}})={U}_{{{{{{{{\rm{M}}}}}}}}}{q}_{{{{{{{{\rm{M}}}}}}}}}\tanh ({{{{{{{\bf{q}}}}}}}}{d}_{s})/{{{{{{{\bf{q}}}}}}}}\) in which \({q}_{{{{{{{{\rm{M}}}}}}}}}=4\pi /\sqrt{3}{L}_{{{{{{{{\rm{M}}}}}}}}}\) is the length of reciprocal lattice vector, and U_{M} = e^{2}/(4πεε_{0}L_{M}) is characteristic Coulomb interaction. It is noted that at the twist angle of θ_{G} = 0. 5^{∘}, the amplitude of U_{M} is only ~28 meV which is far lower than the bandwidth of band II (~80 meV).
In this system, the Coulomb interactions can be divided into the intravalley term \({H}_{{{{{{{{\rm{C}}}}}}}}}^{{{{{{{{\rm{intra}}}}}}}}}\) and the intervalley term \({H}_{{{{{{{{\rm{C}}}}}}}}}^{{{{{{{{\rm{inter}}}}}}}}}\),
Here \({H}_{C}^{{{{{{{{\rm{intra}}}}}}}}}\) represents the process of two electrons created and annihilated in the same valley, and \({H}_{C}^{{{{{{{{\rm{inter}}}}}}}}}\) means that two electrons are created in μ and −μ and get annihilated in −μ and μ valleys with the interaction energy \(V( {{{{{{{\bf{K}}}}}}}}{{{{{{{{\bf{K}}}}}}}}}^{\prime} )\). We only consider the intravalley interaction because the intervalley interaction is two orders of magnitudes weaker than intravalley interaction. With the Coulomb interaction projected onto the band II, we start with 32 possible initial trial wavefunctions in the valleyspinsublattice space, for example, spinpolarized/valleypolarized/IVC cases, and perform the selfconsistent calculations to solve the interacting Hamiltonian with Hartree–Fock approximations. Finally, we find the solution with the lowest total energy as the Hartree–Fock ground state.
Data availability
The data that support the findings of this study are available at Zenodo, https://doi.org/10.5281/zenodo.5078093.
Code availability
The computational codes that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
This work is supported by the National Key R&D Program of China (2017YFA0206302, 2019YFA0307800, 2020YFA0309601, and 2018YFA0306900), and is supported by the National Natural Science Foundation of China (NSFC) with Grants 11974357, U1932151, 51627801, 12074029, U1930402, 52031014, and 11991060. S.Z. and J.L. acknowledge the startup grant of ShanghaiTech University. X.L. acknowledges support from the Joint Research Fund of LiaoningShenyang National Laboratory for Materials Science with Grant No. 2019JH3/30100031. T.Y. acknowledges support from Liaoning Provincial Natural Science Fund with Grant 2021MS006. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan (Grant Number JPMXP0112101001), JSPS KAKENHI (Grant Numbers 19H05790 and JP20H00354) and A3 Foresight by JSPS.
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Z.H. conceived the experiment, and Z.H., Z.Z., J.L., Y.Y. and J.K. supervised the overall project. Z.L., J.K. and T.Y. carried out continuum and tightbinding modellings and simulations. X.S., H.Z., J.H., X.L., and Z.W. fabricated the devices. J.L. and S.Z. carried out band structure calculations of the correlated states. X.S., Z.H., K.Y. and Y.Y. carried out electrical transport measurements; K.W. and T.T. provided high quality hBN bulk crystals. X.S. and Z.H. analysed the data, with J.L., J.K., T.Y., Y.Y., S.Z., J.M. and M.Z. participated in thorough discussions. The manuscript was written by Z.H., X.S., J.L., Z.L. and J.K., with discussion and inputs from all authors.
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Sun, X., Zhang, S., Liu, Z. et al. Correlated states in doublyaligned hBN/graphene/hBN heterostructures. Nat Commun 12, 7196 (2021). https://doi.org/10.1038/s4146702127514y
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DOI: https://doi.org/10.1038/s4146702127514y
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