Correlated states in doubly-aligned hBN/graphene/hBN heterostructures

Interfacial moiré superlattices in van der Waals vertical assemblies effectively reconstruct the crystal symmetry, leading to opportunities for investigating exotic quantum states. Notably, a two-dimensional nanosheet has top and bottom open surfaces, allowing the specific case of doubly aligned super-moiré 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 (h-BN), 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 inter-valley coherence. Our results provide a way to construct intriguing correlations in 2D electronic systems in the weak interaction regime.


Supplementary
. Hofstadter butterfly for sample S12. Landau fan diagram for device S12 plotted in φ /φ 0 and n/n 0 . Data measured at 50 mK.
Supplementary Figure 9. Locating resistive peaks in sample S12. The analysis of twist angle of sample S12 from the phenomenological formula in Eq. Supplementary Figure 11. AFM scan of dual moiré length-scale obtained in another Sample. λ ∼ 16.6 nm (indicating an existing strain, which is in agreement with other reports [10]) was found in the first pick up, while λ ∼ 14.5 nm was seen for the second pick up. Inhomogeneous regions can be seen in the red dashed box, indicating that the doubly aligned area can be rather local, and this is why the device yield is very low (less than 1/20). Scale bars in optical images in (a)-(b) are 10 µm. Clear inhomogeneities in the moiré superlattices can be seen in the red dashed box area, indicating that the well aligned area can sometimes be rather local, with micronmeter sizes or smaller. This kind of inhomogeneities in moiré superlattice is also reported elsewhere. [11] Supplementary Figure 13. Electrical characterization of sample S4. (a). I-V characteristics measured at zero gate-voltage configurations at room temperature. The inset shows an optical image of a typical device, scale bar is 10 µm. (b). Two-terminal resistance versus gate voltage at room temperature. Sample S4 has only two electrodes working, and we could only perform two probe measurements for this sample. Supplementary Figure 17. Thermal activation gaps at integer fillings for samples S4, S12, S60, and S63. It is seen that the thermal activation gaps fitted from the two samples, within the temperature range of 5 to 50 K, are in good agreement.
Supplementary Figure 18. Comparison of field effect curves as a function of temperature for samples S12, S4, S56, S60, and S63. From top to down are field effect curves as a function of temperature measured in different samples, with the optical images of each corresponding data shown on the right side.
Supplementary Figure 19. Magneto-transport of different samples in the alignment regime, for samples S4, S12, and S56. From top to down are magneto-transport recorded in the range of 0 to 2.5 T at 50 mK from S12, S4, and S56, respectively. Notice that sample S56 was measured in its R xy with the absolute value plotted, and the pattern is a bit disturbed due to electron-hole sign switch. Data obtained at 50 mK.
where R i , V (R i ) and t(R i − R j ) are the lattice point at i site, the on-site potential at i site and the hopping integral between i and j site, respectively. We assume that the on-site potential of the carbon atom is equal to 0, then we have which are the on-site potential of the boron atom and the nitrogen atoms, respectively [1]. And the hopping integral can be explained by the Slater-Koster-type function of any atomic position [2][3][4] − Here, R and e z are the vector distance and the unit vector perpendicular to the layers, respectively. a 0 = a/ √ 3 = 0.142 nm is the distance of nearest neighbour A and B atom on graphene and d 0 = 0.335 nm is the interlayer spacing of graphene. V 0 ppπ and V 0 ppσ is the transfer integral between the nearest neighbor atoms contributed by the π bonds and the σ bonds, respectively. Here, we take V 0 ppπ = -2.7 eV and V 0 ppσ = 0.48 eV as the reference [5]. We now consider the intralayer interaction and the low-energy effective theory of graphene.
The electronic properties of graphene is mainly contributed by electronic states at K and K points [6,7]. Then the effective Hamiltonian can be derived from the tight-binding model in the lowenergy limit. In present case, K point and K point are expressed by K ξ = −ξ (2a * 1 + a * 2 )/3 where ξ = ±1 for K and K points, respectively. The Hamiltonian of the monolayer graphene for intralayer coupling near the K ξ points is written as where k and v is the wave vector and the Fermi velocity of graphene, respectively. The Fermi velocity of graphene is equal to 0.8 × 10 6 m/s [8]. Note that in our theoretical calculations, the graphene layer can be twisted by an angle θ with respect to the presumingly exactly aligned top and bottom layers, and this twist angle is manifested as a θ dependent Dirac point K ξ (θ ) = K ξ + [0, 4πθ /3a]. And σ ξ = (ξ σ x , σ y ) with Pauli matrix σ x and σ y describe the different sublattices.
Similar to the graphene layer, the intralayer coupling of the two h-BN layers can be considered This is justified when θ is small, because K ξ and K ξ are close to each other, and the graphene's electronic states near K ξ are coupled only with the h-BN's states near K ξ by the long-range interlayer coupling.
Then we derive the interlayer coupling matrix, and the intralayer coupling Hamiltonian. The Bloch wavefunction can be obtained by the Fourier transformation from the real space basis where R X L is the atom position with different layers and sublattices which X = A, B denotes different sublattices and L = 1, 2 represents different layers. We define the lattice vectors between two atom positions as follows R X = n 1 a 1 + n 2 a 2 + τ X R X = n 1 a 1 + n 2 a 2 + τ X , where τ X and τ X are the lattice vectors between two sublattice atoms A and B in the graphene layer and h-BN layers, respectively. Therefore, the interlayer coupling Hamiltonian in tight-binding model can be expressed by Then we use the in-plane Fourier transformation where r + d z = (τ X − τ X ) and the integral in q is taken over two-dimensional reciprocal space.
The Eq. (9) can be simplified as where G and G are the reciprocal lattice vectors in two nearest neighbor layers. To simplify Eq. (11), we used the transformation The hopping integralt(k) decays rapidly as k increases. We only consider the electronic state near K or K points, the Hamiltonian can be simplified as where we used G i · δ (r) = G M i · r andt(K ξ ) = 0.152 eV. The single particle Hamiltonian of h-BN/graphene/h-BN system which the top and bottom layer of h-BN are aligned is Due to we only consider the low-energy spectrum, we can make H BN as the second-order perturbation of the effective Hamiltonian. The effective Hamiltonian matrix is It is noted that the lattice constant of BN is a BN = 2.504Å, which is slightly larger than the graphene's lattice constant a = 2.46Å. It induces isotropic expansion 1 + ε = a BN /a = 1.018 and makes the moiré lattice constant L become where ε represents the lattice mismatch and θ refers to the twist effect.
On the other band, when the the top and bottom layer of h-BN have relative twist angle about 60 degree, The single particle Hamiltonian of h-BN/graphene/h-BN system can be written as where The effective Hamiltonian of Eq. (17) by second-order perturbation theory is The energy spectrum of different types of h-BN/graphene/h-BN system can be calculated by Eq. (15) and Eq. (19), shown in Fig. 2e in the main text.