Abstract
A van der Waals heterostructure is a type of metamaterial that consists of vertically stacked twodimensional building blocks held together by the van der Waals forces between the layers. This design means that the properties of van der Waals heterostructures can be engineered precisely, even more so than those of twodimensional materials^{1}. One such property is the ‘twist’ angle between different layers in the heterostructure. This angle has a crucial role in the electronic properties of van der Waals heterostructures, but does not have a direct analogue in other types of heterostructure, such as semiconductors grown using molecular beam epitaxy. For small twist angles, the moiré pattern that is produced by the lattice misorientation between the twodimensional layers creates longrange modulation of the stacking order. So far, studies of the effects of the twist angle in van der Waals heterostructures have concentrated mostly on heterostructures consisting of monolayer graphene on top of hexagonal boron nitride, which exhibit relatively weak interlayer interaction owing to the large bandgap in hexagonal boron nitride^{2,3,4,5}. Here we study a heterostructure consisting of bilayer graphene, in which the two graphene layers are twisted relative to each other by a certain angle. We show experimentally that, as predicted theoretically^{6}, when this angle is close to the ‘magic’ angle the electronic band structure near zero Fermi energy becomes flat, owing to strong interlayer coupling. These flat bands exhibit insulating states at halffilling, which are not expected in the absence of correlations between electrons. We show that these correlated states at halffilling are consistent with Mottlike insulator states, which can arise from electrons being localized in the superlattice that is induced by the moiré pattern. These properties of magicangletwisted bilayer graphene heterostructures suggest that these materials could be used to study other exotic manybody quantum phases in two dimensions in the absence of a magnetic field. The accessibility of the flat bands through electrical tunability and the bandwidth tunability through the twist angle could pave the way towards more exotic correlated systems, such as unconventional superconductors and quantum spin liquids.
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Acknowledgements
We acknowledge discussions with L. Levitov, P. Lee, S. Todadri, B. I. Halperin, S. Carr, Z. Alpichshev, J. Y. Khoo and N. Staley. This work was primarily supported by the National Science Foundation (NSF; DMR1405221) and the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4541 for device fabrication, transport measurements and data analysis (Y.C., J.Y.L., J.D.S.Y. and P.J.H.), with additional support from the NSS Program, Singapore (J.Y.L.). Capacitance work by R.C.A., A.D. and S.L.T. and theory work by S.F. was supported by the STC Center for Integrated Quantum Materials, NSF grant number DMR1231319. Data analysis by V.F. was supported by AFOSR grant number FA95501610382. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by MEXT, Japan and JSPS KAKENHI grant numbers JP15K21722 and JP25106006. This work made use of the Materials Research Science and Engineering Center Shared Experimental Facilities supported by the NSF (DMR0819762) and of Harvard’s Center for Nanoscale Systems, supported by the NSF (ECS0335765). E.K. acknowledges support by ARO MURI award W911NF140247. R.C.A. acknowledges support by the Gordon and Betty Moore Foundation under grant number GBMF2931.
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Y.C., J.Y.L. and J.D.S.Y. fabricated the devices and performed transport measurements. Y.C. and V.F. performed data analysis. P.J.H. supervised the project. S.F. and E.K. provided numerical calculations. S.L.T., A.D. and R.C.A. measured capacitance data. K.W. and T.T. provided hexagonal boron nitride devices. Y.C., V.F. and P.J.H. wrote the paper with input from all authors.
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Extended data figures and tables
Extended Data Figure 1 Evolution of the lowenergy band structure of TBG near the magic angle.
a–f, E_{+} dispersion as in equation (6) for different v_{F} and fixed m = 0.5. The k_{x} and k_{y} range in the figures is [−2, 2] and the colour scale (on the right side of the figures) for the dimensionless energy axis is 0 to 10 from bottom to top. The associated winding number of each touching point is labelled. g–l, The evolution of the lowenergy band structure of TBG near the first magic angle in the model. The colour shows the hotspots of the Berry curvature at the touching point of each band. The energy axis spans an extremely small range of [−50, 50] μeV. The momentum axes are measured by k_{θ} ≈ Kθ and the range for both k_{x}/K_{θ} and k_{y}/K_{θ} is [−0.1, 0.1]. The centre of the momentum space is the K_{s} point of the mini Brillouin zone (see Fig. 1d), and the thick lines denotes the K_{s}–M_{s}– directions (there are three inequivalent ones). All results are shown for the Kvalley continuum description of TBG^{6}.
Extended Data Figure 2 Capacitance measurement setup and extraction of the Fermi velocity.
a, Schematic of the lowtemperature capacitance bridge. The X and Y outputs from the lockin amplifier refer to the inphase and outofphase components, respectively. C_{(device)} and R_{series} are the capacitance and resistance of the sample. V_{g} is the d.c. gate voltage, V_{d} is the excitation voltage, V_{ref} is the reference voltage and C_{ref} is the reference capacitance. All connections into and out of the cryostat are made with coaxial cables. b, Capacitance ΔC of device D2 near the charge neutrality point, and fitting curves according to equations (4) and (5) with different Fermi velocities. v_{0} = 10^{6} m s^{−1} is the Fermi velocity in pristine graphene.
Extended Data Figure 3 Quantum oscillations and extraction of the effective mass.
a–c, Temperaturedependent magnetoresistance ΔR of device D1 at gatevoltageinduced carrier densities of n = −2.08 × 10^{12} cm^{−2} (a), n = −1.00 × 10^{12} cm^{−2} (b) and n = 0.19 × 10^{12} cm^{−2} (c). The temperatures are, from dark to bright, 0.3 K, 1.7 K, 4.2 K and 10.7 K. d, Oscillation amplitudes of the most prominent peaks in a–c. The curves are fitted according to the Lifshitz–Kosevich formula (equation (4)). e, Magnetoconductance G of device D1 (measured at 0.3 K) plotted versus n and 1/B. f, The same data with a polynomial background in B removed for each density. The green boxes denote the range of densities for the halffilling states. At densities beyond the halffilling states, the oscillations do not converge at the Dirac point, but instead at the halffilling states.
Extended Data Figure 4 Supplementary transport data in devices D3 and D4.
a, Magnetoconductance G in device D3 (θ = 1.12°) versus n and B. The primary features at the superlattice gaps ±n_{s} and the halffilling states ±n_{s}/2 are essentially identical to those for device D1. b, c, Fourprobe (b; G_{xx}) and twoprobe (c; G_{2}) conductance measured in device D4 (θ = 1.16°) at 0.3 K. The coloured vertical bars and the corresponding numbers indicate the associated integer filling inside each unit cell of the moiré pattern. As well as the halffilling states (±2), we also observe weak drops in the fourprobe conductivity that point to threequarterfilling states at ±3. d, e, Hall measurement in device D4 at various temperatures: the Hall coefficients R_{H} (d) and the Hall density n_{H} = −1/(eR_{H}) (e). The coloured vertical bars and the corresponding numbers are as in b and c. The x axis is the gateinduced total charge density n, whereas the Hall density n_{H} and its sign indicate the number density and characteristic (electronlike or holelike) curve of the carriers being transported.
Extended Data Figure 5 Supplementary transport data in device D1.
a, Temperature dependence of the conductance G of device D1 from 0.3 K to 300 K. b, The conductance versus temperature at five characteristic carrier densities, labelled A_{±} (superlattice gaps), B_{±} (above and below the Dirac point) and D (the Dirac point) in a. The arrow denotes the temperature above which the conductances at B_{±} merge with that of D. The solid lines accompanying the A_{±} traces are Arrhenius fits to the data. The thermal activation gaps of the superlattice insulating states at A_{±} can be obtained by fitting the temperature dependence of the conductance at these densities. See ref. 13 for a detailed discussion about the superlattice gaps in nonmagicangle devices. The fit to the Arrhenius formula exp[−Δ/(2kT)] yields Δ_{−} = 32 meV for the A_{–} gap and Δ_{+} = 40 meV for the A_{+} gap. For comparison, the same gaps measured in θ = 1.8° TBG are slightly larger, Δ_{−} = 50 meV and Δ_{+} = 60 meV for the gaps at negative and positive densities, respectively^{13}. c, Magnetoconductance in device D1 as a function of gateinduced charge density n and perpendicular magnetic field B. d, Magnetoconductance in device D1 measured as a function of n and inplane magnetic field B_{‖}. The inplane measurement is made at a higher temperature of about 2 K. Combined with the degradation of the sample quality that resulted from the thermal cycling that was necessary to change the field orientation, the halffilling states are not as well developed as in the previous measurements. However, the gradual suppression of the halffilling states is still unambiguously observed when B_{‖} is above about 6 T, slightly higher but similar to the approximately 4–6T threshold for the perpendicular field (see c and Fig. 4a, b). The red and blue arrows point to the pside and nside halffilling states, respectively.
Extended Data Figure 6 DOS in magicangle TBG.
a, b, Singleparticle DOS in TBG at θ = 1.08°, on linear (a) and logarithmic (b) scales. The red dashed lines denote the energy at which the lower and upper flat bands are halffilled. The results are obtained numerically using a continuum model^{6}.
Extended Data Figure 7 Determining the twist angle.
a–d, Resistivity ρ_{xx} (resistance R for the θ = 1.08° device) measurements for four samples with different twist angles: θ = 1.38° (a), θ = 1.08° (b), θ = 0.75° (c) and θ = 0.65° (d). The filled arrows highlight superlattice features at ±n_{s} and open arrows highlight ±2n_{s} features that may correspond to features reported in ref. 17. So far, we have observed the halffilling states only in devices that have twist angles within 0.1° of the first magic angle. e, Magnetoconductance data (derivative with respect to n; dG/dn) of device D1 (θ = 1.08°) measured at 4 K. The dashed lines label the main (green) and satellite (blue) Landau fans. From the convergence point of the blue fans, we can accurately determine the superlattice density n_{s} and thus θ, with an uncertainty of about 0.02°. f, Hofstadter’s oscillation manifested as periodic crossings of Landau levels in 1/B. Data shown is the magnetoconductance (derivative with respect to n; dG/dn) of device D3 (θ = 1.12°). The horizontal lines have a uniform spacing of 0.033 ± 0.001 T^{−1}, which corresponds to θ = 1.12° ± 0.01°.
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Cao, Y., Fatemi, V., Demir, A. et al. Correlated insulator behaviour at halffilling in magicangle graphene superlattices. Nature 556, 80–84 (2018). https://doi.org/10.1038/nature26154
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