Correlated insulator behaviour at half-filling in magic-angle graphene superlattices

Abstract

A van der Waals heterostructure is a type of metamaterial that consists of vertically stacked two-dimensional 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 two-dimensional materials1. 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 two-dimensional layers creates long-range 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 nitride2,3,4,5. 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 theoretically6, 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 half-filling, which are not expected in the absence of correlations between electrons. We show that these correlated states at half-filling are consistent with Mott-like insulator states, which can arise from electrons being localized in the superlattice that is induced by the moiré pattern. These properties of magic-angle-twisted bilayer graphene heterostructures suggest that these materials could be used to study other exotic many-body 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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Figure 1: Electronic band structure of twisted bilayer graphene (TBG).
Figure 2: Half-filling insulating states in magic-angle TBG.
Figure 3: Flat bands in magic-angle TBG.
Figure 4: Magnetic-field response of the half-filling insulating phases.

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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; DMR-1405221) 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 DMR-1231319. Data analysis by V.F. was supported by AFOSR grant number FA9550-16-1-0382. 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 (DMR-0819762) and of Harvard’s Center for Nanoscale Systems, supported by the NSF (ECS-0335765). E.K. acknowledges support by ARO MURI award W911NF-14-0247. 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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Correspondence to Pablo Jarillo-Herrero.

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Extended data figures and tables

Extended Data Figure 1 Evolution of the low-energy band structure of TBG near the magic angle.

af, E+ dispersion as in equation (6) for different vF and fixed m = 0.5. The kx and ky 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. gl, The evolution of the low-energy 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 kθ ≈  and the range for both kx/Kθ and ky/Kθ is [−0.1, 0.1]. The centre of the momentum space is the Ks point of the mini Brillouin zone (see Fig. 1d), and the thick lines denotes the Ks–Ms directions (there are three inequivalent ones). All results are shown for the K-valley continuum description of TBG6.

Extended Data Figure 2 Capacitance measurement set-up and extraction of the Fermi velocity.

a, Schematic of the low-temperature capacitance bridge. The X and Y outputs from the lock-in amplifier refer to the in-phase and out-of-phase components, respectively. C(device) and Rseries are the capacitance and resistance of the sample. Vg is the d.c. gate voltage, Vd is the excitation voltage, Vref is the reference voltage and Cref 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. v0 = 106 m s−1 is the Fermi velocity in pristine graphene.

Extended Data Figure 3 Quantum oscillations and extraction of the effective mass.

ac, Temperature-dependent magneto-resistance ΔR of device D1 at gate-voltage-induced carrier densities of n = −2.08 × 1012 cm−2 (a), n = −1.00 × 1012 cm−2 (b) and n = 0.19 × 1012 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 ac. The curves are fitted according to the Lifshitz–Kosevich formula (equation (4)). e, Magneto-conductance 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 half-filling states. At densities beyond the half-filling states, the oscillations do not converge at the Dirac point, but instead at the half-filling states.

Extended Data Figure 4 Supplementary transport data in devices D3 and D4.

a, Magneto-conductance G in device D3 (θ = 1.12°) versus n and B. The primary features at the superlattice gaps ±ns and the half-filling states ±ns/2 are essentially identical to those for device D1. b, c, Four-probe (b; Gxx) and two-probe (c; G2) 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 half-filling states (±2), we also observe weak drops in the four-probe conductivity that point to three-quarter-filling states at ±3. d, e, Hall measurement in device D4 at various temperatures: the Hall coefficients RH (d) and the Hall density nH = −1/(eRH) (e). The coloured vertical bars and the corresponding numbers are as in b and c. The x axis is the gate-induced total charge density n, whereas the Hall density nH and its sign indicate the number density and characteristic (electron-like or hole-like) 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 non-magic-angle 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, respectively13. c, Magneto-conductance in device D1 as a function of gate-induced charge density n and perpendicular magnetic field B. d, Magneto-conductance in device D1 measured as a function of n and in-plane magnetic field B. The in-plane 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 half-filling states are not as well developed as in the previous measurements. However, the gradual suppression of the half-filling states is still unambiguously observed when B is above about 6 T, slightly higher but similar to the approximately 4–6-T threshold for the perpendicular field (see c and Fig. 4a, b). The red and blue arrows point to the p-side and n-side half-filling states, respectively.

Extended Data Figure 6 DOS in magic-angle TBG.

a, b, Single-particle 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 half-filled. The results are obtained numerically using a continuum model6.

Extended Data Figure 7 Determining the twist angle.

ad, Resistivity ρxx (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 ±ns and open arrows highlight ±2ns features that may correspond to features reported in ref. 17. So far, we have observed the half-filling states only in devices that have twist angles within 0.1° of the first magic angle. e, Magneto-conductance 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 ns 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 magneto-conductance (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 half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018). https://doi.org/10.1038/nature26154

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