Abstract
Understanding and tuning correlated states is of great interest and importance to modern condensed-matter physics. The recent discovery of unconventional superconductivity and Mott-like insulating states in magic-angle twisted bilayer graphene presents a unique platform to study correlation phenomena, in which the Coulomb energy dominates over the quenched kinetic energy as a result of hybridized flat bands. Extending this approach to the case of twisted multilayer graphene would allow even higher control over the band structure because of the reduced symmetry of the system. Here we study electronic transport properties of twisted monolayer–bilayer graphene (a bilayer on top of monolayer graphene heterostructure). We observe the formation of van Hove singularities that are highly tunable by changing either the twist angle or external electric field and can cause strong correlation effects under optimum conditions. We provide basic theoretical interpretations of the observed electronic structure.
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Data availability
Data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.
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Acknowledgements
We acknowledge support from NRF (project medium-sized centre programme R-723-000-001-281, Singapore), the EU Flagship Programmes (Graphene CNECTICT-604391 and 2D-SIPC Quantum Technology), European Research Council Synergy Grant Hetero2D, the Royal Society and EPSRC grants EP/N010345/1, EP/P026850/1 and EP/S030719/1.
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S.X., N.B. and N.X. fabricated devices. Y.S., C.M., J.B. and B.A.P. performed transport measurements. Y.S., S.X., M.M.A.E., A.G.-R. and B.T. performed data analysis. M.M.A.E., A.G.-R., B.T., S.A. and V.I.F. developed the theory and performed theoretical calculations. Y.S., S.X., M.M.A.E., S.A., V.I.F., A.C., A.H.C.N. and K.S.N. contributed to the interpretation of data. K.W. and T.T. grew hBN single crystals. Y.S., S.X., M.M.A.E., S.A., V.I.F. and K.S.N. contributed to writing the manuscript. All authors discussed the results and commented on the manuscript.
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Extended data
Extended Data Fig. 1 Brown-Zak oscillations.
σxx maps of sample S1 (a) and S2 (b) versus normalised charge carrier concentration and magnetic field measured at T=1.6K and D=0. Numbers indicate \(\frac{{{\Phi}_0}}{\Phi }\) fractions, where \({\Phi}_0 = \frac{{\it{h}}}{{\it{e}}}\), h is the Planck constant, e is the electron charge, and Φ=BS is magnetic flux through a moiré unit cell.
Extended Data Fig. 2 Two mirror-symmetric configurations of tMBG.
ρxx maps as a function of back gate voltage Vbg and top gate voltage Vtg of samples with twist angle θ ≈ 1.47o (a) and 1.6o (b) measured at T=1.6 K and B=0 T. The schematics above the maps show the stacking configurations. Black and blue balls indicate bilayer and monolayer graphene, respectively. Red arrows define positive D.
Extended Data Fig. 3 Evolution of the band structure of tMBG with potential energy difference U.
The path in k-space is (Γ→K→Γ). Black (red) colour indicates the momentum path (Γ→K1→Γ)(Γ→K2→Γ), with major contribution from the Dirac (parabolic) band inherited from MLG (BLG) located at the corner K1 (K2) of the MBZ. a, Band structure without coupling the two subsystems through the tunnelling matrices T. b, Band structure with non-zero tunnelling matrices for zero interlayer potential energy difference U=0. The coupling of the two subsystems even without applying potential difference U results in shifting the Dirac cone upwards. c-h, Effect of applying positive interlayer potential energy difference U>0. The bandgap at CNP needs U larger than 30 meV to open up. Moreover, electron band c1 is flatter than the hole band v1. i-n, Effect of applying negative interlayer potential energy difference U<0. In contrast to when U>0, it requires larger energy difference U< -50 meV to open a gap at CNP.
Extended Data Fig. 4 Dirac energy shift as a function of twist angle.
Solid black curve is the Dirac energy shift produced by our continuum model and the red dots are the results of tight-binding model in Ref. 20.
Extended Data Fig. 5 Temperature dependence of correlated states under D<0 in sample S2.
a, The same figure as Fig. 3d in the main text, but after subtracting smooth backgrounds. b, Peak amplitudes as a function of temperature at commensurate electron fillings.
Extended Data Fig. 6 Calculation of group velocity and DOS for twist angle θ=1.22o for U = 40meV.
a, Group velocity \({\it{v}}_{\it{g}} = \frac{1}{\hbar }\left| {\nabla _{\it{k}}{\it{E}}\left( {\it{k}} \right)} \right|\) contours of conduction band in the momentum space. Black line shows BZ, and the red dots mark the positions of saddle points, that is vHS. b, Linetrace of DOS as a function of band energy at U = 40 meV. Inset shows the data as a function of electron filling. Blue solid line marks the filling at n/n0 = 0.155, discussed in Fig. 4.
Extended Data Fig. 7 Additional data on the nonlinear I-V behaviour.
a, Differential resistance dV/dI as a function of Ib and displacement field D at n/n0=0.15. The coloured dashed lines indicate four regions discussed in the main text. b, dV/dI as a function of bias current Ib and carrier concentration at D =0.38V/nm. Blue dashed line indicates the region where Ith exhibits two additional sharp peaks. Dark red dashed lines in (a) and (b) mark Ith, at which dV/dI demonstrates a sharp rise. c, Resistivity versus temperature under D = 0.38 V/nm. The insulating response at half filling (red) onsets at T ≈ 12K, and the resistivity near superconducting regime decreases sharply from ρxx ≈ 2.5 kΩ to ρxx≈ 400 Ω at 0.3 K (black). d, Response of the superconductivity-like dV/dI at D=0.38 V/nm, n/n0=0.15 as a function of DC current Ib at 9.3 K (red), 1.2 K (magenta) and 0.3 K (blue). e, dV/dI at D=0.38 V/nm, n/n0=0.15 as a function of perpendicular B field from 0 T to 3 T, increasing by a step of 0.5 T. f, Band structure of conduction band c1 and valence band v1 at U=30, 40, 50 and 60 meV, respectively. The blue dashed lines indicate the Fermi levels for the filling n/n0=0.15. g, Line traces of the coloured dashed lines indicated in Fig. 4b.
Source data
Source Data Fig. 1
Data for Fig. 1c.
Source Data Fig. 2
Data for Fig. 2b.
Source Data Fig. 3
Data for Fig. 3b–f.
Source Data Fig. 4
Data for Fig. 4d,e.
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Xu, S., Al Ezzi, M.M., Balakrishnan, N. et al. Tunable van Hove singularities and correlated states in twisted monolayer–bilayer graphene. Nat. Phys. 17, 619–626 (2021). https://doi.org/10.1038/s41567-021-01172-9
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DOI: https://doi.org/10.1038/s41567-021-01172-9
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