Abstract
Rhombohedral-stacked multilayer graphene hosts a pair of flat bands touching at zero energy, which should give rise to correlated electron phenomena that can be tuned further by an electric field. Moreover, when electron correlation breaks the isospin symmetry, the valley-dependent Berry phase at zero energy may give rise to topologically non-trivial states. Here we measure electron transport through hexagonal boron nitride-encapsulated pentalayer graphene down to 100 mK. We observed a correlated insulating state with resistance at the megaohm level or greater at charge density n = 0 and displacement field D = 0. Tight-binding calculations predict a metallic ground state under these conditions. By increasing D, we observed a Chern insulator state with C = −5 and two other states with C = −3 at a magnetic field of around 1 T. At high D and n, we observed isospin-polarized quarter- and half-metals. Hence, rhombohedral pentalayer graphene exhibits two different types of Fermi-surface instability, one driven by a pair of flat bands touching at zero energy, and one induced by the Stoner mechanism in a single flat band. Our results establish rhombohedral multilayer graphene as a suitable system for exploring intertwined electron correlation and topology phenomena in natural graphitic materials without the need for moiré superlattice engineering.
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Data availability
The data shown in the main figures are available from the Harvard Dataverse Repository at https://doi.org/10.7910/DVN/ISWXLA. The datasets generated during and/or analysed during this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge helpful discussions with F. Zhang, T. Senthil, L. Levitov, L. Fu, Z. Dong and A. Patri. L.J. acknowledges support from a Sloan Fellowship. Work by Tonghang Han was supported by NSF grant number DMR- 2225925. The device fabrication of this work was supported by the STC Center for Integrated Quantum Materials, NSF grant number DMR-1231319. Device fabrication was carried out at the Harvard Center for Nanoscale Systems and MIT.Nano. Part of the device fabrication was supported by the USD(R&E) under contract no. FA8702-15-D-0001. K.W. and T.T. acknowledge support from the JSPS KAKENHI (grant numbers 20H00354, 21H05233 and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan. H.P. acknowledges support by NSF grant number PHY-1506284 and AFOSR grant number FA9550-21-1-0216. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation Cooperative Agreement No. DMR-2128556* and the State of Florida.
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L.J. supervised the project. Tonghang Han, Z.L., G.S., J.S. and J.W. performed the DC magneto-transport measurements. Tonghang Han and Tianyi Han fabricated the devices. K.W. and T.T. grew the hBN single crystals. H.P. contributed to the data analysis. All authors discussed the results and wrote the paper.
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Extended data
Extended Data Fig. 1 Identification of rhombohedral pentalayer graphene.
a, AFM topography map of a pentalayer graphene sample on a SiO2/Si substrate. (The small region on the right corresponds to a graphene tetralayer.) b, Near-field infrared nanoscopy image of the same pentalayer graphene sample as in a, showing different contrast on the pentalayer region. The bright region corresponds to Bernal stacking and the darker region corresponds to rhombohedral stacking. c, Raman spectra taken at rhombohedral and Bernal stacking domains in the pentalayer graphene sample as in b. ω is the Raman shift.
Extended Data Fig. 2 Single particle band structure and density of states of rhombohedral multilayer graphene.
a–f, Tight-binding calculation of single-particle band structure and density of states (DOS) for rhombohedral multilayer graphene (layer number N = 2, 3, 4, 5, 7, 9). Due to the remote hopping terms, the band structure deviates from E ≈ kN at low energy. The rhombohedral pentalayer graphene has the flattest band among all layer numbers.
Extended Data Fig. 3 Correlated insulator at n = D = 0 in pentalayer rhombohedral graphene.
a, The calculated single-particle density of states (DOS) in Bernal bilayer, rhombohedral trilayer (time a factor of 15 and 3 for comparison), and pentalayer graphene at D = 0. The blue and orange shaded areas depict the DOS of the valence and conduction band. b–d, The n-D Rxx map of the bilayer, trilayer and pentalayer graphene measured at 2 K. Pentalayer graphene has a much larger DOS and band overlap at n = 0 compared to the bilayer and trilayer case and expects to be more conducting at D = n = 0. However, both bilayer and trilayer graphene are conducting at D = n = 0, while an insulating state appears at D = n = 0 in pentalayer graphene, indicating the non-single-particle origin of the insulating state. The off-diagonal line in the n-D map of pentalayer graphene is due to the big contact resistance when the bottom gate is zero.
Extended Data Fig. 4 Phase diagram for both electron and hole doping.
a, b, Color plots of four-probe resistance Rxx as a function of carrier density n and displacement field D for the hole doping side and electron doping side measured at B = 0 and a temperature of 100 mK. Colored dots label different phases including band insulator (BI), correlated insulator (CI), spin-polarized half metal (SPHM), isospin-polarized quarter metal (IPQM) and unpolarized metal (UP). c, d, Hall resistance Rxy and longitudinal resistance Rxx as a function of the out-of-plane magnetic field at the red dot in a. e, f, Rxy and Rxx at the yellow dot in a. The quarter metal c & d shows a clear anomalous Hall effect and magnetic hysteresis, indicating a net valley polarization. While the half metal e & f does not show anomalous Hall effect or magnetic hysteresis, indicating the absence of net valley polarization. Therefore, we conclude the half metal to be spin polarized but valley unpolarized.
Extended Data Fig. 5 Additional temperature dependence data of the correlated insulator.
a, Temperature dependence of the four-probe resistance Rxx measured at charge neutrality (n = 0). BI and CI stand for band insulator and correlated insulator. The correlated insulator develops below ~ 30 K. The two white arrows indicate the semimetal phase with an anomalous temperature dependence at the low-temperature region, discussed in Fig. 4c. b, Rxx versus D at even higher temperatures. As the temperature is increased, the resistive state at D = 0 disappears and evolves to a dip in Rxx. Circles trace the center of the bumps in Rxx.
Extended Data Fig. 6 The C = -3 state at D = 0.21 V/nm.
a, b, 2D color plot of Rxx and Rxy versus carrier density n and out-of-plane magnetic field B taken at D = 0.21 V/nm at a temperature of 100 mK. The dashed line indicates the C = -3 state. The C = -3 state is the only visible state on the hole doping side at low magnetic fields, in contrast to the electron side where all Landau levels appear at a similar magnetic field.
Extended Data Fig. 7 Tracing the C = -3 state towards zero magnetic field.
a, 2D color plot of dRxx/dB versus carrier density n and out-of-plane magnetic field B taken at D = 0.11 V/nm at a temperature of 100 mK. The dashed line indicates the C = -3 state. b, 2D color plot of dRxy/dn versus carrier density n and out-of-plane magnetic field B taken at D = 0.11 V/nm at a temperature of 100 mK. The dashed line indicates the C = -3 state. The C = -3 state is the only visible state on the hole doping side at low magnetic field and traces all the way to 0 T. On the electron side, a complete set of Landau levels is observed and disappears at around 1.5 T.
Extended Data Fig. 8 Chern insulators at the negative side of D.
a, 2D color plots of Rxx (upper panel) and Rxy (lower panel) at B = 1 T, revealing three Chern insulator states at the hole-doped side in the gap-closing range of D. The state with a Chern number C = -5 happens at D = -0.15 V/nm, while two states with C = -3 happen at D = -0.093 V/nm and D = -0.2 V/nm, as indicated by the dashed lines. b, c, d, 2D color plots of Rxx (upper panel) and Rxy (lower panel) versus n and B at D = -0.15 V/nm, -0.093 V/nm and -0.2 V/nm respectively. The dashed lines indicate the n-B relation of the Chern insulators as predicted by the Streda formula. These results are consistent with those of the positive D side.
Extended Data Fig. 9 Evolution of the band structure at charge-neutrality with D.
a, Rxx measured at n = 0 at 2 K. b, The band structure schematic of each isospin flavor at charge-neutrality under different D values. The colored dots correspond to the states in a, including LAF (layer antiferromagnet), SM (semimetal), QAH (quantum anomalous Hall) and LP (layer polarized state). The color of each band represents the valley Chern number (black: 5/2, red: -5/2), while the band with a grey color indicates the gap-closing case. c, The layer polarization of the LP, QAH and LAF state. The layer polarization of both conduction and valence band for each isospin flavor is shown, where green and orange color corresponds to the K’ and K valley. At D = 0, the system starts with the LAF state where charges are evenly distributed in the top and bottom layer in a spin-polarized manner (point A). As D increases, the gap of spin up (down) flavor expands (shrinks) due to its layer configuration (point B). It is important to note that the gap sizes of the two valleys with the same spin may differ. Consequently, one of the two gaps closes and reopens first (point C), leading to a situation where the gap of the K valley inverts while the gap of the K’ valley remains the same, resulting in the so-called QAH state (point D). Moreover, the layer polarization becomes partially polarized at this stage. If D continues to increase, the gap of the other valley eventually closes and reopens (point E), leading to the fully layer-polarized state (point F).
Extended Data Fig. 10 Temperature dependence of the SPHM, IPQM and UP states.
Temperature-dependent Rxx at D = 0.16 V/nm and n = 0 (red), D = 0 and n = -2.5*1012cm-2 (light grey, UP), D = 0.26 V/nm and n = -1.9*1012cm-2 (dark grey, SPHM), and D = 0.26 V/nm and n = -0.8*1012cm-2 (black, IPQM).
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Supplementary Figs. 1–8 and Discussion (Sections I–VII).
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Han, T., Lu, Z., Scuri, G. et al. Correlated insulator and Chern insulators in pentalayer rhombohedral-stacked graphene. Nat. Nanotechnol. 19, 181–187 (2024). https://doi.org/10.1038/s41565-023-01520-1
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DOI: https://doi.org/10.1038/s41565-023-01520-1
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