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# Electronic phase separation in multilayer rhombohedral graphite

## Abstract

Of the two stable forms of graphite, hexagonal and rhombohedral, the former is more common and has been studied extensively. The latter is less stable, which has so far precluded its detailed investigation, despite many theoretical predictions about the abundance of exotic interaction-induced physics1,2,3,4,5,6. Advances in van der Waals heterostructure technology7 have now allowed us to make high-quality rhombohedral graphite films up to 50 graphene layers thick and study their transport properties. Here we show that the bulk electronic states in such rhombohedral graphite are gapped8 and, at low temperatures, electron transport is dominated by surface states. Because of their proposed topological nature, the surface states are of sufficiently high quality to observe the quantum Hall effect, whereby rhombohedral graphite exhibits phase transitions between a gapless semimetallic phase and a gapped quantum spin Hall phase with giant Berry curvature. We find that an energy gap can also be opened in the surface states by breaking their inversion symmetry by applying a perpendicular electric field. Moreover, in rhombohedral graphite thinner than four nanometres, a gap is present even without an external electric field. This spontaneous gap opening shows pronounced hysteresis and other signatures characteristic of electronic phase separation, which we attribute to emergence of strongly correlated electronic surface states.

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## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

This work was supported by the EU Graphene Flagship Program, the European Research Council, the Royal Society, and the Engineering and Physical Sciences Research Council (EPSRC). A.M. acknowledges the support of an EPSRC Early Career Fellowship (EP/N007131/1). S.V.M. was supported by the RFBR (20-02-00601). This work was partially supported by the French National Research Agency (ANR) in the framework of a RhomboG grant (ANR-17-CE24-0030).

## Author information

Authors

### Contributions

A.M. conceived the presented idea and directed the project. Y.S., S.V.M., S.O., C.M., J.B., J.Y., A.I.B. and B.A.P. performed transport measurements. S.X., Y.Y. and S.-K.S. fabricated devices. Y.S., S.O., C.M. and J.Y. performed data analysis. S.S. and V.I.F. developed the theory and performed theoretical calculations. Y.S., S.O., A.K.G., V.I.F. and A.M. contributed to the interpretation of data. K.W. and T.T. grew hBN single crystals. A.M., Y.S., A.K.G., S.S., V.I.F. and K.S.N. contributed to the writing of the manuscript. All authors discussed the results and commented on the manuscript.

### Corresponding author

Correspondence to Artem Mishchenko.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Dmitri Efetov and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Effect of stacking sequence on the displacement-field-induced bandgap.

a, b, Raman maps of device 3, a 6.5-nm graphite flake with domains of differing local stacking sequence, shown before (a) and after (b) encapsulation. The colour coding indicates the ratio of the integrated area of the low-frequency component of the graphite Raman 2D band (ranging around 2,670–2,700 cm−1) to that of the high-frequency component (ranging around 2,700–2,730 cm-1). Coloured dots in a and b mark the positions where the Raman spectra shown in c and d respectively were taken. Scale bars, 10 μm. c, d, Typical 2D Raman peaks of HG (black curves A and a), RG (red curves C and c) and graphite of mixed ABA and ABC stacking (blue curves B and b), before (c) and after (d) encapsulation. Laser wavelength, 532 nm. e, ρxx as a function of D for Hall bar devices made using domains at point b (light blue) and point c (red), respectively, T = 1.6 K. Inset, optical micrograph of the devices before shaping them into the Hall bar geometry. f, g, Resistivity maps ρxx(nt, nb) of graphite with ABC stacking (f) and mixed stacking (g); T = 1.6 K, B = 0 T.

### Extended Data Fig. 2 Band structures of multilayer RG with stacking faults.

a, Schematic of twin-boundary defect. b, c, Calculated spectra for the ABCABCABACBACBACBA sequence at positive (b) and negative (c) D. d, Schematic of a buried Bernal stacking fault ABCBCA. e, f, Calculated band structure for such a defect: ABCABCABABCABCABCA sequence at positive (e) and negative (f) D. g, Schematic of surface stacking fault ABCAC. h, i, Spectra for the ABCABCABCABCABCABA sequence at positive (h) and negative (i) D. In all the calculations we used D = ±1 V nm−1. See Methods section ‘Possibility of stacking faults’ for details of nomenclature used in a, d, g. E is the band energy, px is the in-plane momentum, and pc = γ1/v, where v is the Dirac velocity.

### Extended Data Fig. 3 The quantum Hall effect in thin RG.

a, Hall resistivity ρxy (red curve) and longitudinal resistivity ρxx (black curve) as a function of magnetic field B measured at 20 mK for the same device as in main text Fig. 3 (device 1); n = 2.3 × 1012 cm−2, D = 0 V nm−1. b, ρxy and ρxx as a function of n for the same device as in a; D = 0, B = 10 T, T = 20 mK.

### Extended Data Fig. 4 Landau levels from surface states of multilayer RG.

a, Calculated free-particle spectrum. b, Conductivity map σxx(nt, nb) for 3-nm-thick RG (device 1) measured at B = 9 T and T = 20 mK. Note that, for two independent surfaces, LLs should form sets of horizontal and vertical lines. The observed behaviour suggests that top and bottom surfaces of our RG devices are nearly independent 2D systems. c, Landau fan σxx(n, B) for device 5 (3.3-nm-thick RG); D = 0, T = 0.25 K. d, Differential dρxx/dν(ν, B) map on the electron side (same device as in c). The red arrows indicate LL crossings. e, σxx(nb, B) for device 8 (7.2-nm-thick RG) measured at T = 1.7 K. f, σxx(n, B) for device 4 (16.5-nm-thick RG) at D = 0 and T = 0.25 K.

### Extended Data Fig. 5 Single-gate (D ≠ 0) Landau fan diagrams highlighting the robust ν = −N quantum Hall state in N-layer-thick RG.

a, Conductivity map σxy(ν, B) for 7-layer-thick RG (device 7). b, Conductivity map σxx(ν, B) for 9-layer-thick RG (device 1). c, σxx(ν, B) for 11-layer-thick RG (device 6).

### Extended Data Fig. 6 Insulating states and hysteretic behaviour in multilayer RG.

a, b, Additional data for 3.3-nm-thick RG (device 5, same as in main text Fig. 4a, b). a, Main panel, temperature dependence of ρxx(n) around the insulating state, B = 0 T. Inset, Arrhenius plot for the peak resistivity, indicating the presence of a bandgap of about 2–3 meV. The y axis shows peak resistivity, and red line indicates the fit. b, Main panel, histogram of the conductivity values found on the hysteretic curves σxx(n), such as those shown in the inset. Eighty-three such curves were used to make the histogram, where data from forward and backward sweeps are plotted in blue and red, respectively. D = 0, T = 250 mK, B = 0. c, Hysteresis (shaded red) in ρxx(B) observed in device 7 (2.3-nm-thick RG) at the charge neutrality point; d, hysteresis (shaded red) in ρxx(n) at B = 5 T. For c and d, T = 250 mK. eg, Hysteretic behaviour of device 1 for different cooling cycles. No noticeable hysteresis was found for the first cooling event (e). Hysteresis in ρxx(n) (f) and ρxx(D) (g) is clearly seen after another cooling. Solid (dashed) lines indicate positive (negative) sweep directions and the coloured areas highlight the difference between the sweep directions. T = 20 mK; B = 0 T.

### Extended Data Fig. 7 Stacking order of device 6 at different fabrication stages.

a, Optical image of the graphite flake (left) and the corresponding Raman map with a step size of 0.8 μm × 0.8 μm (right). Scale bar, 10 μm. b, Optical image of the graphite flake encapsulated by hBN (left) and the corresponding Raman map with a step size of 0.7 μm × 0.7 μm (right). Scale bar, 10 μm. c, Optical image of the finished Hall bar (left) and the corresponding Raman map with a step size of 0.5 μm × 0.5 μm (right). Scale bar, 4 μm. The colour coding of the Raman maps in ac indicates the ratio of the integrated area of the low-frequency component (ranging around 2,635–2,665 cm−1) to that of the high-frequency component (ranging around 2,665–2,695 cm−1) of the graphite Raman 2D band. The coloured dots in ac mark the positions where the Raman spectra shown in e were taken. d, Optical image showing that the Hall bar device is made from the ABC stacked region. Scale bar, 10 μm. e, Typical 2D Raman peaks of RG (dots a, c and d) and of graphite of mixed ABA and ABC stacking (dot b). Laser excitation wavelength, 633 nm.

### Extended Data Fig. 8 Temperature dependence of resistivity.

a, ρxx as a function of T at zero gate doping for RG device 4 with a thickness of 16.5 nm (blue solid curve), for RG device 5 with a thickness of 3.3 nm (red solid curve), and for 6-nm HG (grey dashed line). b, Same data as a but plotted on a log scale. While cooling down, ρxx of RG first increases for T > Tc, and then decreases, in sharp contrast to the monotonic decrease of ρxx for HG. The critical temperature Tc decreases with increasing thickness of RG. Besides the presence of Tc, ρxx of the 3.3-nm RG device 5 shows a sharp increase for T < 6 K owing to the phase transition to the insulating state.

### Extended Data Fig. 9 Bandgap opening by displacement field.

a, Band dispersion of RG under an applied displacement field; bandgap $$\mathop{\varDelta }\limits^{ \sim }$$ is masked by a bandwidth of 2γ4γ1/γ0 such that only the effective gap $$\varDelta \,=\,\mathop{\varDelta }\limits^{ \sim }-2{\gamma }_{4}{\gamma }_{1}/{\gamma }_{0}$$ is visible in transport measurements. b, Calculated dependence of $$\mathop{\varDelta }\limits^{ \sim }$$ and Δ on displacement field for N = 9 layers.

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Shi, Y., Xu, S., Yang, Y. et al. Electronic phase separation in multilayer rhombohedral graphite. Nature 584, 210–214 (2020). https://doi.org/10.1038/s41586-020-2568-2

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