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Cascade of isospin phase transitions in Bernal-stacked bilayer graphene at zero magnetic field

Abstract

Emergent phenomena arising from the collective behaviour of electrons is expected when Coulomb interactions dominate over the kinetic energy, and one way to create this situation is to reduce the electronic bandwidth. Bernal-stacked bilayer graphene intrinsically supports saddle points in the band structure that are predicted to host a variety of spontaneous symmetry-broken states1,2,3,4,5,6,7. Here we show that bilayer graphene displays a cascade of symmetry-broken states with spontaneous spin and valley isospin ordering at zero magnetic field. We independently tune the carrier density and electric displacement field to explore the phase space of isospin order. Itinerant ferromagnetic states emerge near the conduction and valence band edges with complete spin and valley polarization. At larger hole densities, twofold degenerate quantum oscillations manifest in an additional symmetry-broken state that is enhanced by the application of an in-plane magnetic field. Both symmetry-broken states display enhanced layer polarization, suggesting a coupling to the layer degree of freedom1,7. These states occur in the absence of a moiré superlattice and are intrinsic to natural graphene bilayers. Therefore, we demonstrate that bilayer graphene represents a related but distinct approach to produce collective behaviour from flat dispersion, complementary to engineered moiré structures.

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Fig. 1: Flat dispersion in Bernal-stacked bilayer graphene.
Fig. 2: Degeneracy of symmetry-broken states in B.
Fig. 3: Fermi-surface signatures in quantum oscillations of the symmetry-broken states.
Fig. 4: Phase transitions between spin and valley polarization.

Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We would like to acknowledge helpful discussions with L. Levitov, L. Fu, Y. Zhang and C. Collignon. R.A. acknowledges support by the STC Center for Integrated Quantum Materials and National Science Foundation (NSF) grant no. DMR-1231319 (measurements and data analysis). P.J.-H. acknowledges support by the US Department of Energy, Office of Science, Basic Energy Sciences, under award no. DE-SC0020149 (device fabrication); by the Army Research Office (early nanofabrication development) through grant no. W911NF1810316; and the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF9463. Q.M. acknowledges support by the Center for the Advancement of Topological Semimetals, an Energy Frontier Research Center funded by the US Department of Energy Office of Science, through the Ames Laboratory under contract DE-AC02-07CH11358 (data analysis and manuscript writing). This work made use of the Materials Research Science and Engineering Center Shared Experimental Facilities supported by NSF grant no. DMR-0819762. This work was performed in part at the Harvard University Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the NSF under NSF ECCS award no. 1541959. S.A. is partially supported by NSF Graduate Research Fellowship Program via grant no. 1122374. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan (grant no. JPMXP0112101001), and JSPS KAKENHI (grant nos. 19H05790, 20H00354 and 21H05233).

Author information

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Authors

Contributions

S.C.d.l.B. and S.A. performed the measurements and calculations, analysed the data and wrote the manuscript with input from all the authors. Z.Z. fabricated the samples. K.W. and T.T. grew the hexagonal boron nitride crystals. S.C.d.l.B, Q.M. and Z.Z. conceived of the measurements, and Q.M., P.J.-H. and R.A. supervised the project.

Corresponding authors

Correspondence to Sergio C. de la Barrera, Pablo Jarillo-Herrero or Raymond Ashoori.

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Nature Physics thanks Wenzhong Bao and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Evolution of bilayer and rhombohedral trilayer graphene bands with potential asymmetry.

a Evolution of bilayer and b rhombohedral (ABC) trilayer graphene bands along kx as interlayer asymmetry, Δg, is increased from zero. ‘Flatness’ of the dispersion in both cases is apparent as the number of states near the valence band edge occupies a large extent in kx. c Qualitative comparison of the k-space extent of the valence band edge, kVBE, for bilayer and ABC trilayer graphene as a function of Δg extracted from the matching points in a-b. Bands were calculated using the same tight-binding model and parameters as in Supplementary Fig. 2, following Ref. 41.

Extended Data Fig. 2 Single particle density of states for bilayer graphene.

a Calculated single particle density of states (DOS) versus carrier density for fixed values of interlayer potential asymmetry, Δg, showing a single Van Hove singularity per conduction/valence band. The gap between bands collapses to a single point (n = 0) when plotted versus density, rather than energy. Inset: Band surfaces with isoenergy contours for different carrier densities. b Inverted density of states, DOS−1, showing a sharp minimum (dark feature) at the position of the large Van Hove singularity (VHs) trending toward larger hole density as potential asymmetry increases (in analogy to D-field in experiment). A smaller VHs is also seen for electrons, but broadens with increasing Δg. The DOS was obtained by numerical integration of tight binding bands using the model and parameters of Ref. 41.

Extended Data Fig. 3 Layer polarization in the zero-energy Landau level.

a Map of Cdiff measured in Device II at B = 4 T and 1.7 K. Sloped lines throughout the map arise from cyclotron gaps in the graphite gate electrodes. b Density cuts taken at fixed displacement fields indicated by the dotted lines in a.

Extended Data Fig. 4 Displacement field dependence of Cdiff at high and low temperatures.

Maps of Cdiff measured in Device II at a 1.8 K and b 40 K. In a, regions of enhanced layer polarization at finite displacement field correspond to phase transitions between symmetry- broken states. c Difference map showing the data in b subtracted from those in a. B = 0 in both measurements.

Extended Data Fig. 5 Capacitance bridge amplifier circuit.

Circuit schematic showing the dual-amplifier configuration used to measure Cp and Cdiff in the same device. Combinations of AC and DC voltages are applied to terminals Vtg, Vbl, Vbg in order to gate the bilayer graphene, excite charge for the capacitance measurement, and to simultaneously power the relevant amplifier. Amplifier 2 is used for Cp measurements with an AC excitation on the top gate, while Amplifier 1 is used for Cdiff measurements, with AC excitations applied to the top and bottom gates simultaneously.

Supplementary information

Supplementary Information

Supplementary Figs. 1–5, Discussion and Table 1.

Source data

Source Data Fig. 1

Numerical data.

Source Data Fig. 2

Numerical data.

Source Data Fig. 3

Numerical data.

Source Data Fig. 4

Numerical data.

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de la Barrera, S.C., Aronson, S., Zheng, Z. et al. Cascade of isospin phase transitions in Bernal-stacked bilayer graphene at zero magnetic field. Nat. Phys. 18, 771–775 (2022). https://doi.org/10.1038/s41567-022-01616-w

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