STM spectroscopy of graphene supported on hexagonal boron nitride (hBN) found that the Fermi velocity extracted from the gate-dependent Dirac-point motion differs by about 25% between electrons (1.16×1061.16\times10^6 m/s) and holes (0.94×1060.94\times10^6 m/s), with the sign of the asymmetry depending on the moiré period; the authors could not explain it and suggested next-nearest-neighbour coupling as a possible cause. The open problem is to establish the physical origin of this asymmetry and its moiré dependence, and to build the microscopic theory that quantitatively connects the moiré geometry, higher-order interlayer couplings, and the observed local spectroscopic asymmetries. Later work validated each candidate mechanism in adjacent settings — next-nearest-neighbour interlayer coupling, substrate-induced band asymmetry, and interaction-driven velocity renormalization — but none reconciles the magnitude and moiré-sign pattern of the original observation, so the question of the quantitative origin remains open.

Background

Graphene, a single layer of carbon atoms in a honeycomb lattice, hosts low-energy carriers that behave as massless Dirac fermions: the energy depends linearly on momentum, E=ℏvFkE=\hbar v_F k, with a Fermi velocity vFv_F of about 10610^6 m/s, and the conduction and valence bands touch at the charge-neutrality (Dirac) point, where the density of states ideally vanishes. In scanning tunnelling spectroscopy (STS), the differential conductance dI/dVdI/dV measured by a scanning tunnelling microscope (STM) is proportional to the local density of states; tracking the dI/dVdI/dV minimum while sweeping a back-gate voltage VgV_g yields the Dirac-point energy, whose Vg\sqrt{V_g} dependence gives vFv_F through ED=ℏvFπαVgE_D=\hbar v_F\sqrt{\pi\alpha V_g} (with α\alpha the gate-to-graphene capacitance per unit area). Graphene placed on amorphous SiO2_2 suffers strong electrostatic disorder that fragments it into microscopic electron and hole puddles; hexagonal boron nitride (hBN), an insulating layered crystal with the same honeycomb atomic structure as graphene but a 1.8% larger lattice constant, provides an atomically flat, charge-trap-poor substrate whose small mismatch and relative crystal rotation generate a long-wavelength moiré superlattice in the graphene. Early theory had predicted a ~50 meV band gap for graphene on hBN from sublattice-symmetry breaking, but Xue et al. (Nature Materials 10, 282 (2011)) showed by STM that the lattice mismatch restores sublattice symmetry so the spectrum stays gapless, while the electron–hole charge fluctuations shrink by two orders of magnitude relative to SiO2_2, making the low-density Dirac-point regime accessible in a supported device. In the same measurements, however, the Fermi velocity extracted from the gate dependence came out as vF=1.16±0.01×106v_F=1.16\pm0.01\times10^6 m/s for electrons and 0.94±0.02×1060.94\pm0.02\times10^6 m/s for holes — an asymmetry of about 25% whose sign varies with the moiré pattern (the shorter moiré period gave a higher hole velocity, the longer one a higher electron velocity). The authors' numerical model included nearest- and next-nearest-neighbour interlayer hopping between carbon and boron/nitrogen atoms on finite lattices (up to 600‘ב`\times`600 unit cells) but no next-nearest-neighbour terms in the velocity analysis and no many-body self-energy (the interaction-induced correction to a quasiparticle's energy and dispersion), and they explicitly left the asymmetry's origin open, suggesting next-nearest-neighbour coupling as a candidate. Three known mechanisms can in principle break electron–hole symmetry in this system: (i) substrate interlayer coupling beyond nearest neighbours (next-nearest-neighbour interlayer hopping between carbon and boron/nitrogen sites, the intralayer next-nearest-neighbour hopping t′t' of graphene itself, and the different on-site energies of boron and nitrogen), (ii) electron–electron interactions, which renormalize the Dirac cone (the velocity rises at low carrier density, as later measured by STS and by quantum-capacitance measurements, which read out the finite density of states of a graphene electrode), and (iii) residual extrinsic disorder such as adsorbates or trapped charge, which also produces a finite dI/dVdI/dV at the Dirac-point minimum and a residual doping offset. Subsequent work established the moiré pattern as a weak periodic superlattice potential that reshapes graphene's dispersion — generating additional superlattice Dirac points, Hofstadter physics in magnetic field (the fractal 'butterfly' energy spectrum of electrons in a lattice plus magnetic flux), and moiré-dependent velocity anisotropies — and moiré miniband models that include the symmetry-breaking coupling terms are now standard, yet a quantitative account of the original velocity asymmetry has not been established.

Problem Statement

Establish the physical origin of the electron–hole asymmetry of the low-energy Fermi velocity of graphene on hexagonal boron nitride observed by gate-dependent scanning tunnelling spectroscopy — approximately 25% (vF=1.16±0.01×106v_F=1.16\pm0.01\times10^6 m/s for electrons versus 0.94±0.02×1060.94\pm0.02\times10^6 m/s for holes), with the sign depending on the moiré period (shorter moiré patterns giving a higher hole velocity, longer ones a higher electron velocity) — and provide a microscopic theory that quantitatively connects the measured moiré geometry, the higher-order interlayer couplings (such as next-nearest-neighbour interlayer hopping and the distinct boron/nitrogen on-site energies), and the observed local spectroscopic asymmetries. Establish, jointly: (i) determine which mechanism(s) dominate — substrate interlayer coupling beyond nearest neighbours (including the intralayer next-nearest-neighbour hopping t′t') or electron–electron-interaction self-energy; (ii) reproduce the magnitude and the moiré-wavelength-dependent sign of the asymmetry; and (iii) show how the residual extrinsic perturbations present in the experiment (finite dI/dVdI/dV at the Dirac-point minimum and residual charging that shifts the Dirac-point line away from Vt=0V_t=0 at finite VgV_g) are to be separated from the intrinsic band asymmetry, so that the theory can be confronted with the measured spectra.

References

  1. Jiamin Xue, Javier Sanchez-Yamagishi, Danny Bulmash, Philippe Jacquod, Aparna Deshpande, K. Watanabe, T. Taniguchi, Pablo Jarillo-Herrero, Brian J. LeRoy. Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride. Nature Materials 10, 282–285 (2011). DOI 10.1038/nmat2968; arXiv:1102.2642. https://doi.org/10.1038/nmat2968
  2. Matthew Yankowitz, Jiamin Xue, Daniel Cormode, Javier D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, Pablo Jarillo-Herrero, Philippe Jacquod, Brian J. LeRoy. Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nature Physics 8, 382–386 (2012). DOI 10.1038/nphys2272; arXiv:1202.2870. https://doi.org/10.1038/nphys2272
  3. Matthew Yankowitz, Jiamin Xue, B. J. LeRoy. Graphene on hexagonal boron nitride. Journal of Physics: Condensed Matter 26, 303201 (2014). DOI 10.1088/0953-8984/26/30/303201; arXiv:1401.5145. https://doi.org/10.1088/0953-8984/26/30/303201
  4. Jungseok Chae, Suyong Jung, Andrea F. Young, Cory R. Dean, Lei Wang, Yuanda Gao, Kenji Watanabe, Takashi Taniguchi, James Hone, Kenneth L. Shepard, Phillip Kim, Nikolai B. Zhitenev, Joseph A. Stroscio. Renormalization of the graphene dispersion velocity determined from scanning tunneling spectroscopy. Physical Review Letters 109, 116802 (2012). DOI 10.1103/PhysRevLett.109.116802. https://doi.org/10.1103/PhysRevLett.109.116802
  5. Q. Zou, B. D. Belle, L. Z. Zhang, W. D. Xiao, K. Yang, L. W. Liu, G. Q. Wang, X. M. Fei, Y. Huang, R. S. Ma, Y. Lu, P. H. Tan, H. M. Guo, S. X. Du, H.-J. Gao. Modulation of Fermi velocities of Dirac electrons in single layer graphene by moiré superlattice. Applied Physics Letters 103, 113106 (2013). DOI 10.1063/1.4821178. https://doi.org/10.1063/1.4821178
  6. G. L. Yu, R. Jalil, Branson Belle, Alexander S. Mayorov, Peter Blake, Frederick Schedin, Sergey V. Morozov, Leonid A. Ponomarenko, F. Chiappini, S. Wiedmann, Uli Zeitler, Mikhail I. Katsnelson, A. K. Geim, Kostya S. Novoselov, Daniel C. Elias. Interaction phenomena in graphene seen through quantum capacitance. Proceedings of the National Academy of Sciences 110, 3282–3286 (2013). DOI 10.1073/pnas.1300599110; arXiv:1302.3967. https://doi.org/10.1073/pnas.1300599110
  7. Zhi-Guo Chen, Zhiwen Shi, Wei Yang, Xiaobo Lu, You Lai, Hugen Yan, Feng Wang, Guangyu Zhang, Zhiqiang Li. Observation of an intrinsic bandgap and Landau level renormalization in graphene/boron-nitride heterostructures. Nature Communications 5, 4461 (2014). DOI 10.1038/ncomms5461; arXiv:1406.5571. https://doi.org/10.1038/ncomms5461
  8. D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Morozov, A. A. Zhukov, P. Blake, L. A. Ponomarenko, I. V. Grigorieva, K. S. Novoselov, F. Guinea, A. K. Geim. Dirac cones reshaped by interaction effects in suspended graphene. Nature Physics 7, 701–704 (2011). DOI 10.1038/nphys2049. https://doi.org/10.1038/nphys2049
  9. F. Amet, J. R. Williams, A. G. F. Garcia, M. Yankowitz, K. Watanabe, T. Taniguchi, D. Goldhaber-Gordon. Tunneling spectroscopy of graphene-boron-nitride heterostructures. Physical Review B 85, 073405 (2012). DOI 10.1103/PhysRevB.85.073405; arXiv:1108.2686. https://doi.org/10.1103/PhysRevB.85.073405
  10. Nicholas Dale, Ryo Mori, M. Iqbal Bakti Utama, Jonathan D. Denlinger, Conrad Stansbury, Claudia G. Fatuzzo, Sihan Zhao, Kyunghoon Lee, Takashi Taniguchi, Kenji Watanabe, Chris Jozwiak, Aaron Bostwick, Eli Rotenberg, Roland J. Koch, Feng Wang, Alessandra Lanzara. Correlation-driven electron-hole asymmetry in graphene field effect devices. npj Quantum Materials 7, 9 (2022). DOI 10.1038/s41535-021-00404-8; arXiv:2103.08076. https://doi.org/10.1038/s41535-021-00404-8
  11. Jiseon Shin, Youngju Park, Bheema Lingam Chittari, Jin-Hua Sun, Jeil Jung. Electron-hole asymmetry and band gaps of commensurate double moiré patterns in twisted bilayer graphene on hexagonal boron nitride. Physical Review B 103, 075423 (2021). DOI 10.1103/PhysRevB.103.075423; arXiv:2011.14790. https://doi.org/10.1103/PhysRevB.103.075423
  12. Daniel Rhodes, Sang Hoon Chae, Rebeca Ribeiro-Palau, James Hone. Disorder in van der Waals heterostructures of 2D materials. Nature Materials 18, 541–549 (2019). DOI 10.1038/s41563-019-0366-8. https://doi.org/10.1038/s41563-019-0366-8
  13. Gwanghyun Ahn, Hye Ri Kim, Taeg Yeoung Ko, Kyoungjun Choi, Kenji Watanabe, Takashi Taniguchi, Byung Hee Hong, Sunmin Ryu. Optical Probing of the Electronic Interaction between Graphene and Hexagonal Boron Nitride. ACS Nano 7, 1533–1541 (2013). DOI 10.1021/nn305306n; arXiv:1310.8044. https://doi.org/10.1021/nn305306n