Microscopic origin of the Dirac-point gap and the n=0 Landau-level splitting in graphene on graphite, and electron-electron quasiparticle lifetimes in finite magnetic field

Low-temperature, high-magnetic-field scanning tunneling spectroscopy of graphene layers electronically decoupled from a graphite substrate (Li, Luican, and Andrei, 2009) revealed a gap of roughly 10 meV centered at the Dirac point in zero field and a matching, weakly field-dependent splitting of the n=0 Landau level, together with a linearly energy-dependent inverse quasiparticle lifetime with slope γ of about 9 fs/eV. The authors noted that a sublattice-symmetry-breaking mechanism is implied by the gap being pinned at the Dirac point rather than at the Fermi energy, but left its identification open, and pointed out that zero-field theoretical estimates of the electron-electron interaction-limited lifetime give γ near 20 fs/eV, suggesting that a finite-field calculation is needed. The problem is to identify microscopically the symmetry-breaking field responsible for the Dirac-point gap and n=0 splitting in this system, and to compute electron-electron interaction-limited quasiparticle lifetimes in finite magnetic field that quantitatively account for the measured slope. Later scanning-tunneling work on graphene on graphite has substantially narrowed the first part, but no quantitative microscopic account of the magnitude and field dependence in this geometry, and no dedicated finite-field lifetime calculation matching γ, has been found.

Background

Graphene, a single hexagonal layer of carbon, hosts low-energy carriers that behave as massless Dirac fermions: the bands meet linearly at two inequivalent corners (K and K') of the Brillouin zone, and the energy scale at which they meet is the Dirac point (DP). In a perpendicular magnetic field B the spectrum collapses into Landau levels (LLs) at energies $E_{n}=E_{D}+\mathrm{sgn}(n)\sqrt{2e\hbar v_{F}^{2}|n|B}$, where $v_{F}$ is the Fermi velocity and $n\in\{0,\pm1,\pm2,\dots\}$. The n=0 level is special: it sits at $E_{D}$ and is fourfold degenerate (two spin states times the two valleys K and K'), and its wavefunctions live on a single honeycomb sublattice per valley (A or B), which makes it uniquely sensitive to any perturbation that treats the two sublattices differently. A sublattice-asymmetric potential is a Dirac 'mass' term: it opens a gap $2\Delta$ exactly at $E_{D}$ (inversion-symmetry breaking), rather than at the Fermi energy, and it shifts the K and K' components of the n=0 LL in opposite directions, splitting it into two valley-polarized levels conventionally labeled 0+ and 0-. Scanning tunneling spectroscopy (STS), which measures the tunneling conductance versus sample bias and hence the local density of states, resolves these levels directly. Graphite (often highly oriented pyrolytic graphite, HOPG) presents a practical platform: surface layers that are rotated or otherwise decoupled from the bulk display the single-layer Landau-level sequence. Quasiparticle lifetimes are extracted from the widths of the LL peaks: for Lorentzian peaks the inverse lifetime follows $1/\tau_{n}=E_{n}/\gamma+1/\tau_{0}$, where the slope γ measures the intrinsic, energy-dependent decay rate and is expected, for electron-electron scattering in graphene, to be a non-Fermi-liquid signature — the inverse lifetime grows linearly in energy rather than quadratically as it would in an ordinary Fermi liquid — whose value tests interaction theory. In the 2009 experiment of Li, Luican, and Andrei on decoupled graphene on graphite, Lorentzian line shapes gave γ of about 9 fs/eV, whereas zero-field theoretical estimates of the electron-electron limited lifetime (González, Guinea, and Vozmediano, 1996; Müller, Fritz, and Sachdev, 2008) give roughly 20 fs/eV; the authors suggested that interactions, being enhanced in magnetic field, could reconcile the numbers if the calculation were done at finite field. The same measurements found the ~10 meV gap at the DP in zero-field spectra and a ~10 meV, weakly field-dependent splitting of the n=0 LL, and the authors argued from the pinning of the gap at $E_{D}$ that it is driven by broken sublattice symmetry while leaving the symmetry-breaking mechanism, the splitting mechanism, and the finite-field lifetime calculation open.

Problem Statement

Identify the microscopic mechanism that opens the roughly 10 meV gap centered at the Dirac point — the energy $E_{D}$ at which graphene's conduction and valence bands meet — in electronically decoupled graphene on graphite, and that produces the corresponding, weakly field-dependent splitting of the n=0 Landau level (the fourfold spin- and valley-degenerate level pinned at $E_{D}$); in particular, elucidate the symmetry-breaking field or perturbation (candidate pictures include broken sublattice symmetry driven by the substrate, interlayer hybridization, disorder-induced chiral-symmetry breaking, i.e. random sublattice-asymmetric hopping, and interaction-driven order) that pins the gap at $E_{D}$ rather than at the Fermi energy, and establish quantitatively the magnitude and field dependence of the gap and of the n=0 splitting. In addition, compute the electron-electron interaction-limited quasiparticle lifetimes of Dirac fermions in finite magnetic field, and determine whether they quantitatively account for the measured linear energy dependence of the inverse lifetime in the Landau levels of graphene on graphite, with slope γ of about 9 fs/eV, given that zero-field theoretical estimates give roughly 20 fs/eV. These three asks are jointly the target of the originating experiment.

References

1. G. Li, A. Luican, E. Y. Andrei, 'Scanning Tunneling Spectroscopy of Graphene on Graphite', Phys. Rev. Lett. 102, 176804 (2009). DOI: 10.1103/PhysRevLett.102.176804, https://doi.org/10.1103/PhysRevLett.102.176804 (arXiv:0803.4016, https://arxiv.org/abs/0803.4016)
2. W.-X. Wang, L.-J. Yin, J. Qiao, T. Cai, S.-Y. Li, R.-F. Dou, J.-C. Nie, X. Wu, L. He, 'Atomic resolution imaging of the two-component Dirac-Landau levels in a gapped graphene monolayer', Phys. Rev. B 92, 165420 (2015). DOI: 10.1103/PhysRevB.92.165420, https://doi.org/10.1103/PhysRevB.92.165420 (arXiv:1505.06398, https://arxiv.org/abs/1505.06398)
3. L.-J. Yin, S.-Y. Li, J. Qiao, J.-C. Nie, L. He, 'Landau quantization in graphene monolayer, Bernal bilayer, and Bernal trilayer on graphite surface', Phys. Rev. B 91, 115405 (2015). DOI: 10.1103/PhysRevB.91.115405, https://doi.org/10.1103/PhysRevB.91.115405 (arXiv:1501.01538, https://arxiv.org/abs/1501.01538)
4. S.-Y. Li, Y. Zhang, L.-J. Yin, L. He, 'Scanning tunneling microscope study of quantum Hall isospin ferromagnetic states in the zero Landau level in a graphene monolayer', Phys. Rev. B 100, 085437 (2019). DOI: 10.1103/PhysRevB.100.085437, https://doi.org/10.1103/PhysRevB.100.085437 (preprint posted as arXiv:1904.06902 under the shorter title 'STM Study of Quantum Hall Isospin Ferromagnetic States of Zero Landau Level in Graphene Monolayer', https://arxiv.org/abs/1904.06902)
5. L.-J. Yin, L.-J. Shi, S.-Y. Li, Y. Zhang, Z.-H. Guo, L. He, 'High-Magnetic-Field Tunneling Spectra of ABC-Stacked Trilayer Graphene on Graphite', Phys. Rev. Lett. 122, 146802 (2019). DOI: 10.1103/PhysRevLett.122.146802, https://doi.org/10.1103/PhysRevLett.122.146802 (arXiv:1811.10142, https://arxiv.org/abs/1811.10142)
6. A. L. C. Pereira, 'Splitting of critical energies in the n=0 Landau level of graphene', New J. Phys. 11, 095019 (2009). DOI: 10.1088/1367-2630/11/9/095019, https://doi.org/10.1088/1367-2630/11/9/095019
7. H. Funk, A. Knorr, F. Wendler, E. Malic, 'Microscopic view on Landau level broadening mechanisms in graphene', Phys. Rev. B 92, 205428 (2015). DOI: 10.1103/PhysRevB.92.205428, https://doi.org/10.1103/PhysRevB.92.205428
8. Q. Li, S. Das Sarma, 'Finite temperature inelastic mean free path and quasiparticle lifetime in graphene', Phys. Rev. B 87, 085406 (2013). DOI: 10.1103/PhysRevB.87.085406, https://doi.org/10.1103/PhysRevB.87.085406
9. J. González, F. Guinea, M. A. H. Vozmediano, 'Unconventional Quasiparticle Lifetime in Graphite', Phys. Rev. Lett. 77, 3589 (1996). DOI: 10.1103/PhysRevLett.77.3589, https://doi.org/10.1103/PhysRevLett.77.3589
10. M. Müller, L. Fritz, S. Sachdev, 'Quantum-critical relativistic magnetotransport in graphene', Phys. Rev. B 78, 115406 (2008). DOI: 10.1103/PhysRevB.78.115406, https://doi.org/10.1103/PhysRevB.78.115406
11. A. A. Sokolik, Yu. E. Lozovik, 'Many-body renormalization of Landau levels in graphene due to screened Coulomb interaction', Phys. Rev. B 97, 075416 (2018). DOI: 10.1103/PhysRevB.97.075416, https://doi.org/10.1103/PhysRevB.97.075416
12. A. Luican, G. Li, E. Y. Andrei, 'Quantized Landau level spectrum and its density dependence in graphene', Phys. Rev. B 83, 041405 (2011). DOI: 10.1103/PhysRevB.83.041405, https://doi.org/10.1103/PhysRevB.83.041405