Microscopic origin and complete assignment of two-dimensional Dirac-fermion Landau levels in scanning tunnelling spectroscopy of graphite surfaces

Low-temperature scanning tunnelling spectroscopy (STS) on the surface of highly oriented pyrolytic graphite (HOPG) in magnetic fields up to 12 T reveals Landau-level (LL) peak sequences of both massless Dirac fermions ($E_n \propto \sqrt{|n|B}$) and massive chiral Dirac fermions ($E_n \propto \sqrt{|n|(|n|+1)B}$), together with peaks that follow neither scaling, all extrapolating to a common Dirac-point energy. Li and Andrei (Nature Physics 3, 623 (2007)) identified several issues that remain open: the band-structure assignment of the non-scaling ("unidentified") peaks including surface-versus-bulk contributions; the structural mechanism (stacking faults, rhombohedral stacking, displaced or otherwise decoupled top layers) that lets strictly two-dimensional quasiparticles with sizeable spectral weight appear at a bulk graphite surface, with massless and massive chiral carriers at the same Dirac-point energy; the reconciliation of the tunneling-derived effective mass $m^{*}=(0.028\pm0.003)\,m_{\mathrm{e}}$ with larger momentum-resolved photoemission values; and a predictive theoretical description of the STS response including tip-induced perturbations, finite-layer confinement and realistic stacking. Later work classified decoupled monolayer, Bernal bilayer and Bernal trilayer surface regions and computed stacking-fault two-dimensional bands, their LL spectra and cyclotron masses, but no published model simultaneously accounts for the full set of observed peak sequences and spectral weights, identifies the structural motif in real HOPG samples, or provides the requested STS response theory.

Background

Graphene, a single layer of carbon atoms in a honeycomb lattice, hosts low-energy quasiparticles that behave as massless Dirac fermions: their energy disperses linearly, $E=\pm v_{\mathrm{F}}\hbar k$, and in a perpendicular magnetic field $B$ they form the unusual Landau-level sequence $E_{n}=E_{\mathrm{D}}+\operatorname{sgn}(n)v_{\mathrm{F}}\sqrt{2e\hbar |n|B}$, including a field-independent $n=0$ level at the Dirac-point energy $E_{\mathrm{D}}$ (the energy where the linear bands cross). In a graphene bilayer with Bernal (AB) stacking, the two low-energy bands are instead parabolic, $E=\pm \hbar^{2}k^{2}/2m^{*}$, and the carriers are massive but chiral Dirac fermions whose LLs obey $E_{n}=\operatorname{sgn}(n)\hbar\omega_{c}\sqrt{|n|(|n|+1)}$ with cyclotron frequency $\omega_{c}=eB/m^{*}$ and effective mass $m^{*}$. Bulk graphite consists of Bernal-stacked graphene layers coupled by weak van der Waals interaction; its electronic structure is conventionally parametrized by the Slonczewski-Weiss-McClure (SWM) model, a standard set of tight-binding hopping amplitudes (intra-layer and inter-layer) fitted to graphite band measurements. HOPG (highly oriented pyrolytic graphite) is a bulk graphite form produced by high-temperature decomposition of hydrocarbons, with well-aligned $c$-axes but many stacking defects; Kish graphite, by contrast, is a highly crystalline graphite form grown from carbon-saturated molten iron. Scanning tunnelling spectroscopy (STS) measures the differential conductance $dI/dV$ of a tunneling junction versus bias, which is proportional to the local density of states of the sample surface; it is local and not momentum selective, whereas angle-resolved photoemission spectroscopy (ARPES) resolves band dispersion in momentum space. Earlier STS work (Matsui et al., Phys. Rev. Lett. 94, 226403 (2005)) had reported LLs at graphite surfaces, and ARPES (Zhou et al., Nat. Phys. 2, 595 (2006)) had observed Dirac-like surface states in graphite, while transport (Morozov et al., Phys. Rev. B 72, 201401 (2005)) had found quasi-two-dimensional carrier gases at graphite surfaces. Li and Andrei (Nature Physics 3, 623 (2007)) then reported the direct STS observation, on HOPG in fields up to 12 T, of LL sequences of both massless and massive chiral Dirac fermions, together with a zero-energy LL and additional peaks following neither scaling, all converging on a single Dirac-point energy. This was surprising: in bulk graphite massless and massive Dirac quasiparticles exist but their Dirac points are displaced by about 50 meV, and the spectral weight of the two-dimensional states along the H-K direction of the bulk Brillouin zone is expected to be negligible compared with bulk states. The authors showed that tip-induced charging (the shift of the sample Fermi energy caused by the electrostatic field of the tunnelling tip) is too small (below 1 meV Fermi-energy shift at their tip-sample distances) to explain the two-dimensional behaviour, and noted its absence in Kish graphite, implying a structural, material-dependent origin. They concluded that for HOPG only a few top layers contribute to the observed states and that stacking faults, rhombohedral (non-Bernal) stacking, or related structural motifs probably decouple the surface layers, while explicitly calling for more detailed theoretical modelling to understand the spectra and the two-dimensional nature of the surface quasiparticles.

Problem Statement

Provide a quantitative, predictive microscopic account of the two-dimensional Dirac-fermion Landau-level spectra observed by scanning tunnelling spectroscopy on graphite surfaces, as first reported by Li and Andrei (Nature Physics 3, 623 (2007)) for HOPG in magnetic fields up to 12 T. All four of the following linked results are jointly required for a complete resolution of this single objective: (1) a band-structure calculation and model of surface versus bulk contributions that assigns every observed tunnelling peak, in particular the peaks that follow neither the massless sequence $E_{n}=\operatorname{sgn}(n)v_{\mathrm{F}}\sqrt{2e\hbar|n|B}$ nor the massive sequence $E_{n}=\operatorname{sgn}(n)\hbar\omega_{c}\sqrt{|n|(|n|+1)}$ (with $\omega_{c}=eB/m^{*}$, $e$ the electron charge, $\hbar$ the reduced Planck constant, and $B$ the perpendicular magnetic field), to specific quasiparticle states; (2) a structural and electronic model of the mechanism that effectively decouples a few top layers of graphite, distinguishing stacking faults, rhombohedral (non-Bernal) stacking, displaced or rotated top layers, and other motifs, which explains how massless and massive chiral quasiparticles can coexist at the same surface Dirac-point energy with spectral weight large enough to dominate the tunnelling spectrum, and predicts that weight relative to bulk states, including why the phenomenology appears on HOPG but not on Kish graphite; (3) a reconciliation of the effective mass of the massive Dirac quasiparticles measured by STS ($m^{*}=(0.028\pm0.003)\,m_{\mathrm{e}}$ in the source measurement, $m_{\mathrm{e}}$ the free-electron mass; $(0.041$-$0.046)\,m_{\mathrm{e}}$ in later decoupled-bilayer measurements) with the larger ARPES-derived and transport-derived values, identifying what each technique measures; and (4) a theoretical description of the STS response itself that includes tip-induced perturbations, finite-layer confinement and realistic stacking, sufficient to reproduce the observed Landau-level peak energies, their field and index dependence (including the non-scaling peaks) and their relative spectral weights, and to yield testable predictions of which graphite materials and surface preparations will exhibit the two-dimensional Dirac Landau-level phenomenology. A complete resolution must account, within a single consistent model, for every observed peak group (the massless and massive Landau-level sequences, the field-independent zero-energy level, and the non-scaling peaks) at the measured magnetic fields and level indices, and must yield falsifiable predictions for specified graphite materials and surface preparations beyond the samples already measured.

References

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