Exchange, sigma-orbital, and interaction corrections to the screening and small-energy-scale band structure of ABC-stacked graphene trilayers

The 2010 combined density-functional-theory (DFT), k.p, and tight-binding study by Zhang, Sahu, Min, and MacDonald of ABC-stacked (rhombohedral) graphene trilayers established the low-energy pi-band model still used for this system and left four explicitly acknowledged open issues: (i) how exchange interactions beyond local or semilocal approximations, and polarization of the occupied sigma orbitals, quantitatively modify the screening of an interlayer potential difference and the induced band gap; (ii) experimental confirmation, by angle-resolved photoemission spectroscopy (ARPES) and quantum transport, of the predicted meV-scale band-structure features -- the three displaced Dirac cones, the Lifshitz-transition energies, and the trigonal-warping orientation; (iii) whether electron-electron interactions beyond Hartree and mean-field treatments produce broken-symmetry ground states near the calculated saddle-point van Hove singularities; and (iv) the quantitative generalization to thicker ABC stacks, whose low-energy model validity window shrinks with layer number N. The required objective is the quantitatively open core of these issues -- the beyond-Hartree description of electrostatic screening and field-induced gaps in ABC trilayers (the exchange and sigma-band-polarization corrections to the screening ratio) together with precision meV-level spectroscopic verification of the specific Lifshitz-transition energies and trigonal-warping orientation; the remaining strands are background context, documented in the Background and the References.

Background

Graphene trilayers come in two stable stacking arrangements: Bernal (ABA) stacking, in which one layer is shifted so that the electronic structure roughly combines a monolayer-like Dirac cone with bilayer-like parabolic bands, and rhombohedral (ABC) stacking, in which each successive layer sits over the hollow site of the one below. In ABC stacking, the low-energy sites form a chain across the three layers, so that keeping only the nearest interlayer hopping $\gamma_1$ yields two exceptionally flat bands that disperse as momentum cubed ($p^3$) instead of the linear dispersion of a monolayer. Fan Zhang, Bhagawan Sahu, Hongki Min, and Allan H. MacDonald [Phys. Rev. B 82, 035409 (2010)] fitted a full pi-band tight-binding model -- pi orbitals are the carbon $p_z$ orbitals that form graphene's valence and conduction bands, and 'remote' interlayer hoppings such as $\gamma_2$, $\gamma_3$, $\gamma_4$ couple sites beyond nearest neighbors -- to ab initio DFT bands of ABC trilayers, and studied the model with k.p theory (an expansion of the band Hamiltonian in small momentum around the Brillouin-zone corner K). Their central small-energy-scale findings were: (a) the triple Dirac point of the simplified model is split by remote hopping into three single Dirac points displaced along the three K-M directions; (b) around each displaced Dirac cone the constant-energy surface forms separate electron or hole pockets which merge in a sequence of two Lifshitz transitions -- topological changes of the Fermi surface occurring at 6.7 and 7.2 meV in the conduction band and at -7.9 and -9.9 meV in the valence band -- at which band-structure saddle points produce two-dimensional logarithmic van Hove singularities, i.e. divergences of the density of states; (c) the constant-energy surfaces are trigonally warped (distorted from circular by threefold symmetry) with a weaker distortion of opposite orientation than bulk graphite would predict, becoming nearly hexagonal and strongly nested at 8-9 meV; and (d) an external potential difference between the outer layers is strongly screened by charge transfer within the trilayer, computed with DFT and with a self-consistent Hartree approximation (the classical mean-field treatment of the Coulomb interaction, where the electrostatic potential is determined self-consistently from the Poisson equation), yet still opens a gap. Screening is quantified by a screening ratio, the screened interlayer potential difference U divided by the applied external difference. The authors explicitly cautioned that their treatment neglects two effects: exchange interactions (the Fock part of the Coulomb interaction, attractive and non-local), which are treated only in a local density approximation in their DFT and not at all in the Hartree model and which act against screening; and polarization of the occupied sigma orbitals (the deep in-plane $sp^2$ bonding bands, neglected in any pi-band model), which adds to the polarizable charge that screens the field. They further noted that broken-symmetry ground states are expected near the van Hove singularities, that future ARPES experiments should test the DFT predictions, and that the energy window in which the low-energy effective model applies shrinks as the number of layers N grows. All of these questions are posed within chiral two-dimensional electron systems (C2DESs), the family of multilayer graphene systems whose low-energy bands carry large momentum-space Berry phases, phases accumulated by the pseudospin (the layer/sublattice composition of the wave function) as it winds in momentum space.

Problem Statement

For ABC-stacked (rhombohedral) graphene trilayers, determine quantitatively, and confront with existing measurements, the small-energy-scale electronic structure, its electrostatic screening, and its interaction physics, as posed by Zhang, Sahu, Min, and MacDonald [Phys. Rev. B 82, 035409 (2010), DOI 10.1103/PhysRevB.82.035409]. The task poses two independent sub-questions: (i) Quantify how exchange interactions beyond local or semilocal approximations, and the polarization of the occupied sigma orbitals omitted from pi-band and Hartree treatments, alter the screening ratio of an external interlayer potential difference and the size of the induced band gap, reporting the resulting correction to the screening ratio and to the induced gap. (ii) Confront the predicted Lifshitz-transition energies (6.7 and 7.2 meV in the conduction band, -7.9 and -9.9 meV in the valence band) and the orientation and strength of the trigonal warping of the constant-energy surfaces with the existing angle-resolved photoemission and quantum-transport measurements listed in the References (for example References 2, 6 and 7). (iii) Where the quantitative result of (i) permits, state whether simple screening measurements can indeed serve as a window on this many-body physics, i.e. whether the exchange and sigma-orbital corrections are large enough to be resolved in the measured screening ratio. The source's other strands -- the three Dirac cones displaced along the K-M directions in the broader band structure, interaction-driven broken-symmetry ground states at the saddle-point van Hove singularities, and the quantitative generalization to thicker ABC stacks as the energy range of the low-energy effective model decreases with film thickness N -- are background context.

References

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