Dirac cones for bi- and trilayer Bernal-stacked graphene in a quantum graph model

Abstract

A quantum graph model for a single sheet of graphene is extended to bilayer and trilayer Bernal-stacked graphene; the spectra are characterized and the dispersion relations explicitly obtained; Dirac cones are then proven to be present only for trilayer graphene, although the bilayer has a gapless parabolic band component. Our model rigorously exhibits basic facts from tight-binding calculations, effective two-dimensional models and a π-orbital continuum model with nearestneighbour tunneling that have been discussed in the physics literature.

Effective Quantum Graph Models of Some Nonequilateral Graphyne Materials

“…Our effective QGM of (6,6,12)-graphyne also presents Dirac cones for all values of the parameters, with two different Dirac points, in agreement with [9,14]. This case is technically more involved than the others, so that we have checked that there are no other touch points by looking at the graph of the dispersion relations (instead of just analytical expressions as in the other cases).…”

“…From the Floquet-Bloch theory, the study of the Hamiltonian H is reduced to the study of the family of Bloch Hamiltonians H(θ), where θ = (θ 1 , θ 2 ) is the quasimomentum in the Brillouin zone B := [−π, π] 2 . These operators H(θ) act the same way H does, but in functions that satisfy, in addition to the Neumann vertex conditions, the Floquet condition (see (6)). Moreover, H(θ) has a purely discrete spectrum and the graph of the function θ → σ(H(θ)) is the dispersion relation of H. The range of this function is the spectrum of H, that is, σ(H) is given by the reunion of σ(H(θ)), θ ∈ B.…”

“…• γ-graphyne: we have t 1 = 1 and 0 < t 2 = t 3 < 1. • (6,6,12)-graphyne: we take t 1 = t 2 = 1 and 0 < t 3 < 1.…”

“…Note that the choice associated with the a 1 edge, in all cases, point to a corner of the fundamental domain (there are two possibilities for the (6,6,12) case), but we do not have any justification to take this as a criterion to select graphyne edges to be mapped to the honeycomb fundamental domain.…”

“…The latter approach has two advantages: it is simpler then the previous one and permits an accurate spectral analysis including details of Dirac cones; however, this technique requires that all edges in the graph have the same length, that is, an equilateral graph. Recently, the present authors have proposed adaptations in the boundary conditions so that the AA-and AB-stacked graphene multilayer could (at least qualitatively) be modeled by QGMs [6,7]; there is a work that also covers a few layers of graphene combined with hexagonal boron nitride [8] (again with a QGM).…”