E. McCann, D.S.L. Abergel, and V.I. Fal'ko
Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
Abstract. We employ the tight binding model to describe the electronic band
structure of bilayer graphene and we explain how the optical absorption coefficient
of a bilayer is influenced by the presence and dispersion of the electronic bands, in
contrast to the featureless absorption coefficient of monolayer graphene. We show
that the effective low energy Hamiltonian is dominated by chiral quasiparticles
with a parabolic dispersion and Berry phase 2π. Layer asymmetry produces a gap
in the spectrum but, by comparing the charging energy with the single particle
energy, we demonstrate that an undoped, gapless bilayer is stable with respect to
the spontaneous opening of a gap. Then, we describe the control of a gap in the
presence of an external gate voltage. Finally, we take into account the influence of
trigonal warping which produces a Lifshitz transition at very low energy, breaking
the isoenergetic line about each valley into four pockets.
Following the fabrication of monolayer graphene , the observation of an unusual sequencing
of quantum Hall effect plateaus  was explained in terms of Dirac-like chiral quasiparticles
with Berry phase π [3–6]. Subsequently, bilayer graphene became the subject of intense interest
in its own right. This followed the realisation that the low energy Hamiltonian of a bilayer
describes chiral quasiparticles with a parabolic dispersion and Berry phase 2π  as confirmed
by quantum Hall effect  and ARPES measurements .
The electronic band structure of bilayer graphene has been modelled using both density
functional theory [10–12] and the tight binding model [7,13–17]. It has been predicted 
that asymmetry between the on-site energies in the layers leads to a tunable gap between the
conduction and valence bands. The dependence of the gap on external gate voltage has been
modelled taking into account screening within the tight binding model [12,16,17] and such
calculations appear to be in good agreement with ARPES measurements , observations of
the quantum Hall effect , and density functional theory calculations .
In this paper, we describe the tight binding model of bilayer graphene and the corresponding
low energy band structure in section 2. Section 3 explains how the optical absorption coefficient
of bilayer graphene is influenced by the presence and dispersion of the electronic bands
[18, 19], in contrast to the featureless absorption coefficient of monolayer graphene. We obtain
the effective low energy Hamiltonian of bilayer graphene in section 4 and we show that it is
dominated by chiral quasiparticles with a parabolic dispersion and Berry phase 2π. Section 5
describes the opening of a gap in bilayer graphene due to layer asymmetry with a description
of the band structure in section 5.1, a demonstration that an undoped, gapless bilayer is stable
with respect to the opening of a gap in section 5.2, and a calculation using a self-consistent
Hartree approximation to describe the control of the gap in the presence of external gates in
section 5.3. In section 6 we take into account the effect of trigonal warping on the band structure
and we present our conclusions in section 7.
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