Electron-electron interactions in graphene bilayers

We study the effect of electron-electron interactions in the quasiparticle dispersion of a graphene bilayer within the Hartree-Fock-Thomas-Fermi theory by using a four-bands model. We find that the electronic fluid can be described by a non-interacting like dispersion but with renormalized parameters. We compare our results with recent cyclotron resonance experiments in this system. Since graphene was isolated in 2004 [1], it has attracted attention because of its possible application in all-carbon based electronic devices [2] and its connections to relativistic field theory [3]. While there is strong theoretical [4] and experimental evidence [2,5] that single layer graphene (SLG) behaves as essentially a weakly interacting gas of two-dimensional (2D) Dirac particles, the situation in bilayer graphene (BLG) is much less clear. Early theoretical studies have indicated that the SLG is much less prone towards magnetic states [6], while BLG can become magnetic at low densities [7]. Moreover, while the electronic compressibility of SLG has essentially features of an insulator [5, 8, 9], the BLG compressibility is, unlike the 2D electron gas (2DEG) [10], non-monotonic and strongly dependent on electronic density [11]. It has also been argued that, unlike SLG, BLG should be unstable towards manybody states such as a pseudospin magnet [12], a Wigner crystal [13], and an excitonic superfluid [14]. It has been demonstrated that BLG is a tunable gap semiconductor by application of a transverse electric field [15,16], leading to extra flexibility in dealing with its electronic properties [17,18]. While electrons in BLG have a different topological (Berry’s) phase than electrons in SLG, as evident in integer quantum Hall effect measurements [19], the experimental evidence for electron-electron interaction effects in BLG has been elusive. Nevertheless, recent cyclotron resonance experiments in bilayer graphene [20] have shown departures from the non-interacting bilayer model proposed by McCann and Falko [22]. These disagreements do not seem to be describable in terms of disorder effects alone [23]. The objective of our paper is to clarify these discrepancies. The SLG has a honeycomb lattice structure that leads to a Dirac-like electronic dispersion, E(k) = ±c̃|k|, at the edges (the K and K’ points) of the Brillouin zone. The electrons are described in terms of a 2D “relativistic” Dirac Hamiltonian with zero rest mass, where the velocity of light, c, is replaced by the Fermi-Dirac velocity, c̃. In the BLG (Bernal structure) the two graphene layers are rotated by a relative angle of π/3 that breaks the sublattice symmetry leading to 2 pairs of massive Dirac particles at the K (K’) point. Nevertheless, the system remains metallic because 2 bands, belonging to different pairs, touch in a point. More explicitly, the non-interacting bands have the form: E1(k) = −mc̃2 + E(k), E2(k) = mc̃ − E(k), E3(k) = mc̃ 2 + E(k) and E4(k) = −mc̃2 − E(k), where E(k) = √ (mc̃2)2 + (c̃k)2. Hence, E1(k) and E4(k) (E2(k) and E3(k)) describe a massive relativistic dispersion with rest mass energy given by mc̃. Rotations by other angles do not break the sublattice symmetry and hence do not lead to mass generation [24]. Our results suggest that BLG behaves as a liquid of Dirac quasiparticles with renormalized mass and velocity. The situation described here is unique when compared to standard non-relativistic Fermi liquids such as He [25] and ordinary metals [26], or even to relativistic Fermi liquids such as quark matter in the core of neutron stars [27]. While the electrons in graphene are effectively “relativistic”, in the sense that they obey an effective Lorentz invariance (only true at low energies) with the Dirac velocity playing the role of velocity of light, on the other hand, from the point of view of an external observer, the whole graphene system is Galilean invariant and non-relativistic since the Dirac velocity is much smaller than the actual