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ElelcetrcontircoTrnanicspoTrtrParonpesrtpieosorftFPewr-oLapyeerGrtriaephsenoefMaFteriawls-LayerGrapheneMaterials 145 y z B2 A B1 y2 a1a2 A2 A1 x a) B1 A1 x c) vp/γ 1 1 0 -1 -2 -1 0 1 2 b) d) B2 Fig. 2. Panel (a) and (b) show respectively a top view and a 3D view of the crystal structure of AB-stacked bilayer graphene whose unit cell contains four equivalent carbon atoms -A1, B1, A2 and B2. The hopping parameters γ0, γ1 and γ3 are highlighted in the scheme. The graph in (c) shows the energy dispersion of bilayer graphene without an energetic asymmetry between the sublattices (continuous line) and with various values of the onsite energy difference (dashed lines) (Koshino (2009)). When all the hopping parameters are considered in the tight binding calculation, the low energy dispersion of bilayers is not any more simply parabolic but 4 Dirac cones develop. Panel (d) shows an equi-energy contour plots of the lowest electron band of bilayer graphene for equi-energy onsite symmetry case with the 4 Dirac cones clearly visible (Koshino (2009)). The higher energy bands are instead shifted ±0.4 eV away from the E = 0 Fermi level position. In the low energy limit, the energy dispersion of bilayer graphene is readily obtained from the secular determinant of the effective Hamiltonians HK and HK′ for the high symmetry K and K’ points: h ̄2 0 (kx−iky)2 h ̄2 0 (kx+iky)2 HK=−2m∗ (kx+iky)2 0 andHK′ =−2m∗ (kx−iky)2 0 (10) which operate on the two Bloch wavefunction components φ = (φA1, φB2)T on the sublattices A1 and B2 located in different layers and with m* the effective mass. Therefore, opposed to the case of single-layer graphene, in bilayers the pseudospin characterises the layer degree of freedom. The low energy dispersion in bilayers is parabolic (see Fig. 2c): E(k) = ±h ̄2k2/2m∗, (11) where m* is the charge particles effective mass m∗ = γ1/2v2F. If we lift the energy degeneracy between the carbon sublattices on the two layers the gap-less energy dispersion of bilayers develops an energy gap around E = 0 δ/γ 0 0.05 0.2 0.5 1 E/γ 1PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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