Band model of the graphene bilayer

25EeV0.040.030.020.010-0.1 -0.05 0 0.05 0.1k eV(a)EeV0.0080.0060.0040.0020-0.04 -0.02 0 0.02 0.04k eV(b)Figure 3·5: Comparison between the bands obtained from the full Hamiltonianin Eq. (3.1) and those of the effective model in Eq. (3.8) alongdifferentdirections in the BZ. Solid line – [φ =0,Eq.(3.1)],Dash-dottedline–[φ =0,Eq.(3.8)],Dashedline–[φ = π/6, Eq. (3.1)], Dotted line –[φ = π/6, Eq. (3.8)]. (a) Larger energy scale. (b) Zoom in at low energies.extra band crossings in the directions φ =0andφ = ±2π/3 whichgivesrisetoellipticalDirac cone 3 away from the point p =0. Thisstructureispresentatasmallenergyscaleofthe order of ∼ 1meV, therefore experimental probes that are sensitive to this energy scaleare necessary to be able to detect these features. Moreover, as we will see in Chapter 5,different forms of disorder can easily generate energies of this scale or larger in the realexperimental samples, thus this structure might be hard to detect experimentally.Astudyoffew-layergraphenesystems(includinggraphenebilayers) with plots similarto those in this chapter can be found in (Partoens and Peeters, 2006). Afirst-principlesstudy with both similar scope and results also exists (Latil and Henrard, 2006).3 This means that there are two inequivalent perpendicular axes (1 and 2) in the cone, with two differentvalues of the Fermi velocity v F1 ≠ v F2. Thespectrumisthen± p v 2 F1 p2 1 + v2 F2 p2 2 .