Band model of the graphene bilayer

19with Γ = 1 and γ 2 = γ 5 =0,leadingto:⎛⎞∆ v F pe iφ t ⊥ −v 4 v F pe −iφv F pe −iφ 0 −v 4 v F pe −iφ v 3 v F pe iφH 0 (p) =. (3.1)⎜ t ⊥ −v 4 v F pe iφ ∆ v F pe −iφ⎟⎝⎠−v 4 v F pe iφ v 3 v F pe −iφ v F pe iφ 0Another way of arriving at Eq. (3.1) is to use the tight-binding model directly in the bilayer.Since the system is two-dimensional only the relative position of the atoms projected onto the x-y-plane enters into the model. The projected position of the different atoms areshown in Fig. 3·2. Since the A atoms are sitting right on top of each other in the lattice, theB2B1A1A2Figure 3·2: The real space lattice structure of the graphene bilayer projectedonto the x-y plane showing the relative positions of the differentsublattices.hopping term between the A1 andA2 atomsarelocalinrealspaceandhenceaconstantthat we denote by t ⊥ in momentum space. Referring back to Section 2.1 we note that thehopping B1 → A1 [A1 → B1] gives rise to the factor ζ(k) [ζ ∗ (k)], with ζ(k) definedinEq. (2.6). Since the geometrical role of the A and B atoms are interchanged between plane1andplane2weimmediatelyfindthatinFourierspacethehopping A2 → B2 [B2 → A2]gives rise to the factor ζ(k) [ζ ∗ (k)]. Furthermore, the direction in the hopping B1 → B2