π - ADDI

1.3. Electronic Structure 17Figure 1.3: Graphene is the building block of all graphitic materials. It is considered as the basis for graphite,carbon nanotube and fullerenes (buckyballs). Reproduced from Ref. [8].electron (π) is assumed to be located at the orbital perpendicular to the surface. Thus, for theflat layer the π-state is decoupled from the other C σ-states by symmetry (odd under inversionin the plane) and it can be calculated separately. With the help of Bloch theorem applied tothe two-dimensional periodic lattice shown in Figure 1.6, the Hamiltonian in reciprocal spaceis a 2×2 matrix dependent on the k-vector:H(k) = ε o I − γ o(0 1 + e −ik·a 1+ e −ik·a 21 + e ik·a 1+ e ik·a 20)(1.1)where the hopping energy for nearest neighbors is γ o = 2.7 eV. This equation has two eigenvaluessymmetric around ε o :E(k) = ε o ± γ o |1 + e ik·a 1+ e ik·a 2| (1.2)In cartesian coordinates the lattice vectors can be written as a 1,2 = a( √ 3e x ± e y )/2 witha = √ 3d C−C (see Figure 1.6), where d C−C is the distance C-C, and the energies can beexpressed as: