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26 Electronic Band Structure of Graphene where the superscript j denotes the different atoms per unit cell. The secular equation (2.10) remains valid, in terms of the Hermitian n × n matrices Hij ≡ ψ(i)∗Hψ(j) and Sij ≡ ψ(i)∗ψ(j), (2.11) and it is now an equation of degree n. This means that there are n energy bands, i.e. as many energy bands as atoms per unit cell. In a first step, one often neglects the overlap corrections, i.e. one assumes a quasi-orthogonality of the wavefunctions, S ij = N δij . It turns out, however, to keep track of these overlap corrections in the case of graphene. As is discussed in the following section, they yield a contribution which is on the same order of magnitude as the nnn hopping corrections. Formal solution Before turning to the specific case of graphene and its energy bands, we solve formally the secular equation for an arbitrary lattice with several atoms per unit cell. The Hamiltonian matrix (2.11) may be written, with the help of Eq. (2.6), as Hij = X eik·(Rl−Rm) Z d2r φ(i)∗(r + δi − Rk)Hφ(j)(r + δj − Rm) k kkk kkk Rl ,Rm = NXeik·Rl Z d2rφ(i)∗(r)[Ha +∆V]φ(j)(r+δij −Rl) kl = N ǫ(i)sij +tij kk (2.12) (2.13) (2.14) where δij ≡ δj − δi, sij ≡ Xeik·Rl Z d2rφ(i)∗(r+δi −Rk)φ(j)(r+δj −Rm) = k Sij kN k Rl and we have defined the hopping matrix tij ≡Xeik·Rl Z d2rφ(i)∗(r+δi −Rk)∆Vφ(j)(r+δj −Rm). k Rl The last line in Eq. (2.12) has been obtained from the fact that the atomic wavefunctions φ(i)(r) are eigenstates of the atomic Hamiltonian Ha with thePDF Image | Physical Properties of Graphene
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