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Tight-Binding Model for Electrons on the Honeycomb Lattice 27 a3 A a1 B3 Figure 2.1: Tight-binding model for the honeycomb lattice. atomic energy ǫ(i) for an orbital of type i. This atomic energy plays the role of an onsite energy. The secular equation now reads det tij − ǫλ − ǫ(i) = 0. (2.15) Notice that, if the the atoms on the different sublattices are all of the same electronic configuration, one has ǫ(i) = ǫ0 for all i, and one may omit this onsite energy, which yields only a constant physically irrelevant shift of the energy bands. 2.1.3 Solution for graphene with nearest-neighbour and next-nearest-neighour hopping After these formal considerations, we now study the particular case of the tight-binding model on the honeycomb lattice, which yields, to great accu- racy, the π energy bands of graphene. Because all atomic orbitals are pz orbitals of carbon atoms, we may omit the onsite energy ǫ0, as discussed in the last paragraph. We choose the Bravais lattice vectors to be those of the A sublattice, i.e. δA = 0, and the equivalent site on the B sublattice is obtained by the displacement δB = δAB = δ3 (see Fig. 2.1). The hopping amplitude between nn is given by the expression t≡Z d2rφA∗(r)∆VφB(r+δ3), (2.16) B2 B1 a2 kk δ3PDF Image | Physical Properties of Graphene
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