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28 Electronic Band Structure of Graphene and we also take into account nnn hopping which connects nn sites on the same sublattice tnnn ≡ Z d2r φA∗(r)∆V φA(r + a1) = Z d2r φB∗(r)∆V φB(r + a1). (2.17) Notice that one may have chosen any other vector δj or a2, respectively, in the calculation of the hopping amplitudes. Because of the normalisation of the atomic wavefunctions, we have R d2rφ(j)∗(r)φ(j)(r) = 1, and we consider furthermore the overlap correction between orbitals on nn sites, s≡Z d2rφA∗(r)φB(r+δ3). (2.18) We neglect overlap corrections between all other orbitals which are not nn, as well as hopping amplitudes for larger distances than nnn. If we now consider an arbitrary site A on the A sublattice (Fig. 2.1), we may see that the off-diagonal terms of the hopping matrix (2.14) consist of three terms corresponding to the nn B1, B2, and B3, all of which have the same hopping amplitude t. However, only the site B3 is described by the same lattice vector (shifted by δ3) as the site A and thus yields a zero phase to the hopping matrix. The sites B1 and B2 correspond to lattice vectors shifted by a1 and a3 ≡ a2 − a1, respectively. Therefore, they contribute a phase factor exp(ik · a1) and exp(ik · a3), respectively. The off-diagonal elements of the hopping matrix may therefore be written as4 tAB =tγ∗ = tBA∗, kkk as well as those of the overlap matrix sAB =sγ∗ = sBA∗, kkk (sAA = sBB = 1, due to the above-mentioned normalisation of the atomic kk wavefunctions), where we have defined the sum of the nn phase factors γk ≡1+eik·a1 +eik·a3. (2.19) 4The hopping matrix element tAB corresponds to a hopping from the B to the A k sublattice.PDF Image | Physical Properties of Graphene
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