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24 Electronic Band Structure of Graphene of TRi may be seen from T ψ(r) = ψX(r+R) Rik k i = eik·Rj φ(a) [r − (Rj − Ri)] Rj = eik·Ri X eik·Rm φ(a)(r − Rm) = eik·Ri ψk(r), Rm where the first line reflects the translation by the lattice vector Ri in the argument of the wavefunction, and we have resummed the lattice vectors in the last line with the redefinition Rm = Rj − Ri. 2.1.2 Lattice with several atoms per unit cell If we have several atoms per unit cell, such as in the case of the honeycomb lattice, the reasoning described above must be modified. Notice first that a translation by any vector δj that relates a site on one sublattice to that on a second sublattice is not a symmetry operation, i.e. [Tδj , H] ̸= 0, if we define Tδj ≡ exp(ipˆ · δj/h ̄) in the same manner as the translation operator (2.3). One must, therefore, treat the different sublattices apart. In the case of two atoms per unit cell, we may write down the trial wavefunction as ψk(r) = akψ(A)(r) + bkψ(B)(r), (2.5) kk where ak and bk are complex functions of the quasi-momentum k. Both ψ(A)(r) and ψ(B)(r) are Bloch functions with ψ(j)(r) = X eik·Rl φ(j)(r + δj − Rl), (2.6) k Rl where j = A/B labels the atoms on the two sublattices A and B, and δj is the vector which connects the sites of the underlying Bravais lattice with the site of the j atom within the unit cell. Typically one chooses the sites of one of the sublattices, e.g. the A sublattice, to coincide with the sites of the Bravais lattice. Notice furthermore that there is some arbitrariness in the choice of the phase in Eq. (2.6) – instead of choosing exp(ik · Rl), one may also have chosen exp[ik · (Rl − δj ), as for the arguments of the atomic wavefunctions. The choice, however, does not affect the physical properties kkPDF Image | Physical Properties of Graphene
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