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12 Introduction to Carbon Materials The reciprocal lattice, which is defined with respect to the triangular Bravais lattice, is depicted in Fig. 1.7 (b). It spanned by the vectors ∗ 2π ey ∗ 4π a1 = √3a ex − √3 and a2 = 3aey. (1.4) Physically, all sites of the reciprocal lattice represent equivalent wave vectors. Any wave – be it a vibrational lattice excitation or a quantum-mechanical electronic wave packet – propagating on the lattice with a wave vector differ- ing by a reciprocal lattice vector has indeed the same phase up to a multiple of 2π, due to the relation ai · a∗j = 2πδij (1.5) (for i, j = 1, 2) between direct and reciprocal lattice vectors. The first Bril- louin zone [BZ, shaded region and thick part of the border of the hexagon in Fig. 1.7 (b)] represents a set of inequivalent points in the reciprocal space, i.e. of points which may not be connected to one another by a reciprocal lattice vector, or else of physically distinguishable lattice excitations. The long wavelength excitations are situated in the vicinity of the Γ point, in centre of the first BZ. Furthermore, one distinguishes the six corners of the first BZ, which consist of the inequivalent points K and K′ represented by the vectors 4π ±K = ±3√3aex. (1.6) The four remaining corners [shown in gray in Fig. 1.7 (b)] may indeed be con- nected to one of these points via a translation by a reciprocal lattice vector. These cristallographic points play an essential role in the electronic proper- ties of graphene because their low-energy excitations are centered around the two points K and K′, as is discussed in detail in the following chapter. We emphasise, because of some confusion in the litterature on this point, that the inequivalence of the two BZ corners, K and K′, has nothing to do with the presence of two sublattices, A and B, in the honeycomb lattice. The form of the BZ is an intrinsic property of the Bravais lattice, independent of the possible presence of more than one atom in the unit cell. For com- pleteness, we have also shown, in Fig. 1.7 (b), the three crystallographically inequivalent M points in the middle of the BZ edges.PDF Image | Physical Properties of Graphene
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