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Crystal Structure of Graphene and Graphite 11 (a) y (b) x K M’’ K’ a2 K M’’ K’ aM * 1δ 2 M : A sublattice δ a=0.142 nm 3 Figure 1.7: (a) Honeycomb lattice. The vectors δ1, δ2, and δ3 connect nn carbon atoms, separated by a distance a = 0.142 nm. The vectors a1 and a2 are basis vectors of the tri- angular Bravais lattice. (b) Reciprocal lattice of the triangular lattice. Its primitive lattice vectors are a∗1 and a∗2. The shaded region represents the first Brillouin zone (BZ), with its centre Γ and the two inequivalent corners K (black squares) and K′ (white squares). The thick part of the border of the first BZ represents those points which are counted in the definition such that no points are doubly counted. The first BZ, defined in a strict manner, is, thus, the shaded region plus the thick part of the border. For completeness, we have also shown the three inequivalent cristallographic points M, M′, and M′′ (white triangles). covalent σ bonds, as in the case of benzene. The three vectors which connect a site on the A sublattice with a nn on the B sublattice are given by δ1 = a√3ex +ey, δ2 = a−√3ex +ey, δ3 =−aey, 22 and the triangular Bravais lattice is spanned by the basis vectors a1 =√3aex and a2 = √3aex +√3ey. √ 2 (1.2) (1.3) 2 ’ K K’ M ’ a* a δ1 : B sublattice Γ 1 M Themodulusofthebasisvectorsyieldsth√elatticespacing,a ̃= 3a=0.24 nm, and the area of the unit cell is Auc = 3a ̃2/2 = 0.051 nm2. The density of carbon atoms is, therefore, nC = 2/Auc = 39 nm−2 = 3.9 × 1015 cm−2. Because there is one π electron per carbon atom that is not involved in a covalent σ bond, there are as many valence electrons than carbon atoms, and their density is, thus, nπ = nC = 3.9 × 1015 cm−2. As is discussed in the following chapter, this density is not equal to the carrier density in graphene, which one measures in electrical transport measurements.PDF Image | Physical Properties of Graphene
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