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2142 Graphene – Synthesis, Characterization, Properties anWdill-bAe-psept-lbiyc-IaN-tTiEoCnHs discovered materials, and yet very little is known on few-layer graphene with more than 3 layers. 2. Electronic properties of few-layer graphene materials The electronic properties of a material are intimately related to its energy dispersion. There are several approaches to calculate the electronic energy bands and here we review the current understanding of the graphene materials band structure within the non-interacting tight-binding approximation (Wallace (1947)). When two or more carbon atoms are brought together to form a regular lattice -such as the hexagonal lattice for a single layer graphene- the valence electrons of the different atoms interact. This leads to a broadening of the electronic eigenstates and ultimately to the formation of the continuous bands of a solid. An isolated carbon atom has 6 electrons 1s22s22p2, where the energies of the s-orbital and p-orbitals of the second electronic shell are very similar. Consequently, carbon can form a number of hybridized atomic orbitals characterizing different geometries. In the case of graphene, one s-orbital and two p-orbitals (px and py) undergo a sp2 hybridization with a characteristic planar trigonal symmetry with an angle of 120°between each bond. This is the reason why each carbon atom within graphene has three nearest neighbors at a distance of a0 = 0.142nm. On the other hand, the pz-orbitals overlap sideways with regions of highest electron density above and below the graphene plane and the energy dispersion of these π-bonds determines the electronic transport properties of graphene materials. y a0 -2 -2 02 02 2 0 2 0 B A Fig. 1. Panel (a) shows the crystal structure of monolayer graphene whose unit cell contains two equivalent carbon atoms -A and B. The 3D plot in (b) shows the energy dispersion of graphene (see Eq. 6). The valence and conduction band touch in 6 points, known as valleys. Whenever the onsite energy symmetry between the A and B sublattices is broken a band-gap opens in the energy dispersion of graphene as shown in Panel (d) -energy dispersion obtained considering HAA ̸= HBB with HAA = 0.1γ0 and HBB = −0.1γ0. The hexagonal lattice of graphene is a composite lattice with two carbon atoms in the unit cell -indicated by A and B, see Fig. 1a- and basis vectors: a2 =√3/2a0iˆ+3/2a0jˆ. (1) We consider the linear combination of atomic orbitals (LCAO) of the Bloch wavefunctions corresponding to the sublattice carbon atoms A and B of the form: φ = 1 ∑ eik·d φ(r−d ),i=A,B. (2) i √N n in i in a1 a) b)kx2c)kx a2 -2 x -2 -2 0 ky -2 0 2 ky √ˆˆ a1=− 3/2a0i+3/2a0j E/γ0 E/γ0PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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