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30 Electronic Band Structure of Graphene (a) π (b) 4 3 2 1 -−1 -−2 π∗ π∗ wave vector 3 Γ 1 2M 3K π K’ K K kx K’ K’ K K -−22 -−11 ky Figure 2.2: Energy dispersion obtained within the tight-binding approximation, for tnnn/t = 0.1. One distinguishes the valence (π) band from the conduction (π∗) band. The Fermi level is situated at the points where the π band touches the π∗ band. (a) Energy dispersion as a function of the wave-vector components kx and ky. (b) Cut throught the energy dispersion along characteristic lines (connecting the points K → Γ → M → K. The energy is measured in units of t and the wave vectors in units of 1/a. Energy dispersion of π electrons in graphene The energy dispersion (2.22) is plotted in Fig. 2.2 for tnnn/t = 0.1. It consists of two bands, labeled by the index λ = ±, each of which contains the same number of states. Because each carbon atom contributes one π electron and each electron may occupy either a spin-up or a spin-down state, the lower band with λ = − (the π or valence band) is completely filled and that with λ = + (the π∗ or conduction band) completely empty. The Fermi level is, therefore, situated at the points where the π band touches the π∗ band. Notice that, if tnnn = 0, the energy dispersion (2.22) is electron-hole symmetric, i.e. ǫλ = −ǫ−λ. This means that nnn hopping and nn overlap kk corrections break the electron-hole symmetry. The points, where the π band touches the π∗ band, are called Dirac points, for reasons that are explained in the following chapter. They situated at the points kD where the energy dispersion (2.22) is zero, ǫ λk D = 0 . ( 2 . 2 4 ) Energy Energy [in units of t]PDF Image | Physical Properties of Graphene
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