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Chapter 2 Electronic Band Structure of Graphene As we have discussed in the introduction, three electrons per carbon atom in graphene are involved in the formation of strong covalent σ bonds, and one electron per atom yields the π bonds. The π electrons happen to be those re- sponsible for the electronic properties at low energies, whereas the σ electrons form energy bands far away from the Fermi energy. This chapter is, thus, devoted to a discussion of the energy bands of π electrons within the tight- binding approximation, which was originally calculated for the honeycomb lattice by P. R. Wallace in 1947 [17]. We do not consider the σ electrons, here, and refer the interested reader to the book by Saito, Dresselhaus, and Dresselhaus for a tight-binding calculation of the energy bands formed by the σ electrons [12], which are far away from the Fermi level. The chapter consists of two sections. The first one is devoted to the cal- culation of the π energy bands in graphene, where one is confronted with the problem of two atoms per unit cell. After a brief discussion of Bloch’s theorem, and a formal solution of the tight-binding model, we calculate the energy dispersion of π electrons in graphene, taking into account nearest- neighbour (nn) and next-nearest-neighbour (nnn) hopping and nn overlap corrections. The second section consists of the continuum limit, which de- scribes the low-energy properties of electrons in graphene. 21PDF Image | Physical Properties of Graphene
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