ex05 - Solid State Theory Exercise 5 FS 11 Prof M Sigrist...

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Solid State Theory Exercise 5 FS 11 Prof. M. Sigrist Exercise 5.1 Tight-Binding Model of Graphene a) Compute the low-energy band structure of graphene within a tight-binding descrip- tion taking only nearest-neighbor hopping into account! To get started, consider the electronic configuration of C. C has four valence elec- trons which occupy 2 s 2 and 2 p 2 orbitals. The hexagonal structure of the lattice suggests that three of these valence electrons occupy hybrid sp 2 -orbitals to form covalent σ -bonds with their nearest neighbors (bonding angle 2 π/ 3). Due to the large binding energy, there are no low-energy excitations involving these electrons. The remaining electron occupies the p z orbital that sticks out of the planar lattice forms weaker π -bonds with the neighboring atoms. Based on these considerations, it seems reasonable to focus solely on the electrons in the p z -orbitals, so that the problem reduces to one electron and one orbital per atom.
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