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http://arxiv.org/pdf/1205.5792.pdf The first example in the paper is $C=3$ on a triangular lattice with two orbitals per site. It is essentially three-layers of Haldane's honeycomb lattice model, but stacked together in a clever way so the translation symmetry is restored. UPDATE: In fact two-band free fermion Hamiltonian on a square lattice can realize ...

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Some functionals are known to give closer aggreement with experiment. If you want to stick to the DFT level, you can consider using hybrid functionals like the Tran-Blaha like the Tran-Blaha (TB09) functional which gives bandgap close to experiment for solids. Be aware that the bandwith for example will still be underestimated TB compared with MBPT. If ...

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Your second figure is a simplification of the first one, usually in the $\Gamma$ point, but it could be any other as well. Regarding your questions: There are multiple lines in valence and conduction band because there are several allowed bands or energy eigen states. Technically there is even an infinite number of allowed bands, but usually you would ...

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In any case, for undoped graphene, the Fermi level of electrons and holes are symmetric so the Fermi Energy lies at the Dirac Point, so these two definitions would be equivalent.

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