By Foa Torres L.E.F., Roche S., Charlier J.-C.
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Additional info for Introduction to Graphene-Based Nanomaterials: From Electronic Structure to Quantum Transport
To derive the electronic spectrum of the total Hamiltonian, the corresponding Schr¨odinger equation has to be solved. According to Bloch’s theorem, the eigenfunctions evaluated at two given Bravais lattice points Ri and Rj differ from each other in just a phase factor, exp(ik · (Ri − Rj )). Because of the two-atom basis, the Bloch ansatz 1 Note that the hexagonal shape of the Brillouin zone is a consequence of the triangular Bravais lattice. It is by no means connected with the two-atom basis which does not enter into the deﬁnition of the Brillouin zone.
2012), accurately describing ﬁrst-principles results over the entire Brillouin zone. In contrast to the 1st nn π -π ∗ model which produces a totally symmetric band structure, the 3rd nn TB model recovers the asymmetry between valence and conduction van Hove singularities, and the agreement with ab initio band structures is quite satisfactory, as illustrated in Fig. 6. The 3rd nn parameters of the TB model used to construct these band structures (Fig. 6) are only composed of a single on-site energy term εpz and three hopping terms γ0(1) = γ0 , γ0(2) and γ0(3) corresponding respectively to 1st, 2nd and 3rd nn interactions.
2005, Son, Cohen & Louie, 2006a, Son, Cohen & Louie, 2006b). One notes, however, that the difference in total energy per edge atom between non-spin-polarized and spin-polarized edge states is only of the order of a few tens of meV (Son, Cohen & Louie, 2006a). , 1999). 19 Evolution of the carbon–carbon bond length (dC−C ) across the 8-zGNR ribbon width, obtained after ab initio structural optimization. Both the length of the carbon–carbon parallel (empty circles) and tilted (ﬁlled squares) bonds with respect to the ribbon axis are shown.