Electronic band structure, phonon dispersion, and one-phonon resonant Raman scattering in α-, β-, and γ-graphynes

	The α-, β-, and γ-graphynes have hexagonal symmetry as graphene itself. The three graphynes can be derived from graphene by insertion of an acetylenic group into every carbon bond, two-thirds of the carbon bonds and one-third of the carbon bonds, resp. (Fig. 1). The Brillouin zone of these graphynes is a hexagon (Fig. 2b). The electronic band structure of α-, β-, and γ-graphynes, derived within the non-orthogonal tight-binding (NTB) model with model parameters taken over from an ab-initio study (Porezag) [1,2] is shown in Fig. 2 [3]. The main characteristics of the band structure of the three graphynes are as follows: - α-graphyne has similar band structure as graphene with crossing bands at the Fermi energy at the K point of the Brillouin zone. The splitting of the bands at the M point is in excellent agreement with the previously reported value. - β-graphyne has electronic structure that shows deviation from that of graphene as the crossing of the bands at the Fermi energy takes place along the ΓM direction. Additionally, a tiny gap opens along the ΓK direction. The agreement with the previously reported band energies is very good. - γ-graphyne differs from both α-graphyne and β-graphyne as there is no band crossing at the Fermi energy. The comparison with reported DFT data shows disagreement for the band gaps. It should be noted that DFT systematically underestimates the band gaps and further corrections are needed for the fair prediction of the electronic structure. In summary, the NTB model yields semi-quantitatively correct prediction of the band structure of the graphynes in the vicinity of the Fermi energy. The NTB model is further used for the calculation of the phonon dispersion and the first-order Raman spectra of the studied graphynes [3]. References: 1. V. N. Popov, New J. Phys. 6 (2004) 1-17. 2. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407. 3. V. N. Popov and Ph. Lambin, Phys. Rev. B 88 (2013) 075427.
Fig. 1. Schematic of the atomic structure of (a) graphene, (b) α-graphyne, (c) β-graphyne, and (d) γ-graphyne. The unit cells of the structures are shown by rhombs.
Fig. 2. Electronic band structure of (a) α-graphyne, (c) β-graphyne, and (d) γ-graphyne, calculated along the high-symmetry directions in the Brillouin zone (b). The available reported values of the bands’ energies are shown by solid and empty symbols.

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Valentin Popov

September 8, 2013