Phonon dispersion of graphene
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Fig. 1. Phonon dispersion of graphene calculated within a valence force field (VFF) model (Aizawa). |
The phonon dispersion of graphene can be obtained within the valence-force field model of the lattice dynamics (Fig. 1, Aizawa). Several force field parameters are enough to grasp the main features of the dispersion. Improvement over this simple picture can be done with the expense of introducing of many extra force fields to reproduce satisfactory the overbending of the highest-frequency phonon branch. The available experimental data does not allow for the unambiguous determination of the additional force fields. Alternatively, the phonon dispersion can be derived within a non-orthogonal tight-binding (NTB) model [1,2] in the linear-response approximation [3,4]. The mentioned overbending is predicted fairly well (Fig. 2). However, the NTB results overestimate the experimental data for the bond-stretching bands by about 12% (Fig. 2, crosses). This can be "cured" by downscaling of the NTB points by a factor of 0.9 (Fig. 2, circles). Now the downscaled results fit well the experimental data for the in-plane phonon branches LA, TA, LO, TO), as well as for the acoustic out-of-plane one (ZA), but underestimate the experimental data for the optical out-of-plane branch (ZO). There has been a discussion in the literature about the behavior of the TO branch at the K point (Kresse, Maultzsch, Ferrari). The ab-initio calculations show that the TO branch crosses the LO branch close to the K point and has highest frequency at the K point. There seems to be no experimental evidence for such behavior for there are no measurements of the TO phonon at the K point. On the other hand, the obtained here slope of the TO branch at K can be a signature of a Kohn anomaly at this point (Ferrari). This anomaly originates from strong electron-phonon interactions leading to scattering the electron from one point of the Fermi surface to another. In graphene, an electron can be scattered around K or K' points by a small-wavevector phonon giving rise to anomaly of phonon branches around the Gamma-point. The electron can also be scattered from the K point to the K' point by a phonon with a wavevector qK and, hence, anomaly of phonon branches around the K point (Ferrari). References: 1. 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, L. Henrard, and Ph. Lambin, Phys. Rev. B 72 (2005) 035436. 4. V. N. Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 085407. |
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Fig. 2. Phonon dispersion of graphene calculated within a non-orthogonal tight-binding model in the linear-response approximation [4]. |
Valentin Popov
February 22, 2006