Phonon dispersion, breathing-like and shear modes of fewlayer graphene
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Fewlayer graphene (FLG) consists of several graphene layers, stacked in
various ways. The special cases of stacking are: AA-stacked FLG, which is not
a stable structure, AB-stacked or Bernal FLG, and ABC-stacked or rhombohedral
FLG. Here, we consider only AB- and ABC-stacked FLG (Fig. 1). These
structures have hexagonal symmetry with a hexagonal Brillouin zone.
The electronic band structure of FLG is calculated within a non-orthogonal tight-binding (NTB) model with intralayer parameters, taken over from an ab-initio study (Porezag), and interlayer parameters, taken from another ab-initio study [3]. The phonon dispersion of FLG is calculated within the NTB model using a perturbative approach [4,5]. It has to be noted, that the ab-initio derived parameters account only for part of the electron-electron interactions and additional pair potential has to be introduced in order to obtain electronic band structure, which is in agreement with the ab-initio one. In the case of the interlayer interaction, the additional pair potential was derived by fitting to the breathing and shear modes, and to the cohesive energy of the graphene layers of AB-stacked graphite. Figure 2 shows the phonon dispersion of AB- and ABC-stacked graphite. The phonon branches of AB-stacked graphite along the ΓA direction are in excellent agreement with the available experimental data. It is clearly seen, that the interlayer interaction produces major modification only of the acoustic branches, while leaving the higher-frequency branches almost unchanged, apart from splitting up to a few cm-1 (Fig. 2, inset). Figure 3 shows the obtained low-frequency phonon dispersion of AB-stacked FLG. Apart from the acoustic branches with longitudinal (L), transverse (T), and out-of-plane (Z) character, denoted as LA, TA, and ZA, the interlayer interaction brings about additional optical phonon branches due to lifting of the N-fold degeneracy of the acoustic branches. In particular, N-layer AB-stacked graphene (or NAB graphene) has 3N – 3 additional optical branches, among which there are N – 1 LO branches, N – 1 TO branches, and N – 1 ZO branches. The ZO phonons have breathing-like displacement of the layers (breathing modes or BM), the LO+TO phonons have shear displacement of the layers (shear modes or SM). We note that the low-frequency phonon dispersion of N-layer ABC-stacked FLG is indistinguishable from that of AB-stacked FLG for the same number of layers (not shown here). The BMs and SMs have frequencies between zero and 126 cm-1, the latter being the BM of AB-stacked graphite. 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 Ch. van Alsenoy, Phys. Rev. B 90 (2014) 245429. 4. V. N. Popov, L. Henrard, and Ph. Lambin, Phys. Rev. B 72 (2005) 035436. 5. V. N. Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 085407. |
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Fig. 4. …
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Valentin Popov
January 18, 2015