Kohn anomalies in the phonon dispersion of graphene, non-adiabatic corrections and charge doping effects

Fig. 1. Phonon dispersion of graphene calculated within the NTB model. The phonon dispersion shows Kohn anomalies at the center and around the K point of the Brillouin zone due to strong electron-phonon interactions. The anomalies are manifested as non-zero slopes of the LO and TO branches at Γ and of the TO branch at K.		The phonon dispersion of graphene, calculated within the non-orthogonal tight-binding (NTB) model [1] and adopting the adiabatic approximation [2,3], shows Kohn anomalies for certain phonon branches at the center and at the K point of the Brillouin zone, arising from strong scattering of electrons by phonons. Specifically, the Kohn anomalies are manifested by a nonzero slope of the longitudinal optical (LO) and transverse optical (TO) phonon branches at the Γ point of the Brillouin zone and non-zero slope of the TO branch at the K point of the Brillouin zone (Piscanec) (Fig. 1). Due to the strong electron-phonon coupling, corrections to the adiabatic LO(Γ), TO(Γ), and TO(K) branches, have to be evaluated (Piscanec). These, so-called non-adiabatic (or dynamic) corrections, are derived in first-order quantum-mechanical perturbation theory for single-walled carbon nanotubes [4] and for graphene [5]. The resulting Kohn anomalies of the phonon branches of graphene are significantly modified. In particular: -the slope of the LO and TO branches at Γ becomes zero; the two branches are almost flat before the LO one gets a kink and the TO one gets a dip with increasing the wavevector (Fig. 2). -the slope of the TO branch at K becomes zero and the branch is almost flat over a small range of wavevectors (Fig. 3). -both LO and TO branches attain measurable non-adiabatic broadening of about 8 cm^-1 at Γ, in agreement with previous calculations and experiment (Fig. 4). The broadening of the LO branch decreases monotonously to zero and that of the TO branch passes through a maximum before decreasing monotonously to zero with the increase of the wavevector. -the TO branch attains a measurable non-adiabatic broadening of about 14 cm^-1 at K, in agreement with previous calculations and experiment (Fig. 5). The broadening decreases monotonously to zero away from the K point. The charge doping brings about changes in the phonon dispersion, the most affected being the LO and TO branches around the Γ point and the TO branch around the K point. In particular: -the frequencies of the LO and TO phonons close to Γ increase, the kink of the LO branch and the dip of the TO branch are gradually smeared and disappear with increasing the charge doping level (Fig. 6). The increase of the LO and TO frequencies at Γ reaches 11 cm^-1 for charge doping of 0.3 eV. -the frequency of the TO phonon at K increases with increasing the charge doping level, reaching 32 cm^-1 at charge doping of 0.3 eV (Fig. 7). -the non-adiabatic broadening of the LO and TO branches at Γ and of the TO branch at K gradually decrease with increasing the charge doping level (Figs. 8 and 9). We note that the maximum broadening is already below about 1 cm^-1 at doping level of 0.3 eV. The behavior of the LO/TO phonon frequency and non-adiabatic broadening at Γ and of the TO phonon frequency and non-adiabatic broadening at K are shown in Figs. 10-13. In particular: -the frequency of the LO/TO phonon at Γ and of the TO phonon at K is almost constant for doping levels below 0.1 cm^-1. For doping levels above 0.1 cm^-1, the frequency of these phonons increases almost linearly with the doping level. The slope of the quasilinear parts can be connected to the electron-phonon coupling for these phonons (Piscanec). The derived values of these couplings correspond well to the previously reported ones. -the non-adiabatic broadening of the LO/TO phonon at Γ and of the TO phonon at K decreases monotonously with increasing the doping level. The simulated Raman spectra of the G band, due to the LO/TO phonon, at different charge doping level are shown in Fig. 14. It is clearly seen that the decrease of the non-adiabatic broadening for this phonon with increasing the doping level results also in a significant increase of the peak Raman intensity of this band. This effect is observed experimentally. References: 1. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407. 2. V. N. Popov, L. Henrard, and Ph. Lambin, Phys. Rev. B 72 (2005) 035436. 3. V. N. Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 085407. 4. V. N. Popov and Ph. Lambin, Nano Res. 3 (2010) 822. 5. V. N. Popov and Ph. Lambin, Phys. Rev. B 82 (2010) 045406. .
Fig. 2. Adiabatic (dashed) and non-adiabatic (solid) LO and TO branches close to the Γ point.	Fig. 3. Adiabatic (dashed) and non-adiabatic (solid) branches close to the K point.
Fig. 4. Non-adiabatic broadening for the LO and TO branches close to the Γ point.	Fig. 5. Non-adiabatic broadening of the TO branch close to the K point.
Fig. 6. Modification of the LO and TO branches at Γ at different charge doping levels in eV.	Fig. 7. Modification of the TO branch at K at different charge doping levels in eV.
Fig. 8. Modification of the LO and TO broadening at different charge doping levels.	Fig. 9. Modification of the TO broadening at different charge doping levels.
Fig. 10. LO/TO phonon frequency at Γ vs charge doping level in comparison with available experimental and theoretical data.	Fig. 11. TO phonon frequency at K vs charge doping level.
Fig. 12. Full broadening of the LO/TO phonon at Γ vs charge doping level in comparison with available experimental and theoretical data.	Fig. 13. Non-adiabatic broadening of the TO phonon at K vs charge doping level.
Fig. 14. Raman spectra of the G band, which due to the LO/TO phonon at Γ, at different charge doping levels.

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Valentin Popov

November 21, 2010