Phonon surface mapping of graphite: Disentangling quasi-degenerate phonon dispersions

The two-dimensional mapping of the phonon dispersions around the $K$ point of graphite by inelastic x-ray scattering is provided. The present work resolves the longstanding issue related to the correct assignment of transverse and longitudinal phonon branches at $K$. We observe an almost degeneracy of the three TO-, LA-, and LO-derived phonon branches and a strong phonon trigonal warping. Correlation effects renormalize the Kohn anomaly of the TO mode, which exhibits a trigonal warping effect opposite to that of the electronic band structure. We determined the electron-phonon coupling constant to be $166{\left(\text{eV}/\text{Å}\right)}^2$ in excellent agreement to GW calculations. These results are fundamental for understanding angle-resolved photoemission, double-resonance Raman and transport measurements of graphene-based systems.

Figure 1

(Color online) (a) Scattering geometry for the IXS experiments: spectra are collected along a line starting from the (010) BZ along the (210) (i.e., $\Gamma KM$) direction. (b) IXS spectra of graphite single crystals along the $\Gamma KM$ line. The three phonon modes are fit by Lorentzians. The magenta and green Lorentzians denote the LA and LO modes. The TO mode is denoted by a red Lorentzian. The number in each viewgraph refers to the distance from $K$. (c) The calculated (left panel) and measured (right panel) IXS intensities for LO (green) and TO (red) modes.

Figure 2

(Color online) The interpolated experimental two-dimensional phonon-dispersion relation for (a) the longitudinal-optic and acoustic-phonon branches and (b) the transverse-optical phonon branch that has a Kohn anomaly. Dots denote the experimental data taken from the maxima of Lorentzian fits [see Fig. 1b]. (c) Equienergy contours from the interpolated phonon surfaces for the three highest phonon branches around $K$ for energies 155–170 meV. ○ and △ denote an interpolation to the raw experimental data in (a) and (b). Lines are GW calculations. The present color code follows the diabatic notation: branches with the same color have similar phonon patterns.

Figure 3

(Color online) Phonon dispersion along the $\Gamma KM$ direction. Points are measurements extracted from the maxima of the Lorentzians from Fig. 1b. Full lines are GW calculations. The dashed line is the TO branch from DFT calculations. DFT calculations of the longitudinal branches are not affected significantly by GW corrections and therefore not shown. The color code is the same as in Fig. 2.