Phonons in superconducting ${\mathrm{CaC}}_6$ studied via inelastic x-ray scattering

We investigate the dispersion and temperature dependence of a number of phonons in the recently discovered superconductor ${\mathrm{CaC}}_6$ utilizing inelastic x-ray scattering. Four $\left[00L\right]$ and two ab-plane phonon modes are observed, and measured at temperatures both above and below $T_c$. In general, our measurements of phonon dispersions are in good agreement with existing theoretical calculations of the phonon dispersion. This is significant in light of several discrepancies between experimental measurements of phonon-derived quantities and theoretical calculations. The present work suggests that the origin of these discrepancies lies in the understanding of the electron-phonon coupling in this material, rather than in the phonons themselves.

Figure 1

(Color online) (a) ${\mathrm{CaC}}_6$ crystal structure (Ref. 2). Green circles represent Ca atoms and the gray ring is graphene (Ref. 19). (b) The ${\mathrm{CaC}}_6$ Brillouin zone. (c) An [00L] diffraction pattern taken after the inelastic experiment. The pattern exhibits sharp ${\mathrm{CaC}}_6\left(003\right)$ and (006) peaks. The locations of possible impurity peaks from ${\mathrm{LiC}}_6$ and graphite are indicated. Almost no scattering is observed at these locations.

Figure 2

(Color online) Inelastic spectra along $\Gamma$-$L$ from the fourth Brillouin zone. As discussed in the text, the phonon cross section means that some of the phonon modes reported in Fig. 3 do not appear in these data. Data are labeled by momentum transfer in the reduced zone scheme. The spectra are offset for clarity and are normalized to account for the different efficiencies of the different detectors. The $y$ axis is a ${\log }_{10}$ scale. The blue (dark gray) solid lines are the total fit. Each red (light gray) short-long dashed line is a portion of the total fit, including the elastic line and one phonon mode peak.

Figure 3

(Color online) Phonon dispersion from spectra taken at three different temperatures: $300\mathrm{K}$ in circles, $○$; $50\mathrm{K}$ in triangles, $▵$; and $5\mathrm{K}$ shown in squares, $◻$. (a) The $\left[00L\right]$ spectra. Calculated dispersions are shown as dashed lines (Ref. 6) Different calculated modes are labeled by $A–E$. (b) The ab-plane spectra. The distance to $\chi$ is indicated by a vertical black long dashed line. The $\Gamma$-$X$ calculated dispersions are shown as pink long dashed lines and the $\Gamma$-$\chi$ dispersions are shown as blue short-long dashed lines. Note that $\Gamma$-$X$ is not a strictly ab-plane dispersion. The theoretical curves most closely corresponding to the data shown are marked with thicker lines than other curves.