Phonon dispersion and low-energy anomaly in ${\text{CaC}}_6$ from inelastic neutron and x-ray scattering experiments

We report measurements of phonon dispersion in ${\text{CaC}}_6$ using inelastic x-ray and neutron scattering. We find good overall agreement, particularly in the 50 meV energy region, between experimental data and first-principles density-functional-theory calculations. However, on the longitudinal dispersion along the (111) axis of the rhombohedral representation, we find an unexpected anticrossing with an additional longitudinal mode, at about 11 meV. At a comparable energy, we observe also unexpected intensity on the in-plane direction. These results resolve the previous incorrect assignment of a longitudinal phonon mode to a transverse mode in the same energy range. By calculating the electron susceptibility from first principles we show that this longitudinal excitation is unlikely to be due to a plasmon and consequently can probably be due to defects or vacancies present in the sample.

Figure 1

(Color online) Right panel: rhombohedral crystal structure of ${\text{CaC}}_6$. Large red circles correspond to calcium ions and small black ones to carbon atoms. Left panel: the corresponding hexagonal unit cell. The three-rhombohedral $c$ axis is perpendicular to the honeycomb graphene planes.

Figure 2

(Color online) Top panel: section of the rhombohedral reciprocal space corresponding to the $a^{\ast }{b}^{\ast }$ plane of the graphite sublattice in ${\text{CaC}}_6$. Circle corresponds to fixed $Q$ lines in the plane. Two $\Gamma$ points in different hexagons are shown. The line connecting the two $\Gamma$ points is parallel to $\left(1\overline{1}0\right)$. The large, dashed hexagon indicate the projection of the rhombohedral Brillouin zone, with X the zone boundary along $\left(1\overline{1}0\right)$. Bottom panel: the circles can be shifted in different Brillouin zones along the graphite $c^{\ast }$, adding a component $Q_z\times \left(111\right)$. The L point lies halfway between two planes at $Q_z\times \left(111\right)+\left(0.50.50.5\right)$.

Figure 3

(Color online) Inelastic x-ray (top panel) and neutron (bottom panel) energy loss spectra at $\mathbf{Q}=\mathbf{G}+\mathbf{q}=2\times \left(1,1,1\right)+\left(0.3,0.3,0.3\right)$, corresponding to a longitudinal phonon polarization along 0.6 of the $\Gamma \text{−}L$ line. Data are fitted using a convolution of the instrumental function with an harmonic oscillator. In the top panel, theoretical calculations have been included (dashed lines).

Figure 4

(Color online) Top panel: phonon dispersion along the (1 1 1) direction. Hollow and filled circles indicate, respectively, constant $Q$ and constant energy INS scan. Triangles represent IXS constant $Q$ scan. Lines represents ab initio calculated phonon dispersion. The scans are of type $Q_z\times \left(111\right)+\left(qqq\right)$ with and $Q_z=1,2,3$, in rhombohedral lattice. Using standard notation, we label LA and TA the longitudinal- and transverse-acoustic modes, respectively. We label LO and TO longitudinal- and transverse-optical modes, respectively. Bottom panel: zoom on the low-energy part of the above dispersion data. Circles indicate the experimental points (IXS and INS together). Dashed lines show the dispersion of the ab initio longitudinal-acoustic phonon and of the unknown mode. Continuous lines show the dispersion of the interacting mode described by Eq. (2), fitted to the experimental data.

Figure 5

Real and imaginary part of the susceptibility for ${\text{CaC}}_6$.

Figure 6

(Color online) Constant-$Q$ in-plane phonon density of states obtained using IXS (line with error bars) compared to ab initio calculations (thick lines). The experimental elastic contribution is also shown (thin lines). The scans are obtained at $\mathbf{G}+\mathbf{q}$, where $\mathbf{G}=Q_z\times \left(111\right)$ and $Q_z=1$ (top, left), $z=2$ (top, right), $z=3$ (bottom), while $\mathbf{q}$ lies in plane and we give its modulus along the $\left(1\overline{1}0\right)$ direction in reduced length units for each spectra. For both top panels the energy resolution is 3 meV. For the bottom panels the energy resolution setup is 3 meV (left) and 1.5 meV (right). Calculations are normalized to the mode at about 50 meV in the left panel while normalization is to the lowest-energy mode in the right panel.