Lattice dynamics of the cluster chain compounds $M_2{\mathrm{Mo}}_6{\mathrm{Se}}_6(M=\mathrm{K},\text{Rb},\text{Cs},\text{In},\text{and Tl})$

The lattice dynamics of members of the $M_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$ family of materials with guest ions $M=\mathrm{K}$, Rb, Cs, In, and Tl has been studied using inelastic x-ray scattering and Raman spectroscopy at room temperature, as well as by ab initio calculations. We find a good match between calculations and experiment, both for structure factors (eigenvectors) and for the calculated phonon frequencies. The observed lattice dynamics for ${\mathrm{Tl}}_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$ show no signs of anharmonicity or absence of avoided dispersion crossings, thus ruling out previously hypothesized rattling phonon modes. The reduced mode energies for ${\mathrm{In}}_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$ are identified as only partially responsible for the lower superconducting transition temperature $T_c$ in this material when compared to ${\mathrm{Tl}}_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$.

Figure 1

(a) Crystal structure of $M_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$, seen along $a$ (left) and along the unique axis $c$ (right). (b) Calculated phonon dispersions for ${\mathrm{Tl}}_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$. Colored circles mark the contributions from Tl, Mo, and Se for the transverse (left) and longitudinal (right) phonons. The longitudinal guest-ion and chain modes are marked by a brown and a purple arrow, respectively. Note the different energy scales for the two panels.

Figure 2

(a) Selected IXS spectra of ${\mathrm{Tl}}_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$ close to the $\mathrm{\Gamma }$ point (002). Two observed modes are indicated by brown and purple color in solid symbols for positive energy loss and dashed symbols for negative energy loss. The resolution function, shifted to match the lowest, strongest phonon mode is shown in two copies in each spectrum. The spectra are offset vertically for graphical clarity. (b) Extracted phonon dispersions along the $\left(00l\right)$ direction for longitudinal modes of ${\mathrm{Tl}}_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$. The data are overlaid on a grayscale representation of the expected IXS active modes in our scattering geometry. Error bars represent the statistical uncertainty of peak positions from the numerical fits applied to the raw spectra.

Figure 3

(a) Longitudinal phonon dispersions for several $M_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$. (b) A comparison of the dispersion of the guest-ion modes about the $\mathrm{\Gamma }$ point, normalized against the guest-ion mass.

Figure 4

(a) Raman spectroscopy of the low-energy phonon modes in the series $M_2{\mathrm{Mo}}_6{\mathrm{Se}}_6$. Spectra with the light polarization perpendicular and along the crystal $c$ axis are shown as circles and stars, respectively. (b) Peak positions of all modes observed in (a) as a function of guest ion.