Vacancy ordering and phonon spectrum in the Fe superconductor K$_{0.8}$Fe$_{1.6}$Se$_2$

We have performed Raman-scattering measurements on a high-quality single crystal of the recently discovered Fe-based superconductor K$_{0.8}$Fe$_{1.6}$Se$_2$ ($T_c$ = 32 K). At least thirteen phonon modes were observed in the wave number range 10$-$300 cm$^{-1}$. The spectra possess a four-fold symmetry indicative of bulk vacancy order in the Fe-deficient planes. We perform a vibration analysis based on first-principles calculations, which both confirms the ordered structure and allows a complete mode assignment. We observe an anomaly at $T_c$ in the 180 cm$^{-1}$ $A_g$ mode, which indicates a rather specific type of electron-phonon coupling.