On the phonon dispersion relation of single-crystalline $\beta$--FeSe

We report on the phonon spectrum probed at the $\beta$--FeSe(001) surface by means of high-resolution electron energy-loss spectroscopy (HREELS). Single crystals of $\beta$--FeSe are cleaved under ultra-high vacuum conditions and are subsequently measured below and above the nematic transition temperature. In total we observe five phonon modes and a phonon cutoff energy of about 40 meV. We identify the origin of each phonon mode based on the selection rules of HREELS and by comparing the experimental results to the ones of \emph{ab initio} density functional calculations. The most prominent phonon modes $A_{1g}$, $B_{1g}$, and $A_{2u}$ appear at energies of about $20.5$ and $25.6$ and $40$ meV, respectively. These phonon modes disperse rather weakly while changing the momentum from zero up to the zone boundary, indicating that they are mainly of optical nature. A comparison between our results and the results of \emph{ab initio} calculations indicates that there must be a mutual interplay between magnetism and lattice dynamics in this compound, similar to the other Fe-based superconductors. Finally, we comment on the role of temperature on the phonon modes probed at the $\bar{\Gamma}$--point. It is observed that both the $B_{1g}$ and $A_{2u}$ phonon modes undergo a downward shift while increasing the temperature from 15 to 300 K. In the case of the $A_{2u}$ mode this shift is about 1.5 meV.