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The family of rare-earth tritellurides RTe$_3$ features charge-density-wave(CDW) order related to strongly momentum-dependent electron-phonon coupling.Similar to other CDW compounds, superconductivity is observed when the CDWorder is suppressed via hydrostatic pressure [1]. What sets the heavier membersof the RTe3 series apart is the observation of a second CDW transition at lowertemperatures having an in-plane ordering wavevector $q_{CDW,2}\parallel [100]$of almost the same magnitude but orthogonal to the ordering wavevector$q_{CDW,1}\parallel [001]$ observed at higher temperatures [2]. Here, we reportan inelastic x-ray scattering investigation of the lattice dynamics ofDyTe$_3$. In particular, we show that there are several phonon modes along bothin-plane directions, which respond to the onset of the CDW transition at$T_{CDW,1}=308\,\rm{K}$. Surprisingly, these soft modes close to$q_{CDW,2}=(0.68,0,0)$ show strong softening near $T_{CDW,1}$ but do notexhibit any response to the lower-temperature transition at$T_{CDW,2}=68\,\rm{K}$. Our results indicate that the low-temperature CDW orderis not just the 90{\deg} rotated analogue of the one appearing at hightemperatures.

Competing soft phonon modes at the charge-density-wave transitions in DyTe$_3$