Acoustic phonon softening and lattice instability driven by on-site $f$-$d$ hybridization in CeCoSi

Soft phonon modes in tetragonal CeCoSi, which undergoes a structural transition at $T_0=12$ K followed by antiferromagnetic order at $T_{\text{N}}=9.5$ K, have been investigated using high-resolution inelastic x-ray scattering. Pronounced softening was detected in the transverse acoustic modes corresponding to the $(yz+zx)$-type monoclinic distortion, consistent with the experimentally determined triclinic structure. Remarkably, the softening persists up to the zone boundary along (0, 0, $q$), indicating a short correlation length of the lattice instability. This instability, characterized by a Curie-type strain susceptibility, is interpreted as a consequence of the on-site $4f$-$5d$ hybridization, which is intrinsic to this crystal structure due to the lack of inversion symmetry at the two Ce sites.