The origin of the charge density wave phases in the kagome metal compound AV3Sb5 is still under great scrutiny. Here, we combine diffuse and inelastic x-ray scattering to identify a 3-dimensional precursor of the charge order at the L point that condenses into a CDW through a first order phase transition. The quasi-elastic critical scattering indicates that the dominant contribution to the diffuse precursor is the elastic central peak without phonon softening. However, the inelastic spectra show a small broadening of the Einstein-type phonon mode on approaching TCDW. Our results point to the situation where the Fermi surface instability at the L point is of order-disorder type with critical growth of quasi-static domains. The experimental data indicate that the CDW consists on an alternating Star of David and trihexagonal distortions and its dynamics goes beyond the classical weak-coupling scenario and is discussed within strong-electron phonon coupling and non-adiabatic models.

The long-range electronic modulations recently discovered in the geometrically frustrated kagome lattice have opened new avenues to explore the effect of correlations in materials with topological electron flat bands. Charge density waves (CDW), magnetism and superconducting phases are thought to be - depending on the electron number - the result of either the flat bands, the multiple Dirac crossings or the van Hove singularities close to the Fermi level. Nevertheless, the observation of the lattice response to the emergent new phases of matter, a soft phonon mode, has remained elusive and the microscopic origin of CDWs is still unknown. Here, we show, for the first time, a complete melting of the ScV$_6$Sn$_6$ (166) kagome lattice. The low energy longitudinal phonon with propagation vector $\frac{1}{3} \frac{1}{3} \frac{1}{2}$ collapses at 98 K, without the emergence of long-range charge order, which, remarkably, sets in with a propagation vector $\frac{1}{3} \frac{1}{3} \frac{1}{3}$. The CDW is driven (but locks at a different vector) by the softening of an overdamped phonon flat plane at k$_z$=$\pi$, characterized by an out-of-plane vibration of the trigonal Sn atoms. We observe broad phonon anomalies in momentum space, pointing to \hhy{(1)} the existence of approximately flat phonon bands which gain some dispersion due to electron renormalization, and \hhy{(2)} the effects of the momentum-dependent electron-phonon interaction in the CDW formation. \textit{Ab initio} and analytical calculations corroborate the experimental finding to indicate that the weak leading order phonon instability is located at the wave vector $\frac{1}{3} \frac{1}{3} \frac{1}{2}$ of a rather flat collapsed mode. In particular, we analytically calculate the phonon frequency renormalization from high temperatures to the soft mode, and relate it to a peak in the orbital-resolved susceptibility of the trigonal Sn atoms, obtaining excellent match with both ab initio and experimental results, and explaining the origin of the approximately flat phonon dispersion. Our data report the first example of the collapse of a kagome bosonic mode (softening of a flat phonon plane) and promote the 166 compounds of the kagome family as primary candidates to explore correlated flat phonon-topological flat electron physics.

The nature of the charge density wave phases in the kagome metal compound AV3Sb5 has raised many questions and their origin is still under debate. Here, we combine diffuse scattering (DS) and inelastic x-ray scattering (IXS) to identify a 3-dimensional (3D) precursor of the charge order with propagation vector (0.5 0 0.5), which condenses into a 3D-CDW through a first order phase transition. The quasi-elastic critical scattering indicates that the dominant contribution to the diffuse precursor is the elastic central peak without phonon softening. However, the inelastic spectra show a small broadening of the Einstein-type phonon mode on approaching the phase transition. Our results point to the situation where the Fermi surface instability at the L point is of order-disorder type with a critical growth of quasi-static domains. The results go beyond the classical weak-coupling Peierls transition dynamics and are discussed within models including strong-electron phonon coupling and non-adiabaticity.