A range of unusual emergent behaviors have been reported in the charge-density-wave (CDW) state of the $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ ($A=\mathrm{K}$, Rb, Cs) kagome metals, including a CDW formation process without soft phonons, which points to an unconventional CDW mechanism. Here, we use inelastic x-ray scattering to show that the CDW in ${\mathrm{KV}}_3{\mathrm{Sb}}_5$ forms via phonons that soften to zero energy at the CDW ordering vector ($L$ point) around $T_{\mathrm{CDW}}=78\mathrm{K}$. The intensity of soft phonons exhibits a remarkable in-plane anisotropy, extending over a much larger momentum range along $L\text{−}A$ relative to $L\text{−}H$, which leads to diffuse scattering common among $A{\mathrm{V}}_3{\mathrm{Sb}}_5$. Using first-principles calculations, we find that the momentum-dependent electron-phonon coupling (EPC) peaks at $L$ and exhibits the same in-plane anisotropy as the phonon softening. Conversely, the electronic susceptibility does not peak at $L$ and shows the opposite in-plane anisotropy. Our findings favor momentum-dependent EPC as the driving mechanism of the CDW in ${\mathrm{KV}}_3{\mathrm{Sb}}_5$, with a CDW formation process similar to that of transition metal dichalcogenides.

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• Charge density waves
• Electron-phonon coupling
• Phonons
• Kagome metal
• Density functional theory
• Inelastic X-ray scattering