Competing charge-density wave instabilities in the kagome metal ScV${}_6$Sn${}_6$

Owing to its unique geometry, the kagome lattice hosts various many-body quantum states including frustrated magnetism, superconductivity, and charge-density waves (CDWs), with intense efforts focused on kagome metals exhibiting $2\times 2$ CDWs associated with the nesting of van Hove saddle points. Recently, a $\sqrt{3}\times \sqrt{3}$ CDW was discovered in the kagome metal ScV${}_6$Sn${}_6$ below $T_{\mathrm{C}\mathrm{D}\mathrm{W}}\approx 91$~K, whose underlying mechanism and formation process remain unclear. Using inelastic X-ray scattering, we discover a short-range $\sqrt{3}\times \sqrt{3}\times 2$ CDW that is dominant in ScV${}_6$Sn${}_6$ well above $T_{\mathrm{C}\mathrm{D}\mathrm{W}}$, distinct from the $\sqrt{3}\times \sqrt{3}\times 3$ CDW below $T_{\mathrm{C}\mathrm{D}\mathrm{W}}$. The short-range CDW grows upon cooling, and is accompanied by the softening of phonons, indicative of its dynamic nature. As the $\sqrt{3}\times \sqrt{3}\times 3$ CDW appears, the short-range CDW becomes suppressed, revealing a competition between these CDW instabilities. Our first-principles calculations indicate that the $\sqrt{3}\times \sqrt{3}\times 2$ CDW is energetically favored, consistent with experimental observations at high temperatures. However, the $\sqrt{3}\times \sqrt{3}\times 3$ CDW is selected as the ground state likely due to a large wavevector-dependent electron-phonon coupling, which also accounts for the enhanced electron scattering above $T_{\mathrm{C}\mathrm{D}\mathrm{W}}$. The competing CDW instabilities in ScV${}_6$Sn${}_6$ lead to an unusual CDW formation process, with the most pronounced phonon softening and the static CDW occurring at different wavevectors.