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 $2\times2$ 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 $T_{\rm CDW}\approx91$~K, whose underlying mechanism and formation process remain unclear. Using inelastic X-ray scattering, we discover a short-range $\sqrt{3}\times\sqrt{3}\times2$ CDW that is dominant in ScV$_6$Sn$_6$ well above $T_{\rm CDW}$, distinct from the $\sqrt{3}\times\sqrt{3}\times3$ CDW below $T_{\rm CDW}$. The short-range CDW grows upon cooling, and is accompanied by the softening of phonons, indicative of its dynamic nature. As the $\sqrt{3}\times\sqrt{3}\times3$ CDW appears, the short-range CDW becomes suppressed, revealing a competition between these CDW instabilities. Our first-principles calculations indicate that the $\sqrt{3}\times\sqrt{3}\times2$ CDW is energetically favored, consistent with experimental observations at high temperatures. However, the $\sqrt{3}\times\sqrt{3}\times3$ 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 $T_{\rm CDW}$. 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.