Origin of competing charge density waves in kagome metal ScV$_6$Sn$_6$

Understanding competing charge density wave (CDW) orders in the bilayer kagome metal ScV$_6$Sn$_6$ remains challenging. Experimentally, upon cooling, short-range order with wave vector $\mathbf{q}_{2}=(\frac{1}{3},\frac{1}{3},\frac{1}{2})$ forms, which is subsequently suppressed by the condensation of long-range $\mathbf{q}_{3}=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ CDW order at lower temperature. Theoretically, however, the $\mathbf{q}_{2}$ CDW is predicted as the ground state, leaving the CDW mechanism elusive. Here, using anharmonic phonon-phonon calculations combined with density functional theory, we predict a temperature-driven structural phase transitions from the high-temperature pristine phase to the $\mathbf{q}_{2}$ CDW, followed by the low-temperature $\mathbf{q}_{3}$ CDW, explaining experimental observations. We demonstrate that semi-core electron states stabilize the $\mathbf{q}_{3}$ CDW over the $\mathbf{q}_{2}$ CDW. Furthermore, we find that the out-of-plane lattice parameter controls the competing CDWs, motivating us to propose compressive bi-axial strain as an experimental protocol to stabilize the $\mathbf{q}_{2}$ CDW. Finally, we suggest Ge or Pb doping at the Sn site as another potential avenue to control CDW instabilities. Our work provides a full theory of CDWs in ScV$_6$Sn$_6$, rationalizing experimental observations and resolving earlier discrepancies between theory and experiment.