Abundant lattice instability in kagome metal ScV$_6$Sn$_6$

Kagome materials are emerging platforms for studying charge and spin orders. In this work, we have revealed a rich lattice instability in a $\mathbb{Z}_2$ kagome metal ScV$_6$Sn$_6$ by first-principles calculations. Beyond verifying the $\sqrt3 \times \sqrt3 \times 3$ charge density wave (CDW) order observed by the recent experiment, we further identified three more possible CDW structures, i.e., $\sqrt3 \times \sqrt3 \times 2$ CDW with $P6/mmm$ symmetry, $2 \times 2 \times 2$ CDW with $Immm$ symmetry, and $2 \times 2 \times 2$ CDW with $P6/mmm$ symmetry. The former two are more energetically favored than the $\sqrt3 \times \sqrt3 \times 3$ phase, while the third one is comparable in energy. These CDW distortions involve mainly out-of-plane motions of Sc and Sn atoms, while V atoms constituting the kagome net are almost unchanged. We attribute the lattice instability to the smallness of Sc atomic radius. In contrast, such an instability disappears in its sister compounds $R$V$_6$Sn$_6$ ($R$ is Y, or a rare-earth element) because $R$ has a larger radius. Our work indicates that ScV$_6$Sn$_6$ might exhibit varied CDW phases in different experimental conditions and provides insights to explore rich charge orders in kagome materials.