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The kagome metals display an intriguing variety of electronic and magneticphases arising from the connectivity of atoms on a kagome lattice. A growingnumber of these materials with vanadium kagome nets host charge density waves(CDWs) at low temperatures including ScV$_6$Sn$_6$, CsV$_3$Sb$_5$, andV$_3$Sb$_2$. Curiously, only the Sc version of the $R$V$_6$Sn$_6$HfFe$_6$Ge$_6$-type materials hosts a CDW ($R = $Gd-Lu, Y, Sc). In this studywe investigate the role of rare earth size in CDW formation in the$R$V$_6$Sn$_6$ compounds. Magnetization measurements on our single crystals of(Sc,Lu)V$_6$Sn$_6$ and (Sc,Y)V$_6$Sn$_6$ establish that the CDW is suppressedby substitution of Sc by larger Lu or Y. Single crystal x-ray diffractionreveals that compressible Sn-Sn bonds accommodate the larger rare earth atomswithin loosely packed $R$-Sn-Sn chains without significantly expanding thelattice. We propose that Sc provides the extra room in these chains crucial toCDW formation in ScV$_6$Sn$_6$. Our rattling chain model explains why bothphysical pressure and substitution by larger rare earths hinder CDW formationdespite opposite impacts on lattice size. We emphasize the cooperative effectof pressure and rare earth size by demonstrating that pressure furthersuppresses the CDW in a Lu-doped ScV$_6$Sn$_6$ crystal. Our model not onlyaddresses why a CDW only forms in the $R$V$_6$Sn$_6$ materials with tiny Sc, italso advances to our understanding of why unusual CDWs form in the kagomemetals.

Tiny Sc allows the chains to rattle: Impact of Lu and Y doping on the charge density wave in ScV$_6$Sn$_6$