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Fermi level tuning and double-dome superconductivity in the kagome metals CsV$_3$Sb$_{5-x}$Sn$_x$

The recently reported \textit{A}V$_3$Sb$_5$ (\textit{A} = K, Rb, Cs) family of kagome metals are candidates for unconventional superconductivity and chiral charge density wave (CDW) order; both potentially arise from nested saddle points in their band structures close to the Fermi energy. Here we use chemical substitution to introduce holes into CsV$_3$Sb$_{5}$ and unveil an unconventionalcoupling of the CDW and superconducting states. Specifically, we generate a phase diagram for CsV$_3$Sb$_{5-x}$Sn$_{x}$ that illustrates the impact of hole-doping the system and lifting the nearest vHs toward and above $E_F$. Superconductivity exhibits a non-monotonic evolution with the introduction of holes, resulting in two "domes" peaked at 3.6\,K and 4.1\,K and the rapid suppression of three-dimensional CDW order. The evolution of CDW and superconducting order is compared with the evolution of the electronic band structure of CsV$_3$Sb$_{5-x}$Sn$_x$, where the complete suppression of superconductivity seemingly coincides with an electron-like band comprised of Sb $p_z$ orbitals pushed above E$_F$.