Climbing the ladder: a method for identifying promising copper-lead apatites

We develop a DFT screening procedure for copper-substituted lead apatites of the composition Pb$_9$Cu(XO$_4$)$_6$Y that enforces three design rules: thermodynamic stability, Cu site preference, and symmetry robustness of the near-Fermi electronic structure. A convex-hull analysis over P/V/As as X, O/F/Cl/Br as Y, identifies vanadates as the only members on or beneath the hull. Across the family, Cu substitution at Pb$^{\text{I}}$ ($4f$) preserves flat bands at E$_F$, whereas Pb$^{\text{I}}$ ($6h$) either gaps or severely distorts them. Small symmetry-lowering relaxations ($P3 \rightarrow P1$) are also capable of opening the band gap, motivating symmetry robustness as a filter. Applying these criteria singles out Pb$_9$Cu(VO$_4$)$_6$Br$_2$ (and, possibly, Cl$_2$) as leading candidates. This work motivates experimental study of the selected compounds, as well as a dedicated study of strong correlations.