Phase Stability of Lead Phosphate Apatite Pb$_{10-x}$Cu$_{x}$(PO$_{4}$)$_{6}$O, Pb$_{10-x}$Cu$_{x}$(PO$_{4}$)$_{6}$(OH)$_{2}$, and Pb$_{8}$Cu$_{2}$(PO$_{4}$)$_{6}$

Abstract: Recently, Cu-substituted lead apatite LK-99 was reported to have room-temperature ambient-pressure superconductivity. Here we utilize density functional theory (DFT) total energy and harmonic phonon calculations to investigate the thermodynamic and dynamic stability of two lead phosphate apatites in their pure and Cu-substituted structures. Though Pb${10}$(PO$_4$)$_6$O and Pb${10}$(PO$_4$)$_6$(OH)$_2$ are found to be thermodynamically stable (i.e., on the T=0K ground state convex hull), their Cu-substituted counterparts are above the convex hull. Harmonic phonon calculations reveal dynamic instabilities in all four of these structures. Oxygen vacancy formation energies demonstrate that the addition of Cu dopant substituting for Pb increases the likelihood of the formation of oxygen vacancies on the anion site. We propose a new possible phase in this system, Pb$_8$Cu$_2$(PO$_4$)$_6$, where two monovalent Cu atoms are substituted for two Pb(1) atoms and the anion oxygen is removed. We also propose several reaction pathways for Pb$_9$Cu(PO$_4$)$_6$O and Pb$_8$Cu$_2$(PO$_4$)$_6$, and found that both of these two structures are likely to be synthesized under a 1:1 ratio of reactants Pb$_2$SO$_5$ and Cu$_3$P. Our work provides a thorough foundation for the thermodynamic and dynamic stabilities of LK-99 related compounds and we propose several possible novel synthesis reaction pathways and a new predicted structure for future studies.