  The pursuit of room-temperature ambient-pressure superconductivity in novel
materials has sparked interest, with recent reports suggesting such properties
in Cu-substituted lead apatite, known as LK-99. However, these claims lack
comprehensive experimental and theoretical support. In this study, we address
this gap by conducting ab initio calculations to explore the impact of varying
doping concentrations (x = 0, 1, 2) on the stability and electronic properties
of five compounds in the LK-99 family. Our investigations confirm the isolated
flat bands that intersect the Fermi level in LK-99 (Pb9Cu(PO4)6O:Cu<Pb(1)>). In
contrast, the other four parent compounds exhibit insulating behavior with wide
band gaps. X-ray diffraction spectra based on the DFT simulations at 0K confirm
the presence of Cu substitution on Pb(1) sites in the originally synthesized
LK-99 sample, while an extra peak suggests potential alternative like
Pb8Cu2(PO4)6 phases due to compositional variations in the original LK-99
samples. Furthermore, the LK-99 structure undergoes substantial lattice
constriction, resulting in a significant 5.5% reduction in volume and 6.8% in
area of two mutually inverted triangles formed by Pb(2) atoms. Meanwhile,
energy calculations reveal a marginal energy preference for substituting Cu on
Pb(2) sites over Pb(1) sites, with a difference of approximately 0.010 eV per
atom (roughly 0.9645 k/mol). Intriguingly, at pressures exceeding 73 GPa,
stability shifts towards LK-99 containing Cu substitutions on Pb(1) sites.
Despite exhibiting higher electronic conductivity than parent compounds,
Pb9Cu(PO4)6O:Cu<Pb(1)> falls short of the conductivity levels observed in
metals or advanced oxide conductors with the simulation based on the Boltzmann
transport theory.
