  Room-temperature superconductivity represents a significant scientific
milestone, with the initial report of LK-99, a copper-substituted lead apatite
$\mathrm{Pb}_{10-x}\mathrm{Cu}_{x}(\mathrm{PO}_{4})_{6}\mathrm{O}$, offering a
potential breakthrough. However, other researchers have encountered numerous
challenges in replicating the original experimental results. In recent studies,
Wang et al. successfully observed signs of a possible superconducting phase,
such as smaller resistance and stronger diamagnetism, upon doping S into the
samples. This indicates that the introduction of S is of significant importance
for achieving an appropriate structure. To further investigate the role of S,
we have considered the
$\mathrm{Pb}_{10-x}\mathrm{Cu}_{x}(\mathrm{PO}_{4})_{6}\mathrm{S}$,
systematically discussing its thermodynamic stability, as well as the influence
of S on the distribution, concentration, and electronic properties of Cu. We
find that $\mathrm{Pb}_{10-x}\mathrm{Cu}_{x}(\mathrm{PO}_{4})_{6}\mathrm{S}$
maintains thermodynamic stability, with S primarily influencing the
distribution of Cu. The critical element dictating the electronic
characteristics of the material post-synthesis is Cu, while the impact of S on
the electronic properties is relatively minor. Our work provides valuable
insights into the synthesis of potential apatite based room-temperature
superconductors and the role of S in facilitating Cu doping.
