Single crystal synthesis, structure, and magnetism of Pb$_{10-x}$Cu$_x$(PO$_4$)$_6$O

The recent claim of superconductivity above room temperature in Pb$_{10-x}$Cu$_x$(PO$_4$)$_6$O with 0.9 < $x$ < 1 (referred to as LK-99) has sparked considerable interest. To minimize the influence of structural defects and impurity phases on the physical properties, we have synthesized phase-pure single crystals with $x \sim 1$. We find that the crystals are highly insulating and optically transparent. X-ray analysis reveals an uneven distribution of the substituted Cu throughout the sample. Temperature ($T$) dependent magnetization measurements for $ 2 \leq T \leq 800$ K reveal the diamagnetic response characteristic of a non-magnetic insulator, as well as a small ferromagnetic component, possibly originating from frustrated exchange interactions in Cu-rich clusters in the Pb$_{10-x}$Cu$_x$(PO$_4$)$_6$O structure. No anomalies indicative of phase transitions are observed. We therefore rule out the presence of superconductivity in Pb$_{9}$Cu(PO$_4$)$_6$O crystals, and provide some considerations on the origin of anomalies previously reported in experiments on polycrystalline specimen.