Reflecting on the LK-99 fervour: insights and future prospects.

A Copper-substituted lead apatite, named LK-99 by its authors, has recently emerged as a candidate for the world's first ambient-conditions superconductor, but has since not demonstrated the anticipated properties when subject to independent scrutiny. In this review we cover the experimental and theoretical studies that have been dedicated to this unusual material, while briefly discussing the original arXiv papers. So far, all the synthesis methods result in the formation of a multiphase material with unpredictable structure and unstable stoichiometry. This is a core reason why the studies discussed in the review yield different results when it comes to magnetic, electrophysical properties and even visual appearance of the material. DFT studies of the electronic structure of Pb₉Cu(PO₄)₆O are reviewed and compared with our original calculations, and the limitations of the DFT approach are discussed. While reviewing theoretical publications on the matter, we bring forward the flat-band superconductivity model and discuss the nature of flat bands and their ability to sustain superconducting transport in the model material. The possibility of the compound being a Mott insulator, given the specifics of its electronic structure, as well as the role of doping, is highlighted. Finally, we raise the question of viability of achieving superconductivity in doped apatites beyond the LK-99 formula.