Transition from antiferromagnetic to ferromagnetic fluctuation under charge doping in Pb₉Cu(PO₄)₆O

The hope that copper doped lead apatite Pb₉Cu(PO₄)₆O is a room-temperature superconductor has largely been dashed by global research efforts. Nevertheless, the material has interesting magnetic properties, and research groups around the world have prepared high quality samples. We use a fluctuation exchange approximation (FLEX) approach to study the magnetic tendencies in Pb₉Cu(PO₄)₆O. We find antiferromagnetic fluctuations in the stoichiometric compound. Magnetic fluctuations transfers to ferromagnetic fluctuations with slight charge doping. This ferromagnetic situation is similar to the one-band triangular lattice Hamiltonian at three-quarter filling. Interestingly, the special k_z dependence of the Pb₉Cu(PO₄)₆O band structure makes it very sensitive to doping. Only slight charge doping switches between antiferromagnetic and ferromagnetic fluctuations. The half levitation observed in some samples can be explained by our results. Slight inhomogeneity of samples makes slight charge doping, and the samples show ferromagnetism.