No superconductivity in Pb$_9$Cu$_1$(PO$_4$)$_6$O found in orbital and spin fluctuation exchange calculations

TL;DR

This study uses advanced electronic structure calculations to investigate the potential for superconductivity in LK-99, concluding that spin and orbital fluctuations are too weak to support superconductivity near room temperature.

Contribution

It provides a detailed ab initio analysis of LK-99's electronic interactions and demonstrates the absence of superconductivity under realistic conditions using fluctuation exchange calculations.

Findings

Spin and orbital fluctuations are too weak for superconductivity near room temperature.

Superconductivity can be induced in RPA calculations near magnetic instability.

No superconductivity observed in FLEX calculations under optimized conditions.

Abstract

Finding a material that turns superconducting under ambient conditions has been the goal of over a century of research, and recently Pb$_{10-x}$Cu$_x$(PO$_4$)$_6$O aka LK-99 has been put forward as a possible contestant. In this work, we study the possibility of electronically driven superconductivity in LK-99 also allowing for electron or hole doping. We use an $\textit{ab initio}$ derived two-band model of the Cu $e_g$ orbitals for which we determine interaction values from the constrained random phase approximation (cRPA). For this two-band model we perform calculations in the fluctuation exchange (FLEX) approach to assess the strength of orbital and spin fluctuations. We scan over a broad range of parameters and enforce no magnetic or orbital symmetry breaking. Even under optimized conditions for superconductivity, spin and orbital fluctuations turn out to be too weak for superconductivity anywhere near to room-temperature. We contrast this finding to non-self-consistent RPA, where it is possible to induce spin-singlet $d$-wave superconductivity at $T_{\mathrm{c}}\geq300$ K if the system is put close enough to a magnetic instability.