Effective tight-binding Hamiltonian for the low-energy electronic structure of the Cu-doped lead apatite and the parent compound

TL;DR

This study develops analytical tight-binding models to understand the low-energy electronic structure of LK-99 and its parent compound, revealing how Cu doping alters band dispersions and discussing potential superconductivity mechanisms.

Contribution

The paper provides explicit analytical tight-binding Hamiltonians for both the parent and Cu-doped LK-99, offering a new effective model to study quantum phenomena including superconductivity.

Findings

Parent material is an insulator with well-separated oxygen bands.

Cu doping significantly alters band dispersions due to Cu-O interactions.

A four-band model effectively describes the Cu-O interactions on the buckled honeycomb lattice.

Abstract

We examine the origin of the formation of narrow bands in LK-99 (Pb$_{9}$Cu(PO$_4$)$_6$O) and the parent compound without the Cu doping using density functional theory calculations and model Hamiltonian studies. Explicit analytical expressions are given for a nearest-neighbor tight-binding (TB) Hamiltonian in the momentum space for both the parent and the LK-99 compound, which can serve as an effective model to study various quantum phenomena including superconductivity. The parent material is an insulator with the buckle oxygen atom on the stacked triangular lattice forming the topmost bands, well-separated from the remaining oxygen band manifold. The $C_3$ symmetry-driven two-band TB model describes these two bands quite well. These bands survive in the Cu-doped, LK-99, though with drastically altered band dispersion due to the Cu-O interaction. A similar two-band model involving the Cu $xz$ and $yz$ orbitals broadly describes the top two valence bands of LK-99. However, the band dispersions of both the Cu and O bands are much better described by the four-band TB model incorporating the Cu-O interactions on the buckled honeycomb lattice. We comment on the possible mechanisms of superconductivity in LK-99. even though the actual T$_c$ may be much smaller than reported, and suggest that interstitial Cu clusters leading to broad bands might have a role to play