Electronic structure and vibrational stability of copper-substituted lead apatite LK-99

Two recent preprints [Lee et al., arXiv:2307.12008 and Lee et al., arXiv:2307.12037] have received attention because they claim experimental evidence that a Cu-substituted apatite material (dubbed LK-99) exhibits superconductivity at room temperature and pressure. If this proves to be true, LK-99 will be a holy grail of superconductors. In this work, we use density functional theory $+$ $U$ calculations to elucidate some key features of the electronic structure of LK-99. We find two different phases of this material: (i) a hexagonal lattice featuring metallic half-filled and spin-split bands, nesting of the Fermi surface, and a remarkably large electron-phonon coupling that is vibrationally unstable and (ii) a triclinic lattice, with the Cu and surrounding O distorted. This lattice is vibrationally stable, and its bands correspond to an insulator. In a crystal, the Cu atoms should oscillate between equivalent triclinic positions, with an average close to the hexagonal positions. We discuss the electronic structure expected from these fluctuations and whether it is compatible with superconductivity.

Figure 1

(a) In-plane view of the crystal structure of lead apatite, ${\mathrm{Pb}}_{10}{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$. Nonequivalent Pb atoms are colored gray and dark green. O is red, and P is pink. The lattice vectors are blue arrows. (b) Top and side views of the two hexagonal-like patterns formed by Pb atoms. The inner hexagonal pattern has three Pb atoms in different layers; these layers are not equivalent since one of them has an O at its center. (c) Band structure of lead apatite. The color intensity reflects the projection of the wave functions onto Pb atoms (i.e., within its Wigner radius).

Figure 2

Possible LK-99 atomic structures. The unit cell of lead apatite was duplicated along the $c$ axis. To keep the figure as simple as possible, P and most O atoms are omitted. The Cu atoms are yellow and large. (a) and (b) show two possible stacking sequences, A and B, with Cu atoms forming a triangular or hexagonal sublattice.

Figure 3

Band structure of ${\mathrm{Pb}}_9{\mathrm{Cu}}_1{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$ with stacking A and a FM ground state. The color scale represents the projection of wave functions onto the Cu atom, where a positive or negative value denotes the spin value.

Figure 4

(a) Zoom of the bands at the Fermi level. Isosurfaces of the bands at (b) $E_F-0.01$ eV ($E_F$ is the Fermi energy), (c) $E_F$, and (d) $E_F+0.01$ eV. Different bands are in different colors; the two spherical-like surfaces, centered and $\mathrm{\Gamma }$ and $A$, are apparently nested.

Figure 5

Contour plots of the ELF (dimensionless) on planes that cross the Cu atoms and are parallel to (1,0,0) (bottom) and (0,0,1) (top). The coloring of the atoms follows Figs. 1 and 2; i.e., the Cu atom is yellow.

Figure 6

Phonon band structure of arrangement A (triangular Cu sublattice, space group 174). The right panel shows the total (red) and partial (Cu projection, blue) phonon densities of states.

Figure 7

Band structure of LK-99 under the distortion from a frozen phonon. The color scale shows the projection of the wave function onto the Cu atom; the positive (negative) values denotes the majority (minority) spin. The inset shows the actual phonon at $\mathrm{\Gamma }$ with a purple arrow. It is mostly localized in the Cu atom (blue). The $d$ orbitals are no longer degenerate, with one of them buried $\sim 0.5$ eV below $E_F$ (red-orange band). The amplitude of the frozen phonon was 0.04Å, mostly involving a shift of the Cu atom from its equilibrium position [59].

Figure 8

(a) Unit cell of the triclinic ${\mathrm{Pb}}_9{\mathrm{Cu}}_1{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$ obtained after relaxation. (b) Lateral view, with emphasis on the O atoms closer to Cu. The distances shown are in angstroms, and the other distances are very close to 2.0 Å. (c) Top view, removing atoms other than Cu, Pb, and the central O. The colors follow Fig. 1; Cu is yellow.

Figure 9

Phonon band structure for the triclinic LK-99-like crystal structure. The right panel shows the total (red) and partial (Cu projection, blue) phonon densities of states.

Figure 10

Band structure of the LK-99-like system with the triclinic lattice. The color scale shows the projection of the wave function onto the Cu atom; the positive (negative) values denote the majority (minority) spin. There is a single band within the fundamental band gap; it is not spin degenerate.

Figure 11

Energy levels at $\mathrm{\Gamma }$ of a unit cell of ${\mathrm{Pb}}_9\mathrm{Cu}{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$ at room temperature. The color intensity is the projection of each wave function on Cu atoms.

Figure 12

Two-dimensional contour plots of the Fermi surface of the hexagonal lattice of LK-99. The bands are those plotted in Fig. 4. (a) and (b) show different planes. The arrows represent a pair of phonons with momenta ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$ connecting two Fermi surfaces. The length of the arrow is the same in both panels.

Figure 13

XRD of LK-99 obtained in this work (blue line). For comparison, Fig. 2 of [5] is included (black and red lines).

Figure 14

Bayesian estimation of the error on the forces, or statistical uncertainty. Only the largest error for each time step is plotted. (a) Fitting the force field by mixing DFT and the force field, the plot corresponds to the last refinement of the force field. (b) Molecular dynamics of a $3\times 3\times 3$ supercell at room temperature, using only the force field.

Figure 15

Projected density of states on the Cu atom in LK-99 with stacking A and an FM ground state. Labels $p/3$ and $d/5$ indicate that the sum of the PDOS is divided by 3 and 5, respectively.

Figure 16

Band structure of the hexagonal lattice phase of LK-99 of some values of the Hubbard-like term $U$. The color bar denotes the projection of Cu atoms onto the state.

Figure 17

Phonon band structure of the optimized geometry for different values of the Hubbard-like term $U$. The total (red) and partial (projected on Cu, blue) phonon densities of states are shown in the right panel in different scales.