Introduction

Copper-doped lead apatite Pb10−xCux(PO4)6O with 0.9 < x < 1.1, known as LK-99, has been recently claimed to exhibit superconductivity above room temperature and at ambient pressure1,2. This remarkable claim is backed by magnetic (half-)levitation on a permanent magnet and by a sudden drop in resistivity at the claimed superconducting transition temperature. However, subsequent extensive experimental efforts by other groups have failed to confirm the superconductivity3,4,23,24. Different estimates of critical superconducting temperatures have so far delivered values significantly lower than room temperature25,26,27.

These state-of-the-art electronic structure calculations all assume a specific structural model as a starting point, often suggested by experiments. However, other theoretical works have questioned the suitability of these structural models both in terms of the thermodynamic feasibility of copper do**28 or the dynamical stability of the experimentally proposed structures28,29,30,31,32,33. Indeed, one of the most basic quantities used to characterize a material is its dynamic stability. A dynamically stable structure corresponds to a local minimum of the potential (free) energy surface, and its phonon frequencies are real. A dynamically unstable structure corresponds to a saddle point of the potential (free) energy surface, and some of its phonon frequencies are imaginary with associated eigenvectors that encode atomic displacement patterns that lower the energy of the system. Only dynamically stable structures can represent real materials. Puzzlingly, recent computational works have claimed that the experimentally reported structures of the parent lead apatite28,29,30,31 and of the copper-doped lead apatite compounds28,29,30,32,33 are dynamically unstable at the harmonic level, which would imply that they cannot be the true structures of the materials underpinning LK-99, and would question the validity of most electronic structure calculations to date.

In this work, we demonstrate that both parent lead apatite and copper-doped lead apatite compounds are dynamically stable at room temperature. The parent compounds are largely stable at the harmonic level, with some exhibiting very slight instabilities which are suppressed by quartic anharmonic terms. For the copper-doped compounds, dynamical stability at the harmonic level depends on the do** site, but even those that are dynamically unstable at the harmonic level are overall stable at room temperature with the inclusion of anharmonic phonon–phonon interactions.

Results

Lead apatite

Lead apatite is a compound that was first experimentally reported over 70 years ago34. Figure 1 depicts an example of lead apatite, with a hexagonal lattice (space group P63/m) and general formula Pb10(PO4)6X2, where X is either a halide atom or an OH group. The variant Pb10(PO4)6O, which is claimed to be the parent structure of LK-991,2, has also been known experimentally for decades34,35,36,37. The X site corresponds to Wyckoff position 4e, giving a multiplicity of four in the unit cell, but these sites are only partially filled with an occupation of \(\frac{1}{2}\) for halide atoms and the OH group, and an occupation of \(\frac{1}{4}\) for O. We consider two representative cases, Pb10(PO4)6O and Pb10(PO4)6(OH)2, where the specific distribution of species on the X site results in space groups P3 and P63, respectively.

The phonon dispersion of the parent Pb10(PO4)6O and Pb10(PO4)6(OH)2 compounds is shown in Fig. 1. At the harmonic level, Pb10(PO4)6O exhibits imaginary phonon frequencies at the zone boundary points M and K of the kz = 0 plane, and at the zone boundary point H of the \({k}_{z}=\frac{\pi }{c}\) plane. However, the absolute values of these imaginary frequencies are less than 2.3 meV and the resulting anharmonic potentials have a dominant quartic term that strongly suppresses the instability to about 0.2 meV per formula unit (see Supplementary Figure 4). As a result, the calculation of self-consistent phonons including anharmonic interactions fully stabilizes the structure at the relatively low temperature of 50 K, and potentially lower. Earlier works reported that Pb10(PO4)6O is dynamically unstable at the harmonic level28,29,30, a result we confirm, but our work further demonstrates that Pb10(PO4)6O is overall dynamically stable at room temperature driven by higher-order anharmonic terms.

Fig. 1: Crystal structure and phonon dispersion of parent lead apatite.
figure 1

a Crystal structure of lead apatite Pb10(PO4)6O or Pb10(PO4)6(OH)2. b Harmonic and anharmonic (50 K) phonon dispersions of Pb10(PO4)6O. c Harmonic phonon dispersion of Pb10(PO4)6(OH)2.

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Pb10(PO4)6(OH)2 is dynamically stable at the harmonic level. Earlier works reported that Pb10(PO4)6(OH)2 is dynamically unstable28,29, an opposite conclusion that we attribute to unconverged harmonic calculations (see Supplementary Fig. 3). We note that to fully converge the harmonic calculations of Pb10(PO4)6(OH)2, the required coarse q-point grid includes all of Γ, M, K, and A points, which can only be accomplished with a regular grid of minimum size 6 × 6 × 2 or alternatively a nonuniform Farey grid38 of minimum size (2 × 2 × 2) ∪ (3 × 3 × 1). In our calculations, we use the latter as it is computationally more efficient. We highlight that our strategy, combining a nonuniform Farey grid38 with nondiagonal supercells39,40, offers a significant computational advantage in phonon calculations using the finite displacement method compared to the conventional diagonal supercell method with a regular grid used by most earlier works. Our approach drastically reduces the supercell sizes required to reach convergence, with the number of atoms decreasing from 3168 to only 132.

Overall, we find that both lead apatite compounds Pb10(PO4)6O and Pb10(PO4)6(OH)2 are dynamically stable, a conclusion that is in full agreement with multiple experimental reports of the structure of lead apatite over the past 70 years34,35,36,37,41.

Copper doped lead apatite

The claim of room temperature superconductivity in LK-99 is based on copper do** of lead apatite, with copper replacing about 1 in 10 lead atoms leading to a Pb9Cu(PO4)6O stoichiometry. There are two symmetrically distinct lead sites, labeled Pb(1) and Pb(2) in the literature (see Fig. 1a), and do** at these sites results in structures with the space groups P3 and P1, respectively (Fig. 2a, b). The original LK-99 work suggested that the do** site is Pb(1)1,2, but subsequent experimental works have suggested that both Pb(1) and Pb(2) sites can be doped13,29. Computational works find that the relative energy between the two do** sites depends on the exchange-correlation functional and the magnitude of the Hubbard U parameter used on the copper atom, with most choices favouring do** at the Pb(2) site, a prediction we confirm with our own calculations. For completeness, in this work we explore do** at both sites.

Fig. 2: Electron band structure and phonon dispersion of Pb9Cu(PO4)6O.
figure 2

a, b Optimized crystal structures of Pb9Cu(PO4)6O for copper do** at the (a) Pb(1) and (b) Pb(2) sites. c, d Electronic band structures of Pb9Cu(PO4)6O for copper do** at the (c) Pb(1) and (d) Pb(2) sites. NM, FM, SOC refer to non-magnetic, ferromagnetic, and spin-orbit coupling, respectively. For do** at the Pb(2) site, the initial non-magnetic configuration converges to a ferromagnetic configuration in the presence of spin-orbit coupling. e Harmonic and anharmonic (300 K) phonon dispersions of the Pb9Cu(PO4)6O structure with P3 symmetry for do** at the Pb(1) site. f Harmonic phonon dispersion of the Pb9Cu(PO4)6O structure with P1 symmetry for do** at the Pb(2) site. The phonon dispersions for the copper doped cases are obtained using the NM state without SOC (see SI for phonon dispersions of the FM state). The data are obtained with U = 3 eV on the copper 3d orbital.

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The electronic structures of Pb9Cu(PO4)6O with copper on the Pb(1) and Pb(2) sites are shown in Fig. 2c, d. For do** at the Pb(1) site, a non-magnetic calculation leads to a metallic state in which the Fermi energy crosses four relatively flat bands (a pair of doubly degenerate bands). The inclusion of spin–orbit coupling while maintaining the non-magnetic configuration leads to a splitting of the pair of doubly-degenerate bands, and the Fermi energy crosses a pair of singly-degenerate relatively flat bands. A calculation including spin-orbit coupling and allowing a non-zero magnetic moment leads to a ferromagnetic configuration in which the system is gapped. We note that the spin-orbit coupling is not essential for a gap opening in the presence of ferromagnetic ordering. The actual role of spin-orbit coupling, which lifts the orbital degeneracy, can be replicated by selecting a suitable level of theory, as evidenced by hybrid20 or GW21 calculations showcasing a gapped state in the presence of ferromagnetic ordering. The ferromagnetic configuration is the most energetically favorable, but may not be directly relevant for room temperature experiments as single crystal measurements suggest the material is a non-magnetic insulator exhibiting a diamagnetic response with potentially a small ferromagnetic component13. Additionally, the ferromagnetic ordering may be an artifact of the DFT calculations, as dynamical mean-field theory calculations22,23,24 suggest a gap opens due to a Mott-like band splitting without the need for ferromagnetic ordering. For do** at the Pb(2) site, we also find a metallic state with a single band crossing the Fermi level in non-magnetic calculations, and a gapped state in ferromagnetic calculations. For both do** sites, we find that the phonon dispersion is only weakly affected by the level of electronic structure theory used (see Supplementary Notes 3.1 and 3.2), so the discussion below should be largely independent of the precise electronic structure of the system. We hereafter present the phonon dispersions of the non-magnetic state calculated using PBEsol+U without spin-orbit coupling.

The phonon dispersions of Pb9Cu(PO4)6O with copper on the Pb(1) and Pb(2) sites are shown in Fig. 2e, f. Do** at the Pb(2) site leads to a dynamically stable structure at the harmonic level of theory. By contrast, do** at the Pb(1) site leads to a dynamically unstable structure at the harmonic level that exhibits two imaginary phonon branches of frequencies about 15i meV across the entire Brillouin zone. This harmonic instability is present irrespective of the level of theory used, including a Hubbard U parameter on the copper d orbitals, spin-orbit coupling, and ferromagnetic ordering (see Supplementary Fig. 5). Importantly, anharmonic phonon–phonon interactions strongly suppress the instability and the structure becomes dynamically stable at 300 K. We reach similar conclusions for copper do** of Pb10(PO4)6(OH)2 (see Supplementary Fig. 6). Overall, copper-doped lead apatite is dynamically stable at room temperature for do** at either site.

The original paper claiming superconductivity in LK-99 suggested that copper do** of lead apatite occurs on the Pb(1) site1,2. As the associated structure exhibits a dynamical instability at the harmonic level, we further explore its properties by considering the potential energy surface along the imaginary phonon modes at high symmetry points in the Brillouin zone (Fig. 3a). The dominant instability is driven by a Γ point phonon mode, and fully relaxing the structure along this instability leads to a distinct structure of P1 symmetry (Fig. 3b; see also the link in Data availability statement for the optimized structure file). We find that the P1 structure is dynamically stable at the harmonic level of theory (Fig. 3c), consistent with an earlier report33. We ascribe the harmonic stability of the P1 structure to a downward shift of the occupied part of the density of states compared to the P3 structure (Fig. 3e), a trend similarly observed in the density of states of the harmonically stable Pb(2) do** case in Fig. 2f (see also Supplementary Fig. 10 for the density of states of the Pb(2) do** case). This indicates that the harmonic stability is dominated by copper-derived orbitals. The four bands (a pair of doubly-degenerate bands) that cross the Fermi level in the P3 structure (Fig. 2c) split under the distortion, such that the resultant P1 structure has a metallic state with a single doubly-degenerate band crossing the Fermi level in the non-magnetic configuration (Fig. 3d). The distorted P1 structure becomes an insulator in the presence of ferromagnetic ordering, which is consistent with previous results33,42, similar to lead apatite with copper do** at the Pb(2) site (Fig. 2d).

Fig. 3: Γ-distorted P1 structure and its electron band structure and phonon dispersion.
figure 3

a Potential energy surface along the imaginary phonon modes at high symmetry points in the Brillouin zone of the P3 structure for do** at the Pb(1) site (see its harmonic phonon dispersion in Fig. 2e). b Crystal structure distorted along the Γ mode of the P3 structure. c, d. c Harmonic phonon dispersion and (d) electron band structure of the Γ-distorted P1 structure. In (a–c), the data are obtained using the NM state without SOC. e Non-magnetic density of states (DOS) and Cu partial DOS of both the P3 and Γ-distorted P1 structures. See also partial DOS of other atoms in Supplementary Fig. 10. The data are obtained with U = 3 eV on the copper 3d orbital.

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Interestingly, the relative energy of the P3 structure compared to the Γ-distorted P1 structure is strongly dependent on both the volume and the electronic correlation strength as measured by the Hubbard U parameter. Specifically, we find that harmonic instabilities favouring the P1 phase occur for large values of U and large volumes, while the harmonic instability of the P3 phase completely disappears for small values of U and small volumes. This is evident from the phonon dispersion of the P3 phase for different U values and volumes (Fig. 4a) and from the enthalpy difference between P3 and P1 phases indicated by the color bar in the phase diagram in Fig. 4b. These observations suggest that controlling volume, for example through hydrostatic pressure or strain, and controlling the degree of electronic correlation, for example by applying a gate voltage or do**, can be used to navigate the structural phase diagram of compounds based on lead apatite. Specifically, it may be possible to observe a temperature-driven structural phase transition between a low temperature P1 phase and high temperature P3 phase in a regime with a large harmonic dynamical instability. Finally, we note that electronic correlation beyond the static description provided by a Hubbard U correction may play an important role on this phase diagram22,23,24, so further work is required to fully characterize it.

Fig. 4: Volume-Hubbard U phase diagram.
figure 4

a Representative harmonic phonon dispersions of the non-magnetic P3 structure doped at the Pb(1) site for different U values and volumes. b Volume-Hubbard U phase diagram of Pb9Cu(PO4)6O for do** at the Pb(1) site. The volume of the P3 structure is presented on the horizontal axis as a reference, corresponding to a pressure range of 0 − 25 GPa. We find slightly larger volume changes in the Γ-distorted P1 structure. The color bar indicates the enthalpy difference between the P3 and P1 structures (in meV per atom). SOC is not considered in (a) and (b).

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Discussion

Since the original claim of room temperature superconductivity in LK-99, seven phonon dispersion calculations have been reported in the literature. Of these, parent28,29,30,31 and copper-doped lead apatite with a P3 space group28,29,30,32,33 are claimed to be dynamically unstable at the harmonic level, while another work claims that the copper-doped lead apatite is dynamically stable at the harmonic level27.

We attribute these puzzling and contradictory conclusions about the dynamical stability of lead apatite to the complexity of harmonic phonon calculations in this system, with a unit cell containing at least 41 atoms, and to the subtle interplay between volume, electronic correlation strength, and phonons. First, we find that fully converged phonon calculations for the parent compound Pb10(PO4)6(OH)2 require relatively large coarse q-point grids, specifically incorporating the Γ, M, K, and A points. However, none of the previously reported calculations include all these points, and as a result they incorrectly conclude that this compound is dynamically unstable at the harmonic level. Second, we find that dynamical stability for the copper-doped compounds at the harmonic level depends on both the value of the Hubbard U parameter and the volume of the system (Fig. 4; see also Supplementary Note 3.3). In this context, we rationalize the seemingly contradictory conclusions about dynamical stability of the copper-doped compounds by suggesting that different works use different volumes and different choices for the Hubbard U parameter.

Beyond clarifying the dynamical stability of lead apatite at the harmonic level, we have shown that anharmonic phonon–phonon interactions play a key role in stabilizing multiple lead apatite compounds. Overall, our calculations indicate that both parent and copper-doped lead apatite compounds are dynamically stable at room temperature.

We believe that lead apatite is a nice example to illustrate the ability of state-of-the-art first principles methods to fully characterize a complex and experimentally relevant system. However, our work also demonstrates that reliable results and conclusions can only be reached with a careful consideration of convergence parameters, such as the size of the q-point grid, and physical models, such as the inclusion of anharmonic phonon–phonon interactions.

We show that the experimentally suggested structures of lead apatite and copper-doped lead apatite are dynamically stable at room temperature. Most structures are dynamically stable at the harmonic level, but some key structures, including the structure claimed to be responsible for superconductivity at ambient conditions, only becomes dynamically stable with the inclusion of anharmonic phonon–phonon interactions. Our results resolve a puzzling suggestion by multiple earlier computational works that claimed that the experimentally reported structures of both parent and copper-doped lead apatite compounds were dynamically unstable, and fully reconcile the current experimental and theoretical description of the structure of lead apatite.

Methods

Electronic structure calculations

We perform density functional theory (DFT) calculations using the Vienna ab initio simulation package vasp43,44, which implements the projector-augmented wave method45. We employ PAW pseudopotentials with valence configurations 5d106s26p2 for lead, 3d104s1 for copper, 3s23p3 for phosphorus, 2s22p4 for oxygen, and 1s1 for hydrogen. For the exchange-correlation energy, we use both the generalized-gradient approximation functional of Perdew–Burke–Ernzerhof (PBE)46 and its modified version for solids (PBEsol)47. We find that experimental lattice parameters agree well with those predicted by PBEsol, and the data presented in the main text has been obtained using PBEsol (see the comparison between PBE and PBEsol in Supplementary Fig. 5a). An on-site Hubbard interaction U is applied to the copper 3d orbitals based on the simplified rotationally invariant DFT+U method by Dudarev and co-workers48. We have checked that DFT+U gives almost identical lattice parameters to DFT. Converged results are obtained with a kinetic energy cutoff for the plane wave basis of 600 eV and a k-point grid of size 4 × 4 × 5 and 6 × 6 × 8 for the primitive cell of the parent and copper-doped lead apatite, respectively (see convergence test results in Supplementary Note 1). The geometry of the structures is optimized until all forces are below 0.01 eV per Å and the pressure is below 1 kbar.

Harmonic phonon calculations

We perform harmonic phonon calculations using the finite displacement method in conjunction with nondiagonal supercells39,40. We find that a coarse q-point grid of size 2 × 2 × 2 leads to converged phonon dispersions for the parent compound Pb10(PO4)6O and the Cu-doped compounds. However, for the parent compound Pb10(PO4)6(OH)2, a converged calculation requires a minimum coarse q-point grid including the high symmetry points Γ, M, K, and A, which we accomplish by means of a Farey nonuniform grid38 of size (2 × 2 × 2) ∪ (3 × 3 × 1). To evaluate the force derivatives, we use a three-point central formula with a finite displacement of 0.02 bohr. The underlying electronic structure calculations are performed using the same parameters as those described above. We have also cross-checked the phonon dispersions by performing additional calculations with castep49 and Quantum Espresso50 within the finite displacement method and density functional perturbation theory, respectively (see Supplementary Fig. 2).

Anharmonic phonon calculations

We perform anharmonic phonon calculations using the stochastic self-consistent harmonic approximation (SSCHA)51,52,53, which accounts for anharmonic effects at both zero and finite temperatures. The self-consistent harmonic approximation54 is a quantum variational method on the free energy, and the variational minimization is performed with respect to a trial harmonic system. In its stochastic implementation, the forces on atoms are calculated in an ensemble of configurations drawn from the trial harmonic system. We use vasp to perform electronic structure calculations using the same parameters as those described above, and consider configurations commensurate with a 2 × 2 × 2 supercell. The number of configurations needed to converge the free energy Hessian is of the order of 4000 for the parent lead apatite structure and of the order of 8000 configurations for the copper-doped structure.