# Electronic Structure and Vibrational Stability of Copper-substituted Lead
Apatite (LK-99)

J. Cabezas-Escares  Nicolás F. Barrera  Departamento de Física, Facultad de
Ciencias, Universidad de Chile, Santiago, Chile  Center for the Development of
Nanoscience and Nanotechnology (CEDENNA), Santiago, Chile  Robert H. Lavroff
Department of Chemistry and Biochemistry, University of California Los
Angeles, Los Angeles, California, USA  Anastassia N. Alexandrova  Department
of Chemistry and Biochemistry, University of California Los Angeles, Los
Angeles, California, USA; Department of Materials Science and Engineering,
University of California Los Angeles, Los Angeles, California, USA; and
California Nanoscience Institute (CNSI), Los Angeles, California, USA  C.
Cardenas  F. Munoz<EMAIL_ADDRESS>Departamento de Física, Facultad de
Ciencias, Universidad de Chile, Santiago, Chile  Center for the Development of
Nanoscience and Nanotechnology (CEDENNA), Santiago, Chile

###### Abstract

Two recent preprints in the physics archive (arXiv) have called attention as
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 used Density Functional Theory (DFT+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, a nesting of the Fermi surface, a remarkably
large electron-phonon coupling, but this lattice is vibrationally unstable.
(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 if it is compatible with
superconductivity.

##  I Introduction

Since the discovery of superconductivity in 1911 by H. Kamerlingh Onnes [1,
2], the phenomenon has fascinated scientists. Many efforts have been made to
find materials capable of this property under conditions of temperature and
pressure that allow practical applications. This search has resulted in the
discovery of materials such as the conventional, or BCS, superconductor
$\mathrm{MgB}_{2}$ with $T_{c}=39$ K [3] or unconventional (beyond BCS)
superconductor Y-Ba-Cu-O [4] ($T_{c}=93\,K$). Nevertheless, a room-temperature
and pressure superconductor remains elusive.

The material dubbed LK-99, an apatite-like crystal with the approximate
formula unit CuPb9(PO4)6O, has been in the spotlight of the condensed matter
physics community, as two preprints articles claim it has a superconducting
critical temperature over 400 K at atmospheric pressure [5, 6]. If these
claims prove to be true, this discovery could be one of the major advancements
in the field of superconductivity.

Most of the recent literature on this material points against it as a
superconductor, with several apparently successful replication attempts,
capable of reproducing results such as X-ray patterns, but without exhibiting
superconductivity[7, 8, 9, 10]. It now seems generally accepted that the
abrupt drop in electrical resistance below $T=104^{\circ}$C is originating
from CuS2 impurities[11, 12, 13, 14, 15, 16, 17]. Interestingly, one study
found that CuS2 impurities cannot reproduce the colossal magnetoresistance at
room temperature found in LK-99[18], so this material could be promising as a
magnetic sensor. An experimental study describes a strange metal phase in this
system, without discarding the observation of the Meissner effect[19]. Another
study suggests the possibility of interfacial superconductivity at grain
boundaries[20]. A theoretical paper supports this last idea, with a model of
excitonic superconductivity[21]. Recent theoretical studies have focused on
understanding the defects contained in this material or its parent
compounds[22, 23], and the structural stability of the crystal structure
associated with LK-99[24, 25, 26]. Even one theoretical study has shown that
this material could be one of the few that hosts Weyl fermions while breaking
the time-reversal symmetry[27].

In this article, we explore the possibility of LK-99 being such a
superconducting material.

We will begin by discussing, in Sec. II the main properties and crystal
structure of lead apatite. Then, in Sec. III we will elaborate on the possible
crystal structure and magnetic order of the so-called LK-99 system, as well as
its electronic structure. This is the structure we have seen in other
theoretical studies [28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 38, 23, 39],
and we will show it is vibrationally unstable. In Sec. IV we will introduce a
related crystal structure without imaginary phonon frequencies, but it is a
band insulator. We will discuss our findings in Sec. V. Finally, we will close
this contribution with our conclusions, in Sec. VI

##  II Lead-apatite

Lead apatite materials have a hexagonal lattice with space group P63/m and
formula unit Pb10(PO4)6X2, with X=Cl, OH, F, Br [40]. However, the apatite-
like LK-99 phase has a slightly different composition, Pb10-xCux(PO4)6O. There
are reports of a related (Ca-based) oxyapatite crystal with the desired
composition [41]. Its crystal structure is very close to other apatite
materials. It has a hexagonal lattice with space group number #174 and point
group P$\bar{6}$. We used this geometry as our basis to computationally
characterize Pb apatite, Pb10(PO4)6O. We obtain lattice parameters $a=10.00$
Å, $c=7.44$ Å, slightly larger than experimental reports of similar systems
[42, 43]. Theoretical works with similar methods show a close[30], slightly
smaller [26], or even larger cells[44]. However, after including Cu atoms, the
differences in lattice parameters among theoretical studies are minimal. The
geometry used is shown in Fig. 1a, with Pb atoms forming hexagonal patterns,
as shown in Fig 1b. This Pb apatite is predicted to be an insulator, as can be
seen from its band structure diagram (see Fig. 1c). A more sophisticated
hybrid XC-functional, HSE06, gives a band gap of $\sim 3.8$ eV, see Section
VII. The conventional lead hydroxyapatite, Pb10(PO4)6(OH)2, has a similar band
structure, but with a larger band gap.

Figure 1: (a) In-plane view of the crystal structure of lead-apatite,
Pb10(PO4)6O. 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 view 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 into Pb atoms (i.e. within its Wigner radius).

##  III LK-99: Hexagonal lattice

###  III.1 Crystal Structure

According to Lee et al. [5, 6], the LK-99 phase has a unit formula
Pb10-xCux(PO4)6O, with $0.9<x<1.1$, with Cu atoms replacing a specific Pb
sublattice, the green atoms in Fig. 1a-b. For simplicity, we set $x=1$, i.e. a
single Cu atom per unit cell. This substitution Pb $\to$ Cu implies an odd
number of electrons per unit cell, suggesting a metal with a spin-split ground
state, or a doubling of the unit cell. Fig. 2 shows two possible arrangements
when doubling the unit cell along the $c$-axis, denoted stacking A and B. In
stacking A (B) the Cu atoms form a triangular (hexagonal) sublattice. The
space group of stacking A (B) is #174 (#143)

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. Panels (a), (b)
show two possible stacking sequences, A and B with Cu atoms forming a
triangular or hexagonal sub-lattice.

Overall six possible arrangements are considered, stacking sequences A and B,
each with a ferromagnetic (FM), antiferromagnetic (AFM), or non-magnetic (NM)
ground state. In the AFM stacking, the in-plane spins are parallel with an
antiparallel out-of-plane order The relative energies, without including any
Hubbard-like $U$ electron-electron repulsion correction, are given in Table 1.
It is evident that a magnetic order is preferred over NM. Also, the stacking A
(triangular Cu sublattice) is slightly more stable than the stacking B. The
energies of all different magnetic orders being almost the same is a first
hint that the Cu atoms do not form extended states. A caveat: the designation
as FM and AFM is questionable due to the absence of an effective exchange
interaction among Cu atoms preventing a long-range magnetic order (i.e. in a
Heisenberg model, the exchange parameters should be practically zero),
however, is relevant to include spin-polarization to correctly describe the
system.

Table 1: Relative energies of different stacking and magnetic orders of Pb9Cux(PO4)6O. The minimum energy configuration is taken as 0. These are the only results with the Hubbard-like term $U=0$ in the article. The units are per formula unit (f.u.), eV/f.u.  Stacking/spin |  | A |  | B
---|---|---|---|---
NM |  | 0.13 |  | 0.17
FM |  | 0.00 |  | 0.02
AFM |  | 0.00 |  | 0.04

We tested the effect of a Hubbard-like term $U=2.0$ eV on the relative energy
of the stacking A for the NM and FM orders, and the results are similar to
those of Table 1, with the NM being 0.18 eV/f.u. higher in energy than the FM
order. Such larger stability of a spin-split state is expected from adding
electron-electron interaction.

###  III.2 Electronic properties

Figure 3: Band structure of Pb9Cu1(PO4)6O with stacking A and a FM ground
state. The color scale represents the projection of wave functions onto the Cu
atom, where a positive (negative) value denotes the spin value.

Provided the larger stability of the stacking A (triangular Cu sublattice, see
Fig. 2), we calculated its band structure. In the NM and FM cases, the unit
cell of Pb9Cu1(PO4)6O suffices to describe the system (i.e. no supercell is
needed). Fig. 3 shows the band structure of the FM order. Two Cu d-orbitals
form almost flat bands at the Fermi level: their width is $\approx 0.1$ eV,
see Fig. 4a. These bands are half-filled, and the system is metallic. These
bands are very similar to other reports, with a $U=4-7$ eV, with and without
spin-orbit interaction [28, 29, 30, 31, 32, 34, 36, 37, 31], see Appendix E
for the effect of changing the value of $U$.

Figure 4: (a) Zoom-in to the bands at the Fermi level. (b-d) Isosurfaces of
the bands at $E_{F}-0.01$ eV ($E_{F}$ is the Fermi energy), $E_{F}$, and
$E_{F}+0.01$. Different bands have different colors, particularly the two
spherical-like surfaces, centered and $\Gamma$ and A, are apparently nested.

The Fermi surface and other isovalues of the bands are shown in Fig. 4b-d.
Here a nesting of the Fermi surface seems likely: two bands have an almost
spherical shape, but one is centered at $\Gamma$ and the other at A. Fermi
nesting is often associated with a charge density wave [45] as suggested by
the original preprint attempting to explain the superconductivity of LK-99
[6]. However, the link between nesting and a charge density wave is the
Peierls distortion or a related mechanism [46]. This mechanism can induce some
other effects, including an actual change in geometry, such as dimerization.
In fact, there are theoretical arguments to expect a reduction in the symmetry
of the system [47]. A deeper discussion, including the possible nesting
vector, is provided in Appendix A.

The electron localization function (ELF) is large in those regions of space
where it is likely to find electron pairs of opposite spin [48]. Hence, ELF is
large in regions associated with covalent chemical bonds, lone pairs of
electrons, and the inner shells of atoms. Contour plots of ELF along planes
containing the Cu atoms, see Fig. 5, suggest that Cu is not strongly
covalently bonded to its neighbor oxygen atoms. Indeed, the shape of the ELF
around Cu resembles the one of atoms confined in wells [49, 50]. This
observation is consistent with the small hybridization d and s orbitals of Cu
with the p orbitals of the oxygen, as shown by the angular momentum
decomposition of the density of states of the Cu atom (see Fig. 15 in Appendix
D). A better understanding of the bonding of Cu may benefit from including
static correlation in the bonding analysis, such as in the hybridization
function [51, 52], but this is beyond the scope of this study.

Interestingly, Belli et al. [53] found an empirical positive correlation
between the critical temperature of high-pressure-hydrogen-based
superconductor and what they called the networking value, $\phi$, of the ELF.
$\phi$ is “the highest value of the ELF that creates an isosurface spanning
through the whole crystal in all three Cartesian directions”. If these
criteria would apply to LK-99, its critical temperature would be less than 50
K (see Fig. 4 in [53]). It is worth noting that the quality of the ELF is only
as good as its underlying DFT calculations, so its interpretation should be
taken with a grain of salt.

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, 2, i.e. the Cu atom is yellow.

###  III.3 Phonons and electron-phonon coupling

Figure 6: Phonon bandstructure of the arrangement A (triangular Cu sublattice,
space group #174). The left panel is the total (red) and partial (Cu
projection, gray) phonon density of states.

At the harmonic level, the phonon band structure of arrangement A (see Fig. 6)
has well-defined groups of bands, including two completely imaginary branches.
These branches indicate that the geometry with group #174 is dynamically
unstable, at least under the conditions of our calculations. In the next
section, Sec. IV, we will explore a related but vibrationally stable
structure. Anharmonic effects were studied elsewhere [25], and we will discuss
them in Sec. V.1

Other features of the phonons include (i) the absence of dispersive phonons,
indicating that the system is constructed from molecular-like blocks, and (ii)
the displacement of Cu atoms is confined to low-energy phonons (up to 30 meV),
falling within the energy width of the Cu bands at the Fermi level. More
energetic modes involve the rearrangement of O atoms surrounding the Cu.

Ignoring the imaginary frequencies, it could be instructive to have a crude
idea about electron-phonon coupling (EPC) of the system. The degenerate flat
bands at the Fermi level imply a potentially large EPC. Its evaluation with
DFT is highly expensive, especially for such a large unit cell, and EPC within
GGA$+U$ is not included in the DFT codes known to us. Nevertheless, we have
adopted a much simpler approach to obtain insights on how large the EPC could
be.

According to the isotropic Migdal-Eliashberg theory [54], the overall EPC is a
weighted integral of the $\alpha^{2}F(\omega)$ function, which adds all the
electron-phonon matrix elements ($|g_{mn\nu}(\mathbf{k},\mathbf{q})|^{2}$)
compatible with momentum and energy conservation at the Fermi level.
Explicitly,

$g_{mn\nu}(\mathbf{k},\mathbf{q})=\langle
u_{m,\mathbf{k+q}}|\Delta_{\mathbf{q\nu}}v^{KS}|u_{n,\mathbf{k}}\rangle,$ (1)

where $\mathbf{k},\mathbf{q}$ refers to electron and phonon momentum,
respectively. $m,n$ are electronic band indexes and $\nu$ is a phonon index.
$u_{n\mathbf{k}}(r)$ is the lattice-periodic part of the wavefunction and
$v^{KS}$ is the Fourier transformed Kohn-Sham potential.
$\Delta_{\mathbf{q}\nu}$ means the (first order) changes of the potential due
to phonon $\mathbf{q}$ of branch $\nu$.

In short, a prerequisite for a large EPC are large matrix elements
$g_{mn\nu}$. This is particularly relevant in the case of LK-99, where only
two electronic bands are relevant. They correspond to the $d_{xz},d_{yz}$
orbitals [28]. These orbitals are nearly degenerate (see Fig. 4), leading us
to anticipate the most substantial effect when a phonon disrupts the
degeneracy between these orbitals. Although the lifting of a degeneracy does
not invariably result in a large EPC, it does occur in certain systems, such
as MgB2. In this system, lattice phonons that split the degeneracies along the
$\Gamma-\mathrm{A}$ line are the most influential [55, 56]. It is important to
note, however, that MgB2 and LK-99 are very different systems [57], and this
example is used solely to highlight the potential impact of lifting a
degeneracy on $g_{nm\nu}$. This mechanism also has been used to explain the
larger critical temperature of YB6 with respect to nearly identical LaB6 [58].

Figure 7: Band structure of LK-99 under the distortion from a frozen phonon.
The color scale shows the projection of the wave function into the Cu atom,
the positive (negative) values denotes the majority (minority) spin. The inset
shows the actual phonon at $\Gamma$ 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].

The effect of such a frozen phonon is dramatic, see Fig. 7. Not only is the
degeneracy lifted along $\Gamma-A$, but one of the d-bands is now buried $\sim
0.5$ eV below $E_{F}$ for every point of the reciprocal space. These changes
notably turn LK-99 into an insulator. We are unaware of any similar effect in
other materials. It is worth commenting on a similar approach involving frozen
phonons, with a larger amplitude, which has been used in MgB2 (see Fig. 3 of
Ref. [55]) to shed light on the EPC due to the $E_{2g}$ phonons, and similarly
to predict a BCS superconductor with large EPC [60]. In both cases, a
splitting of degenerate bands of about $\sim 2$ eV was observed. However, they
continued being metals.

##  IV LK-99: Triclinic lattice

###  IV.1 Crystal Structure and phonons

Figure 8: (a) Unit cell of the triclinic Pb9Cu1(PO4)6O obtained after
relaxation. (b) Lateral view, with emphasis on the O atoms closer to Cu. The
distances shown are in Å, and the other distances are very close to 2.0Å. (c)
Top view removing atoms other than Cu, Pb, and the central O. The colors
follow Fig. 1, Cu is yellow.

By following the distortion associated to one of the imaginary phonon branches
(with zero momentum, $\Gamma$ point) we obtained a crystal structure of
Pb9Cu(PO4)6O stable at the harmonic level, with triclinic lattice and space
group #1, see Fig. 8. It is very close to a hexagonal lattice, see Fig. 2(a) .
Since the atomic rearrangements are centered in the Cu atom, we predict that
(i) following the same phonon with a different momentum, plus an adequate
supercell, should result in different final geometries, (ii) following the
other imaginary phonon branch should yield a similar result to the one
obtained, related by a symmetry operation. In a large system, both types of
distortion could coexist, and studying them in a unit cell is unrealistic. The
triclinic unit cell is 0.55 eV/f.u. lower in energy than the hexagonal. Even
though this value may seem large, it is only 13 meV per atom. We do not study
different magnetic arrangements in this section.

Figure 9: Phonon band structure for the triclinic LK-99-like crystal
structure. The right panel is the total (red) and partial (Cu projection,
gray) phonon density of states.

The phonon band structure of the triclinic system has no imaginary branches,
see Fig. 9. Otherwise, it shows the same features found in the hexagonal
lattice.

###  IV.2 Electronic Structure

Figure 10: Band structure of the LK-99-like system with triclinic lattice. The
color scale shows the projection of the wave function into the Cu atom, the
positive (negative) values denotes the majority (minority) spin. There is a
single band within the fundamental band gap; it is not spin-degenerate.

The triclinic system’s band structure (Fig. 10) resembles that of our frozen
phonon calculation (Fig. 7) and a recent study [61]. This is not surprising
since in the hexagonal lattice, the degeneracy along $\Gamma$-A, plus the
Fermi surface nesting suggest a Jahn-Teller-like distortion to lower the total
energy [47].

Regarding superconductivity, the lowest energy structure of apatite-like
Pb9Cu(PO4)6O is an insulator. However, a small electron doping (e.g. defects)
should turn the system into a metal.

##  V Discussion

We have studied two Cu-doped lead apatite systems, with hexagonal and
triclinic lattice. The hexagonal lattice has promising electronic properties
for superconductivity, but it is not stable against vibrations (at the
harmonic level). The triclinic system has the lowest energy and is
vibrationally stable; however, it is a band insulator. Now, we are in a
position to comment on two relevant questions.

###  V.1 What is the actual crystal structure of LK-99?

Does the LK-99 material assume the hexagonal or the triclinic lattice? We are
convinced the picture is not so simple, even assuming the simplest
stoichiometry, Pb9Cu(PO4)6O. Even at low temperatures of $\sim 60$ K, the
system should fluctuate between the different symmetry-related distortions,
resulting in a larger effective symmetry than the triclinic system. In our
study we neglected anharmonic contributions, but they could be able to
stabilize the hexagonal structure at high-enough temperature. Indeed, during
the revision of this article, a preprint showed that anharmonic effects remove
the imaginary phonon branches of the hexagonal structure at room temperature
[25]. In the Appendix B, we calculated the averaged XRD pattern along a
molecular dynamics simulation, which agrees with a thermal stabilization of
the hexagonal lattice.

On top of the previous discussion, we need to mention that up to now, we are
unaware of any successful replication of the crystal, at least as proposed in
the original preprints. Experimental samples suffer from Cu-clustering [62],
as evidenced by an inhomogeneous magnetism[13].

###  V.2 Is LK-99 a superconductor?

From the previous discussion, a BCS superconducting state is unclear. Unless
the crystal has a perfect hexagonal lattice, the system should be a band
insulator. Let us assume the flat band of Fig. 10 is half-occupied (which it
is not); the EPC should not induce large changes in the electronic structure.
The band is flat (atomic-like) and should remain so, unless another Cu d
orbital gets close in energy, returning to something like the hexagonal
crystal (i.e. nearly degenerate half-filled bands), see Fig. 3. In BCS
superconductors, usually, the EPC splits degenerate bands instead of inducing
degeneracies.

Even if the hexagonal lattice is stable at room temperature [25], its
electronic structure at room temperature could be different from the two-flat
bands picture of Figs. 3, 4. To get some insight into the electronic structure
of this material at room temperature, we carried out a DFT-based molecular
dynamics (MD) simulation of one unit cell. Fig. 11 shows the energy levels as
a function of the time. Since the band(s) of interest is (are) very flat, we
only plotted the values at $\Gamma$. In the figure, the bands are colored
according to their projection into Cu atoms. Along most of the dynamics, only
one Cu-like level exists close to the Fermi level. The next Cu level lies
about 1 eV below the Fermi level and it is hybridized with other atoms (pale
purple stripe). Only at $t\approx 3500$ fs two Cu bands are at the Fermi
level, making it an infrequent event. Despite the finite size errors from our
MD simulations, we are confident that the “two flat-bands picture” does not
hold at room temperature. Then we can discard the system as a high-temperature
superconductor, even if doped to be metallic. Of course, if the experimental
crystal structure differs from the one calculated here (e.g. Cu clustering, Cu
atoms in nonequivalent lattice positions) this analysis would need to be
reassessed.

Figure 11: Energy levels at $\Gamma$ of a unit cell of Pb9Cu(PO4)6O at room
temperature. The color intensity is the projection of each wave function on Cu
atoms.

##  VI Conclusions

The so-called LK-99 material is a Cu-doped lead oxyapatite system. We studied
specifically the system Pb9Cu(PO4)6O, with two possible crystal structures: a
hexagonal and a triclinic lattice.

The hexagonal lattice has attracted much attention, and it features
practically flat, half-filled bands. They are spin-split. The Fermi surfaces
show a nesting around $\Gamma$ and A. These bands are unstable against
vibrations (i.e. imaginary phonon frequencies) at least at the harmonic level.

By following one of the imaginary phonon branches, we found a very similar
triclinic lattice. It is lower in energy and vibrationally stable. However, it
is a band insulator, with a single spin-split unoccupied band within the
fundamental band gap.

Since the energy barrier between both lattices is much smaller than room
temperature, in a crystal the Cu atom should oscillate between the different
phases, giving an overall symmetry larger than triclinic. Along these
oscillations insulating and metallic states could take place. However, having
half-filled almost flat bands is an infrequent event.

A superconductor is unlikely, at least from the BCS picture. If the material
is a superconductor, its crystal structure probably differs from the one being
reported.

##  VII Computational Methods

We employed DFT as implemented in the VASP package [63, 64, 65, 66] using the
projector augmented wave method [67] and Perdew-Burke-Ernzerhof (PBE)
exchange-correlation (XC) functional [68]. For the lead apatite we made some
calculations with the HSE06 hybrid functional [69, 70, 71], but since all
relevant physics is captured by simpler methods we kept PBE across the
manuscript. To ensure the completeness of the basis, we set the energy cutoff
to 520 eV. In the structural optimization, the k-points grid was set to
$3\times 3\times 4$. The smearing used for k-points integration was 10 meV.

For phonon calculations, we used a classical force-field created with machine
learning of a DFT molecular dynamics simulation [72, 73]. We used a supercell
of size $3\times 3\times 3$. The phonopy [74, 75, 76] software was employed
for the post-processing of phonons. The mesh for the phonon density of states
was set to $10\times 10\times 10$. More details about the force field
generation are in the Appendix C.

PyProcar [77, 78] was employed for analyzing the electronic structure. The
IFermi [79] software was used to plot Fermi surfaces. Unless otherwise stated
in the text, the calculations used the GGA$+U$ approach in the rotationally
invariant scheme of Dudarev et. al [80]. We used a value of $U=2.0$ eV for the
d-orbitals of Cu. Electronic band structures without this correction or with a
different value are qualitatively similar, see Appendix E.

To build the crystal structure of LK-99, we used as a starting point the
crystal structure of apatite available on the website of the Materials
project.[81] Then the composition was adjusted to the one of LK-99, and the
cell was fully relaxed. Structures were built with VESTA [82].

## ACKNOWLEDGEMENTS

This research was funded by FONDECYT projects 1220366, 1231487, 1220715 and by
the Center for the Development of Nanosciences and Nanotechnology, CEDENNA AFB
220001. JCE and NFB gratefully acknowledge ANID for their national doctorate’s
scholarship year 2023 number 21231429 and national master’s scholarship year
2022 number 22220676, respectively. CC acknowledges ANID for the grant
ECOS210019. FM is supported by Conicyt PIA/Anillo ACT192023. ANA thanks the
Brown Science Foundation Award. Powered@NLHPC: This research was partially
supported by the supercomputing infrastructure of the NLHPC (ECM-02). RHL
performed calculations on UCLA’s IDRE Hoffman2 cluster and acknowledges
support by the United States National Science Foundation Graduate Research
Fellowship under Grant No. 2034835. The authors want to thank Pablo Díaz and
Jirka Hlinka for the discussions about this subject.

##  Appendix A Fermi Surface and Fermi Nesting

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. Panels (A) and (B)
show different planes. The arrows represent a pair of phonons with momentum
$\mathbf{q}_{1},\mathbf{q}_{2}$ connecting two Fermi surfaces. The length of
the arrow is the same in both panels.

The phonon self-energy can be written as:

$\displaystyle\Pi(\mathbf{q},\omega)=$ (2)
$\displaystyle-2\sum_{\mathbf{k},n,n^{\prime}}\frac{f(\epsilon_{n,\mathbf{k}})-f(\epsilon_{n^{\prime},\mathbf{k+q}})}{\epsilon_{n,\mathbf{k}}-\epsilon_{n^{\prime},\mathbf{k+q}}-\hbar\omega-i\delta}M^{2}_{n\mathbf{k},n^{\prime}\mathbf{k+q}},$
(3)

where the band momentum and indexes are $\mathbf{k},n,n^{\prime}$,
respectively. The occupation of a given state with energy $\epsilon$ is given
by $f(\epsilon)$. The phonon momentum and energy are $\omega$ and
$\mathbf{q}$. $M_{n\mathbf{k},n^{\prime}\mathbf{k^{\prime}}}$ are the
effective matrix elements of the deformation potential[83]. A divergence of
the self-energy may arise at $\omega=0$, if for a given vector $\mathbf{q}$
the bands are degenerate in a large region of the reciprocal space; in other
words, the Fermi surfaces are nested.

Due to the difficulty in calculating $M$, it is often approximated as a
constant, and the imaginary part of the bare electronic susceptibility with
$\omega\to 0$, $\chi^{\prime\prime}_{0}(\mathbf{q})$, is used to quantify the
Fermi nesting[46]:

$\lim_{\omega\to
0}\frac{\chi^{\prime\prime}_{0}(\mathbf{q})}{\omega}=\sum_{n\mathbf{k}}\delta(\epsilon_{n,\mathbf{k}}-\epsilon_{F})\delta(\epsilon_{n^{\prime},\mathbf{k+q}}-\epsilon_{F}),$
(4)

with $\epsilon_{F}$ the Fermi energy. Although the calculation of the nesting
can, in principle, be done with DFT [84, 46], our attempt was impractical for
a system this large that needs a dense grid of k-points to describe its Fermi
surface.

Instead of an actual calculation of $\chi^{\prime\prime}_{0}(\mathbf{q})$, in
Fig. 12, we provide a section of the Fermi surface and the vectors
$\mathbf{q}$ connecting two Fermi surfaces. Both Fermi surfaces have little
dispersion near their crossing of the $\Gamma-A$ line, favoring an overlap of
$\epsilon_{\mathbf{k}}$ and $\epsilon_{\mathbf{k+q}}$.

##  Appendix B XRD

Figure 13: XRD of LK-99 obtained in this work (blue line). For comparison,
Figure 2 of [5] is included (black and red lines).

We ran a DFT-machine-learned classical molecular dynamics simulation of this
system at room temperature, with a $3\times 3\times 3$ supercell ($\sim 1200$
atoms); see Fig. 13. The calculated XRD pattern corresponds to the average
along the last 500 fs of the simulation. In general, this is in good agreement
with the experimental results. However, our peaks are shifted to larger
angles, which is explained by the thermal expansion of the lattice in our
simulation. This picture agrees with a hexagonal lattice stabilized by
temperature[25].

##  Appendix C Machine Learning Force Fields

The training of a force field was performed on-the-fly: if the statistical
uncertainty of the forces is above a certain threshold, the calculation is
made with DFT, otherwise, the force field is used. This allows the use of DFT
only as it is needed (i.e. new configurations outside the validity domain of
the force field). The threshold to switch between DFT or the force field is
defined on the fly to refine the force field accuracy along the calculation.

We started the training with a fixed lattice and low-temperature. Then we
allowed the lattice to relax to improve the force field. Finally, we increased
the temperature to 400 K. Fig. 14a shows the convergence of the Bayesian error
in the forces for this last stage. For the simulation of a larger cell, we
only used the force field; the statistical error in forces lies close to the
training set, Fig. 14b. It is worth noting that this level of error provides a
qualitatively correct picture. For instance, the force field allowed us to
identify the triclinic structure as the minimum energy. A posterior relaxation
with DFT of the triclinic structure resulted in minimal changes.

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 figure corresponds to
the last refinement of the force field. (b) Molecular dynamics of a $3\times
3\times 3$ supercell at room temperature, only using the force field.

##  Appendix D Projected Density of States

We used the VASPKIT code [85] to obtain the projected density of states (PDOS)
on Cu atom in LK-99 with stacking A and an FM ground state from VASP
calculated data. Fig. 15 shows the PDOS of the s, p, and d orbitals of Cu
atom.

Figure 15: Projected density of states on Cu atom in LK-99 with stacking A and
an FM ground state. Labels $p/3$ and $d/5$ indicate that the sum of PDOS is
divided by 3 and 5, respectively.

##  Appendix E Dependency of the value of $U$

To explore the dependency of the calculations with respect to the Hubbard-like
term $U$ we calculated the band structure of the system in the hexagonal
lattice with $U=0-4$ eV, see Fig. 16. The changes close to the Fermi energy
are minimal. However, the participation of Cu in the valence band ($\sim-1$
eV) decreases when increasing the value of $U$.

The phonons of the triclinic lattice, see Fig. 17, are practically identical
regardless of the value of $U$. The geometry was fully optimized for each $U$.
The changes in the phonon density of states are likely due to the mesh size.

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 to 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, grey) phonon density of states is at the left of
each panel, both are in different scales.

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