Chemical bonding in lead oxide phosphate Pb₁₀(PO₄)₆O and in supposed superconductor Pb₉Cu(PO₄)₆O Open Access

Recently, in three articles,^1–3 it has been claimed that the compound Pb_10−xCu_x(PO₄)₆O (0.9 < x < 1.1) is a superconductor at room temperature and atmospheric pressure. The parent compound for the above-mentioned superconductor is lead oxide phosphate,⁴ Pb₁₀(PO₄)₆O. These publications were received with enthusiasm by a wide audience due to their potential applications. On the contrary, experts met them with a degree of skepticism due to the lack of convincing experimental results presented.⁵ As follows from the articles,^1–3 the replacement of one Pb atom located in the 4f position in the crystal structure described by the space group P6₃/m⁶ with a Cu atom leads to the insulator–metal transition and the appearance of superconductivity at room temperature and atmospheric pressure.

Shortly after the publications of Refs. 1–3, a number of reports appeared on the arXiv site cond-mat.supr-con; the authors, in most cases, could not reproduce the results of Korean scientists (see, for example, Ref. 7). The only paper⁸ known to us so far reported the discovery of superconductivity at 110 K in a Pb_10−xCu_x(PO₄)₆O sample. The first theoretical studies using the density functional theory (DFT) found flat bands and high density of electronic states (DOS) on the Fermi surface in the Pb₉Cu(PO₄)₆O compound (see, for example, Refs. 9 and 10), which was regarded by the authors as evidence of the possibility of high-temperature superconductivity (HTSC) in compounds with a similar electronic band structure.

Due to the great attention to the potential perspectives of the Pb_10−xCu_x(PO₄)₆O compound in the field of HTSC, it was interesting to compare the type of chemical bonding in the parent Pb₁₀(PO₄)₆O and copper-doped Pb₉Cu(PO₄)₆O compounds with the type of chemical bonding that had been previously found by us, both in monoclinic α-Bi₂O₃ and orthorhombic La₂CuO₄—the parent compounds for obtaining cuprate HTSCs—and in HTSCs themselves.^11,12

To achieve this goal, we applied the approach that we had used earlier to calculate the electronic band structure and analyze the features of the charge density distribution in bismuth and antimony chalcogenides, iron pnictides, cuprates, and cuprate-based HTSCs,^11–14 as well as in trigonal Se and Te¹⁵ and in molecular crystals of chalcogens.¹⁶ The WIEN2k program^17,18 based on the “augmented plane waves plus local orbitals” method of DFT was used in this approach for calculations of the electronic band structure and for finding the charge density distribution in crystal lattices. The main results of this article were obtained using the Perdew–Burke–Ernzerhof (PBE)¹⁹ exchange-correlation functional, which is a variant of the Generalized Gradient Approximation (GGA). For comparison, the calculations were also performed using the Becke and Johnson exchange potential modified by Tran and Blaha²⁰ (mBJ) and the correlations in the Local Density Approximation (LDA).²¹ 

For valence states, relativistic effects have been taken into account using the second variational method including spin–orbital coupling. The core electron states were calculated fully relativistically.¹⁷ To achieve the required accuracy, the following parameters were used: the expansion in l-orbital momentum inside atomic spheres carried out up to l_max = 10, the largest values of the wave vector k_max used for expansion in plane waves found using the product R_mt · k_max = 5.5, where R_mt is the smallest of all atomic sphere radii in the unit cell. The total number of k-points in the full Brillouin zone was taken to be equal to 800. The convergence criterion of the iterative calculation procedure for the total energy was 10⁻⁴ Ry, and for the charge, the difference between the charge density integrated over the unit cell for two successive iterations of the self-consistency cycle was taken to be equal to 10⁻⁴ e (e is the positive value of the elementary charge).

The charge density distribution ρ(r) in crystal lattices of the Pb₁₀(PO₄)₆O and Pb₉Cu(PO₄)₆O compounds was taken from the results of calculations of the electronic band structure. To study the peculiarities of ρ(r), we used the program CRITIC2,²² developed for the topological analysis of scalar fields in periodic structures based on the “Quantum Theory of Atoms in Molecules and Crystals” (QTAIM) method.^23–25 According to Bader’s QTAIM, the points r_c, at which the charge density gradient is equal to zero ∇ρ(r_c) = 0, are called critical points. The types of critical points in the charge density distribution are characterized by the rank ω [the number of nonzero eigenvalues λ_i—the main values of curvature of the Hesse matrix of the second derivatives of the charge density over coordinates (∂²ρ/∂x_i∂x_j)] and signature ϭ—the algebraic sum of λ_i signs: (ω, ϭ). There are four types of stable critical points:^23–25 (3, −3) – nucleus (local maximum), (3, +3) – cage (local minimum), (3, +1) – ring (the first saddle critical point), and (3, −1) – bond (the second saddle critical point). Critical points of the bond type [bond critical points (BCPs)] play an essential role in the classification of the chemical bonding types. Covalent bonding is characterized by the following Bader’s parameters for BCPs:^23–25 the negative sign of the charge density Laplacian ∇²ρ_b < 0, two negative values of curvature λ_1,2 < 0, which are large in absolute values $λ1,2>λ3$⁠, and large values of the charge density ρ_b. While for the ionic bonding, ∇²ρ_b > 0, $λ1,2≪λ3$⁠, ρ_b is small, and the charge density is concentrated mainly on atoms.

In our paper,¹¹ we have revealed a correlation between the critical temperature T_c of HTSC and the value of Laplacian ∇²ρ_b at the BCP with the highest charge density ρ_b. In addition, it has been shown^11,12 that in the parent compounds of cuprate HTSCs, monoclinic α-Bi₂O₃ and orthorhombic La₂CuO₄, as well as in HTSCs themselves, there is a specific mechanism of chemical bonding. This mechanism is characterized by positive values of the charge density Laplacian ∇²ρ_b for all BCPs and by the number of BCPs per atom, N_at, exceeding the number of valence electrons in atoms. This mechanism can cause charge density fluctuations in crystals. Recently, short-range dynamic charge density fluctuations have been detected in YBa₂Cu₃O_7−δ and Nd_1+xBa_2−xCu₃O_7−δ using Resonance Inelastic X-ray Scattering (RIXS).^26,27 It was noted in Refs. 26 and 27 that these fluctuations have a characteristic energy of several millielectronvolts, arise at temperatures above the temperature T*, at which the pseudogap begins to appear, and are observed in a wide range of hole concentrations p in the electron subsystem on the T-p phase diagram, almost for all cuprate HTSCs.

It is shown below that, despite the differences in chemical compositions and crystal structures, the parameters of BCPs in lead oxide phosphate Pb₁₀(PO₄)₆O and in the doped compound Pb₉Cu(PO₄)₆O are similar to the parameters of BCPs of HTSCs and their parent compounds.

Recently, one of the authors of the article⁴ (Krivovichev) has revised²⁸ the previously published data on the crystal structure of the Pb₁₀(PO₄)₆O compound after x-ray diffraction experiments on a single crystal fabricated by Merker and Wondratschek.²⁹ The main differences between the crystal structure presented in Ref. 28 and the results of the paper⁴ are as follows: the symmetry of the crystal structure of the Pb₁₀(PO₄)₆O compound is described by trigonal space group $P3̄$ (N 147)⁶ with a double value of the lattice parameter c and the changed environment of the oxygen atom O4, which is not part of the PO₄ tetrahedra. These data are essential for the calculations of the electronic bands’ structure, especially for the question of the presence of flat bands near the Fermi surface of Pb₉Cu(PO₄)₆O, but they are not fundamentally important for the main purpose of this work, which is the classification of the type of chemical bonding in Pb₁₀(PO₄)₆O and in the doped compound Pb₉Cu(PO₄)₆O. Due to the fact that studies of the crystal structure of Pb₁₀(PO₄)₆O are still ongoing, we performed calculations with published values of the crystal lattice parameters: a = b = 9.865 Å, c = 7.4306 Å for Pb₁₀(PO₄)₆O,⁴ and a = b = 9.843 Å, c = 7.428 Å for Pb₉Cu(PO₄)₆O^1–3 without optimizing their values. Our studies of the electronic structure of trigonal Se and Te have shown¹⁵ that optimization of crystal lattice parameters does not change the type of chemical bonding in the crystal. For the same reason, we did not optimize the values of the parameters of the positions occupied by atoms in the unit cell of the crystal in order to reduce the forces acting on the atoms. The experimental positions of the atoms and their coordinates in the unit cell of Pb₁₀(PO₄)₆O⁴ (176 space group P6₃/m⁶) are given in Table I.

In Table I, the coordinates of one position of each type are given as fractions of the corresponding lattice parameters. Due to the fact that only one of the 4e positions is occupied by the O4 atom and the need to replace one of the four Pb2 atoms with a Cu atom in the Pb₉Cu(PO₄)₆O compound, it turned out that it is more convenient to carry out the calculations using the specification of the positions of another trigonal space group,^6, P3 (143), allowed for minerals of the apatite supergroup.³⁰ The space group P3 has four types of positions⁶ with coordinates: 1a (0, 0, z), 1b (1/3, 2/3, z), 1c (2/3, 1/3, z), and 3d (x, y, z), (−y, x − y, z), (−x + y, −x, z). Obviously, the 1a position can be used for the O4 atom, the six atoms occupying the 6h positions (see Table I) can be placed in two 3d positions, 12 O2 atoms from the 12i position occupy four 3d positions, and 4 Pb2 atoms in the 4f positions can be represented by the two 1b and the two 1c positions.

To classify the types of chemical bonding in the Pb₁₀(PO₄)₆O and Pb₉Cu(PO₄)₆O crystals, we will use Bader’s parameters of BCPs (the eigenvalues λ_i—main curvature values of the Hesse matrix, sign and magnitude of the charge density Laplacian ∇²ρ_b, and value of the charge density ρ_b). In addition, we will also use the dimensionless parameter flatness f (the ratio of the minimum charge density at the cage-type critical points $ρcmin$ to the maximum charge density at BCPs $ρbmax$⁠), which characterizes the uniformity of the charge density distribution in the crystal. Table II shows BCPs parameters of the charge density distribution in the Pb₁₀(PO₄)₆O and Pb₉Cu(PO₄)₆O compounds, calculated using the CRITIC2 program,²² based on the results of calculations of the electronic band structure performed using the WIEN2k program^17,18 and the exchange-correlation functional PBE.¹⁹ In particular, Table II shows the following data: a compound, bonded atoms, a number of BCPs of each type for non-equivalent atoms (N BCPs), interatomic distances d for the atoms involved in bonding, calculated Bader’s characteristics of BCPs (ratio of main eigenvalues of the Hesse matrix |λ_1,2|/λ₃, sign and magnitude of Laplacian ∇²ρ_b, charge density at BCPs ρ_b), total number of BCPs per atom N_at, and flatness f.

Let us note the most significant features of the characteristics of BCPs in Pb₁₀(PO₄)₆O and Pb₉Cu(PO₄)₆O crystals presented in Table II. First of all, the type of chemical bonding in both crystals is similar to that found by us^11,12 in the parent compounds of cuprate HTSCs, monoclinic α-Bi₂O₃, and orthorhombic La₂CuO₄, as well as in HTSCs themselves. Moreover, the values of the charge density Laplacian ∇²ρ_b and the charge density ρ_b for the first three BCPs in Pb₁₀(PO₄)₆O are significantly larger than in α-Bi₂O₃, La₂CuO₄, and cuprate HTSCs.^11,12 This allows us to consider lead oxide phosphate, Pb₁₀(PO₄)₆O, as a potential parent compound for obtaining superconductors with high values of the critical superconducting transition temperature T_c. Large positive values of the charge density Laplacian ∇²ρ_b indicate that the charge density is pushed out of the regions near BCPs.^23–25 It means that PO₄ tetrahedra create a high pressure in the electronic subsystem of the crystal, having a significant impact on the transport properties of the doped compound. The large numbers of BCPs per atom N_at shown in Table II suggest the presence of charge density fluctuations in the parent and doped compounds. It is also important that the doping of lead oxide phosphate with Cu atoms only slightly changes the values of some BCP parameters without affecting the type of chemical bonding inherent in the parent compound, Pb₁₀(PO₄)₆O.

The total density of electronic states (DOSs) calculated by us for the Pb₁₀(PO₄)₆O and Pb₉Cu(PO₄)₆O compounds using the WIEN2k program^17,18 and the exchange-correlation functional PBE¹⁹ are shown in Figs. 1(a) and 1(b).

As follows from Fig. 1(a), the bandgap for the Pb₁₀(PO₄)₆O compound calculated using the PBE exchange-correlation functional is equal to 1 eV. It is well known that PBE functional usually results in underestimated bandgaps. Our calculations using the mBJ²⁰ exchange potential and LDA²¹ correlations gave a value of 1.8 eV for the bandgap. The doped by Cu Pb₉Cu(PO₄)₆O compound proved to be a metal. The total DOS for the Pb₉Cu(PO₄)₆O compound shown in Fig. 1(b) is similar to that given in the paper.⁹ In our opinion, flat electron bands and very large DOS values on the Fermi surface in Pb₉Cu(PO₄)₆O can be due to the features of the experimental crystal structure^1–3 used in the theoretical calculations. It is quite possible that revised crystallographic data²⁸ of the parent Pb₁₀(PO₄)₆O compound may change the band structure and DOS near the Fermi surface for both compounds. Due to the lack of experimental data on the magnetic state of Cu atoms in the Pb₉Cu(PO₄)₆O compound, the calculations of its electronic band structure were carried out without taking into account spin polarization.

Basing on the results of this article, it is possible to conclude that the Pb₁₀(PO₄)₆O compound is a potentially promising parent compound for producing HTSCs. However, a number of experiments are required to elucidate its physical properties and find similarities with the properties of the parent cuprate HTSC compounds—orthorhombic La₂CuO₄ and monoclinic α-Bi₂O₃. First of all, accurate x-ray and neutron diffraction experiments must be carried out over a wide temperature range to determine the crystal structure of Pb₁₀(PO₄)₆O reliably. The electronic band structure of Pb₁₀(PO₄)₆O should also be studied by measuring the x-ray photoemission spectroscopy (XPS) and ultraviolet photoemission spectroscopy (UPS) spectra, which provide information on valence electrons. Data on unoccupied states in the conduction band can be obtained by experimental studies of optical spectra (electron energy loss function, real part of optical conductivity, and reflection coefficient) that are sensitive to interband transitions. In our recent paper,¹² a good agreement was found between the experimental literature data and the results of our calculations of the electronic band structure of orthorhombic antiferromagnetic La₂CuO₄. RIXS experiments can answer the question about the presence of charge density fluctuations in Pb₁₀(PO₄)₆O. It is also desirable to measure magnetization M(H) and M(T) for Pb₁₀(PO₄)₆O at low temperatures T and in weak external magnetic fields H using a SQUID magnetometer. Previously, in the paper,³² it was found that, despite the absence of magnetic atoms in the composition of α-Bi₂O₃, its magnetization in fields H < 4 kOe at 4.2 K was paramagnetic, anisotropic, and significantly dependent on the magnetic prehistory of the single crystal. Moreover, a linear magnetoelectric effect at 4.2 K was found in Ref. 32—the electric polarization P appeared in α-Bi₂O₃ due to the application of an external magnetic field H. Both the paramagnetism and the magnetoelectric effect in α-Bi₂O₃ were explained in Ref. 32 by the presence of holes in the electronic structure of oxygen atoms, which served as paramagnetic centers. The discovery of the paramagnetic properties of the parent compound Pb₁₀(PO₄)₆O will allow us to understand its electronic structure better.

For the Pb_10−xCu_x(PO₄)₆O compound, first of all, it is desirable to clarify the type of positions occupied by Cu atoms in the crystal structure during doping of the parent compound and to vary the concentration of Cu atoms in order to find the concentration at which the insulator–metal transition occurs. The magnetic properties of Cu atoms in Pb_10−xCu_x(PO₄)₆O are also of great interest. In our recent paper,¹² spin-polarized calculations of the electronic band structure of orthorhombic La₂CuO₄ revealed the presence of an antiferromagnetic ground state with a bandgap E_g = 2 eV in the case of using the exchange potential mBJ²⁰ and LDA correlations.²¹ 

Some other methods of doping the parent compound Pb₁₀(PO₄)₆O can also be used. Apparently, the simplest variant of doping consists of adding oxygen atoms to the 4e positions, which are only a quarter filled. It is well known that the compound La₂CuO_4+δ becomes a superconductor at small values of δ (see Ref. 33 and references therein). Although the valence of lead in Pb₁₀(PO₄)₆O is currently unknown, some of the Pb atoms can be replaced by divalent Sr or Ba atoms.³⁰ 

The lack of experimental data on the electronic band structure and other physical properties of Pb₁₀(PO₄)₆O and Pb_10−xCu_x(PO₄)₆O compounds does not currently allow us to conclude that superconductivity could exist at room temperature and atmospheric pressure in the Pb_10−xCu_x(PO₄)₆O compound. However, if we assume that the correlation between the critical temperature T_c of the HTSC and the value of Laplacian ∇²ρ_b at the BCP with the highest charge density ρ_b we found earlier¹¹ is valid for the Pb₁₀(PO₄)₆O compound, then we can expect T_c of about 150–160 K for the superconductor Pb₉Cu(PO₄)₆O (see Fig. 2).

Although this temperature range is far from room temperature, nevertheless, it is also of great interest, both from scientific and practical points of view. The study of the physical properties of lead oxide phosphate can contribute to the understanding of the HTSC mechanism. Other minerals in the apatite supergroup³⁰ may also be of interest as parent compounds for the production of HTSC.

This work has been carried out using the computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at the NRC “Kurchatov Institute.”³⁴ 

The authors have no conflicts to disclose.

V. G. Orlov: Conceptualization (lead); Writing – original draft (equal); Writing – review & editing (equal). G. S. Sergeev: Conceptualization (supporting); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.