Ab initio Calculations of the Electronic-Energy Structure and Optical Properties of Lanthanum and Neodymium Pyrozirconates

Introduction. Lanthanum and neodymium pyrozirconates — La₂Zr₂O₇ and Nd₂Zr₂O₇ — belong to the pyrochlores group. The general formula of these materials is A₂B₂O₇. A and B are metallic cations that can be trivalent (like La and Nd), tetravalent (like Zr), divalent, and pentavalent [1]. Pyrochlores have high dielectric constant. They exhibit unique magnetic [2], chemical, mechanical and electronic [3] properties. Due to this, they can be used as:

• ceramic coatings of thermal barriers, gas sensors, metal oxide transistors;
• solid electrolytes in fuel cells [4];
• immobilization carriers of actinides in nuclear waste;
• catalysts of oxidation reactions [5];
• elements of magnetic devices.

The research described in this paper was conducted taking into account the development of new technologies in the laser technology, optics and materials science [6]. The results of the work may open the way to the creation of more efficient, reliable and functional devices [7]. The studied oxides of complex chemical composition have significant stability, high melting point, high coefficient of thermal expansion [8], low thermal conductivity, excellent ionic conductivity, and resistance to defects [9]. From the practical perspective, it is important to use pyrochlores Ln₂Zr₂O₇ as coatings to provide thermal insulation of metal components from hot gases [10] in turbines of marine electric generators and in aircraft engines [11].

Numerous papers (e.g., [12]) investigated physical properties of pyrochlores, including mechanical and thermal ones. However, some of their properties are very difficult to evaluate and explain due to their strong dependence on the stoichiometry of the samples [10]. Calculations of the electron-energy structure of various pyrochlores were performed within the framework of the density functional theory (e.g., [13]). These calculations used exchange-correlation potentials obtained in the local density approximation, the generalized gradient approximation, and pseudopotentials.

In [13], the importance of the Hubbard correction in calculating the unoccupied d- and f-states of heavy atoms is noted [14]. Nevertheless, even taking into account the correction, the width of the forbidden band obtained in the calculations often turns out to be smaller than the experimentally observed one [15]. Additional corrections must be taken into account, and this is exactly what the authors of the presented work have done.

Thus, additional experimental and theoretical studies on the electron-energy structure and physical properties of pyrochlores are quite relevant and have practical significance.

Let us consider the crystal structure of pyrochlore La₂Zr₂O₇ (space group Fd-3m, Z = 8) with the general formula Ln₂³⁺Zr₂⁴⁺O1₆O2 (O1 and O2 are oxygen atoms located in different crystallographic positions). It can be described as a structure of defective fluorite, in which the cations form a face-centered cubic (fcc) lattice, and 1/8 of the oxygen atom positions are unoccupied to provide charge neutrality (Fig. 1).

Atoms in the crystal structure of La₂Zr₂O₇ are distributed over four unique crystallographic positions:

• La cations are in Wyckoff positions 16d;
• Zr cations are in positions 16c;
• oxygen O1 is in position 48f;
• oxygen O2 is in position 8b.

The positions of 8а nodes (1/8, 1/8, 1/8) are not occupied at all. Oxygen ions O2 in 8b nodes (3/8, 3/8, 3/8) are stable and tetrahedrally coordinated by the rare earth element cations La.

Oxygen ions О1 in positions 48f (x, 1/8, 1/8) are shifted towards the neighboring empty nodes 8а, and they are surrounded by two La cations and two Zr cations (Fig. 1) [16]. The immediate environment of La cations consists of six oxygen atoms O1 (positions 48f) and two oxygen atoms О2 (positions 8b). La-O2 interatomic distance is shorter than
La-O1 distance. Zr cations (Fig. 1) are surrounded by six О1 atoms (positions 48f), located at equivalent distances in trigonal antiprisms with 3m (D_3d) point symmetry.

The crystal structures of Nd₂Zr₂O₇ and La₂Zr₂O₇ compounds coincide. Table 1 shows the crystal lattice parameters of the studied pyrochlores Ln₂Zr₂O₇ (Ln = La, Nd) with the space group Fd-3m, for which a = b = c, α = β = γ = 90°. For La₂Zr₂O₇, parameter a and the coordinates of oxygen O1 are taken from [15], and for Nd₂Zr₂O₇, they are calculated. The total energy of the crystal with different values of a was calculated, and the optimal value corresponding to the minimum of the total energy was determined. Then, the oxygen atoms were displaced within the unit cell and the position, for which the forces acting on the atoms became minimal, was determined.

Thus, some physical properties of pyrochlores, as well as the structure of compounds Nd₂Zr₂O₇ and La₂Zr₂O₇, have been considered in scientific literature in sufficient detail. However, physical properties of La₂Zr₂O₇ and Nd₂Zr₂O₇ have not been sufficiently studied experimentally. The presented research is intended to fill this gap. The objective of the work is to perform model calculations of the electronic structure and optical properties of La₂Zr₂O₇ and Nd₂Zr₂O₇.

Materials and Methods. Ab initio calculations of the electron-energy structure (EES) of La₂Zr₂O₇ and Nd₂Zr₂O₇ were performed within the framework of the density functional theory. The method of augmented plane waves with the addition of local orbitals APW+lo was used. For the implementation, WIEN2k [17] software package was applied, which used a full potential that did not have a predetermined form, such as muffin-tin potential.

When constructing the attached plane wave, we used the expansion by l inside the atomic sphere up to l_max = 10. The following radii of the atomic spheres were used in the present calculations: R_(La) = 2.24 a.u., R_(Nd) = 2.26 a.u., R_(Zr) = 1.96 a.u., R_(O) = 1.78 a.u. (1 a.u. = 0.529117 Å). The series of the expansion in plane waves was interrupted at the values of the wave vector determined by the relation = 7, where R_min — radius of the minimal atomic sphere. The charge density decomposed into a Fourier series up to value G_max = 12 (a.u)^–1. The densities of electron states were obtained through integrating over 1,000 points in the irreducible Brillouin zone (BZ) using the tetrahedron method [18]. The self-consistency procedure was carried out until the change in the integral charge q = ∫|ρ_n – ρ_n–1|dr became less than q ≤ 0.0001. Here, ρ_n–1(r) and ρ_n(r) are the electron densities obtained at iterations n–1 and n, respectively.

To calculate the exchange-correlation potential, the following were used:

• generalized gradient approximation (GGA) in the parameterization proposed in [19];
• modified Becke-Johnson potential, mBJ [20].

In addition to the above exchange-correlation potentials, the EES calculations took into account the strong Coulomb interaction of f- electrons at one node Nd [21] in PBE+U approximation [22] with U = 5eV. Thus, in the final version, the exchange-correlation potential models were used PBE+U and mBJ+U [23].

Nd₂Zr₂O₇ has an incomplete 4f-shell with four f-electrons, therefore, a spin-polarized EES calculation was performed. For La, Nd and Zr atoms, spin-orbit coupling (SOC) was taken into account. It resulted in splitting:

• 5p-states of La and Nd into 5p_1/2 and 5p_3/2 states;
• 4p⁶-states of Zr into 4p_1/2 and 4p_3/2

Research Results. The paper calculates the total and partial densities of electron states (DOS). Atom La has
no f-electrons, while atom Nd has four f-electrons. Despite this difference, in the first approximation, the calculated total densities of electron states and the experimental X-ray photoelectron spectra of the valence bands of the studied compounds demonstrate a similar structure — four regions reflecting the contributions of s-, p-, d- and f-electrons of different elements [23].

Figure 2 provides comparing the experimental X-ray photoelectron spectrum (XPS) to the calculated total and partial densities DOS for compound La₂Zr₂O₇, and Figures 3–5 — for Nd₂Zr₂O₇. Zero of the energy scale corresponds to the top of the valence band E_V. The spectra were obtained at the Frantsevich Institute for Problems of Materials Science, National Academy of Sciences of Ukraine (Kyiv). The experimental features and equipment are described in [24].

In Figure 2, in compound La₂Zr₂O₇, region 1 is shown— the upper part of the valence band from 0 to 4 eV. This region is formed mainly by 2p-states of oxygen with a small admixture of 4d-states of Zr, 5s-states of Zr, and 6s- and 5d-states of La.

Here, in Nd₂Zr₂O₇, at the top of the valence band, f-states of Nd with spin up are located. X-ray photoelectron spectrum (XPS) confirms the calculation. It is seen that the broad peak closest to the top of the valence band with elements A₁ and А₂ in La₂Zr₂O₇ and A in Nd₂Zr₂O₇ corresponds to 2p-states of oxygen atoms O1 and O2.

In regions 2, 12–18 eV from E_v in XPS, there is also a broad peak with features B₁, B₂ and C for La₂Zr₂O₇ and B for Nd₂Zr₂O₇. The theoretical calculation shows the entire fine structure that forms the peak in XPS. 5p⁶-states of La and Nd are split into 5p_1/2- and 5p_3/2-states. This splitting is already evident in free atoms La and Nd [25].

The spin-orbit splitting in a free atom is also present in a solid (Fig. 2) in the third panel from the bottom for the partial states of La. It is the spin-orbit splitting of the 5p-states of La that leads to the splitting of the 2s-states of oxygen. This is clearly seen in the very bottom panel of Figure 2, where the partial states of oxygen are shown. As Figure 2 shows, in the energy region ~12–18 eV, the deep-lying 5p-states of La interact with the 2s-states of oxygen. This interaction of deep-lying states in a solid is unusual and is related primarily to:

• spin-orbit splitting of the 5p-states of La;
• the fact that the 2s-wave function of oxygen is spatially and energetically strongly stretched.

The presence of structural elements B₁, B₂ and С in the X-ray photoelectron spectrum coincides well with the calculations of the peak in the partial densities of the electron states of La and O.

The third energy region (from 24 to 27 eV) from E_v — peaks D₁ and D₂ in the X-ray photoelectron spectrum of La₂Zr₂O₇, С — in the spectrum of Nd₂Zr₂O₇. This region corresponds to the 4p-states of Zr, which are split in the atom into:

• 4p_1/2-state with energy 35 eV (N₂);
• 4p_3/2-states with energy 33 eV (N₃).

In the XPS curve of La₂Zr₂O₇, the splitting of the 4p-states of Zr is manifested as an asymmetry of the lines with elements D₁ and D₂.

The deepest states of the valence bands of La₂Zr₂O₇ and Nd₂Zr₂O₇ are already semi-core states. The fourth energy region in the XPS spectrum (small peak E in Fig. 2, 3) is the 5s-states of La and Nd. Note that La 5s-state in La₂Zr₂O₇ is not split compared to the 5s-states of Nd with different spin directions (Fig. 3). The splitting of the 5s-states of Nd for spin up and spin down occurs under the action of the internal magnetic field. It is created by four 4f-electrons, which are aligned identically with spin up according to Hund's rule [26]. The broad influx of D in the XPS spectrum (Fig. 3) corresponds to the 5s-states of Nd.

The energy distribution of the electron states in the valence band La₂Zr₂O₇ correlates well with the electronegativity (EN) values of the elements [27] included in this compound (Table 2).

Thus, oxygen has the highest EN (3.44), therefore, it is quite natural that the upper part of the valence band is formed by the 2p-state of O. The admixture of the 4d- and 5s-states of Zr to the 2p-states of oxygen is insignificant, since the Zr-O1 bond is predominantly ionic in nature. The electronegativity of oxygen (EN = 3.44) is significantly higher than that of Zr (EN = 1.33). Due to this, the 4d- and 5s-electron densities of Zr are apparently attracted to the oxygen atom (O1), which is typical for the octahedral environment of the Zr atom by О1 atoms. There are also six О1 atoms in the environment of the La atom. The electronegativity of La (EN = 1.1) is significantly less than EN^S = 3.44. Admixture of 5d- and 6s-states of La is almost not observed. The bond between La and О1 atoms is predominantly ionic in nature, the share of covalence in this bond is very small.

The bottom of the conduction band in both compounds is formed mainly by unoccupied f- and d-states of La/Nd, as well as d-states of Zr (Fig. 2, 4, 5).

A well-known problem of calculations using the exchange-correlation potential in the GGA-approximation is the underestimation of the obtained value of the band gap. For some non-conducting compounds, the calculations even give a conducting state or, as in the case of Nd₂Zr₂O₇ in this calculation, a zero value of the band gap. Taking into account the strong interaction of f- electrons in Nd atom, e.g., within the framework of the LDA+U (or GGA+U) approximation in the EES calculations of Nd₂Zr₂O₇, leads to the appearance of a small forbidden band, but the value close to the experimental one can be obtained either in calculation schemes that consider multi-electron phenomena, or by using hybrid or metapotentials, such as the modified Becke-Johnson potential (mBJ) [28].

Table 3 gives the values of the widths of the forbidden bands E_g. They were calculated taking into account the spin-orbit splitting (SOC) of the electron states in the La and Nd atoms in the GGA–PBE approximations for La₂Zr₂O₇ (GGA–PBE+SOC) and GGA+PBE+U+SOC (with U = 5eV for 4f-states of Nd) for Nd₂Zr₂O₇.

The complex dielectric function ε(ω) = ε₁(ω) + iε₂(ω) — the most important characteristic for calculating the optical response of materials to electromagnetic impact. The dielectric function, in principle, should include both transitions between bands and transitions within a band. Interband transitions are divided into direct and indirect. In these calculations, two factors are ignored. The first is intraband transitions, since they are important for metals, and the compounds under study are semiconductors. The second is the contribution of phonons and other quasiparticles included in indirect interband transitions. Only direct transitions between occupied and unoccupied states are considered. The cubic symmetry of the pyrochlore crystal structure determines only three non-zero (diagonal) elements of the dielectric tensor, and the values of all three of these elements are the same.

The calculated curves of the actual ɛ₁(ω) and imaginary ɛ₂(ω) parts of the permittivity for La₂Zr₂O₇ and Nd₂Zr₂O₇ are shown in Figure 6.

The spectral peaks of the absorbing part of the dielectric function correspond to the allowed dipole transitions between the valence band and the conduction band. To identify the fine structure elements, it is required to compare the values of the optical matrix elements. The observed structures correspond to those transitions that have large values of the optical matrix dipole transition elements. When calculating ɛ₂(ω), only dipole transitions within the atom, i.e., without crossover transitions, were taken into account. The interpretation of peaks A, B, C, D, E, F in Figure 6 for ɛ₂(ω) is given in Table 4.

Figure 6 shows the process when electrons of p-symmetry in the upper part of the valence band (peak А in XPS) pass to states of s- and d-symmetry in the conduction band of the oxygen atom. This is how the highest peak of the curve of the imaginary part of the dielectric function ɛ₂(ω) is formed, covering the range 5–8 eV.

The second and third highest peaks of the curve ɛ₂(ω) — B and C with energies 8.28 and 9.86 eV, respectively. They are caused by transitions of oxygen valence p- electrons to free states of oxygen s-symmetry. Let us consider the wide maximum D on the curve ɛ₂(ω). The “picket-fence” of small additional peaks appeared due to transitions of oxygen s-symmetry electrons to free states of oxygen p-symmetry. In addition, in peak D (energy ̴ 20 eVВ), there is a contribution from the transitions of the valence 5p-electrons of La to the vacant d-states of La in the conduction band. Finally, small peak Е (energy ̴ 30 eV) corresponds to the transition of the valence 4p-electrons of Zr to the unoccupied d-states of Zr. At zero energy, the calculated value ɛ₂(0) for La₂Zr₂O₇ is 8.4334, and for Nd₂Zr₂O₇ — 8.501.

All other optical properties can be derived from ɛ₁(ω) and ɛ₂(ω). These include, for example: absorption coefficient α(ω), refractive index n(ω), extinction coefficient k(ω), optical reflectivity R(ω), and energy loss spectrum L(ω) [28].

The high optical absorption coefficient α(ω) ˃ 10⁵ cm⁻¹ [29] is recorded in three clearly defined regions: from 5 to 14 eV, from 14 to 28 eV and from 28 to 40 eV (Fig. 7). Apparently, such absorption may indicate the potential of using thin-film elements from La₂Zr₂O₇ and Nd₂Zr₂O₇.

In addition, the optical absorption spectrum is characterized by a large number of peaks. They are formed due to transitions between occupied levels of the valence band and free levels of the conduction band, allowed by the selection rules (Δl ≠ 0) and related to one atom (cross-transitions between neighboring atoms are unlikely). Such a picket-fence of small peaks on the curve а(ɷ) is undoubtedly related to the features of the above-described electron-energy structure of La₂Zr₂O₇ and Nd₂Zr₂O₇. Probably, it should be taken into account when using these compounds in optoelectronics.

The complex refractive index N(ω) = n(ω) + ik(ω) can be obtained from the complex dielectric function ɛ(ω), where the refractive index n(ω) depends mainly on the actual part ɛ₁(ω), and the extinction coefficient k(ω) depends on the imaginary part of the dielectric function ɛ₂(ω) [30].

At zero energy, the calculated value n(0) (static refractive index) for La₂Zr₂O₇ is 2.904, and for Nd₂Zr₂O₇ — 2.916.

The extinction coefficient k(ω), responsible for the absorption of the electromagnetic wave incident on the crystal, is obtained from the imaginary part of the dielectric function ɛ₂(ω) (Fig. 6). Therefore, there are also three regions where value k(ω) increases from 5 to 13 eV, from 14 to 28 eV and from 28 to 40 eV. The attenuation of the intensity, represented by the extinction coefficient k(ω), starts with a decrease in the function n(ω) (Figс. 8). La₂Zr₂O₇ and Nd₂Zr₂O₇ crystals can absorb photons in a wide energy range (4–10 eV).

In general, the extinction coefficient of La₂Zr₂O₇ and Nd₂Zr₂O₇ is not isotropic and has three well-defined regions (energies are given above) where isotropic behavior of k(ω) is not observed. In these regions, the difference in k(ω) values is 25–50%.

The reflection coefficient R(ω) is shown in Figure 9. The spectrum R(ω) consists of three clearly defined energy regions. The same feature was noted for other optical characteristics. At zero energy, the calculated value R(0) for La₂Zr₂O₇ is 23.786%, and for Nd₂Zr₂O₇ — 23.935%.

It is evident from Figure 8 that incident photons with energy lower than the energy of the forbidden gap (̴ 4 eV) are not absorbed. Photons with energy from 4 eV to ̴ 40 eV are absorbed by La₂Zr₂O₇, Nd₂Zr₂O₇ crystals and excite electrons in the conduction band, while positively charged holes are formed in the valence band. However, part of the photon energy will be lost during thermalization, and it is reflected in the electron energy loss spectrum L(ω) shown in Figure 10.

The energy loss of the electron becomes significant at 14 eV (small peak), ̴ 27 eV, ̴ 33.5 eV. The largest value L(ω) is reached at 39 eV, which corresponds to a sharply decreasing edge of the absorption coefficient а(ɷ) (Fig. 7).

Discussion and Conclusion. The calculation of the electron density states of the valence bands of La₂Zr₂O₇, Nd₂Zr₂O₇ gave two results:

• allowed us to explain all the main features of the experimental X-ray photoelectron spectra of these compounds in the energy range of ~35 eV from the top of the valence band;
• showed which electron symmetry determined the main features of the experimental spectra.

Together with the obtained densities of unoccupied states, the calculated densities of occupied states allowed us to define the optical coefficients of the compounds studied. It is extremely important to take into account the corrections in the exchange-correlation part of the potential (GGA+U, mBJ). Only with them it is possible to obtain the values of the widths of the forbidden bands that correspond to the experimental data. For Nd₂Zr₂O₇, calculations without corrections generally show a non-zero density of states at the Fermi level.