Highly anisotropic electronic properties of the GdBa₂Ca₂Fe₅O₁₃ oxide: a DFT+U study of a potential air electrode for solid oxide fuel cells

Abstract

Profound knowledge of the electronic structure of functional solids is essential to understand and optimize their properties. The current advances in electronic structure theory, together with the improvements in computing power, allow realization of affordable calculations of the electronic structure of complex solids with the aim of explaining or predicting properties of singular materials. This work presents a density functional theory study of the GdBa₂Ca₂Fe₅O₁₃ oxide, a potential air electrode for solid oxide fuel cells with a layered-perovskite-related structure, which presents ordering of three different coordination-polyhedra around the Fe³⁺ ion (FeO₆ octahedra, FeO₅ squared pyramids and FeO₄ tetrahedra). The existence of these three different Fe³⁺-environments significantly impacts the electronic properties of this oxide leading to a narrow band gap. The band structure calculations of GdBa₂Ca₂Fe₅O₁₃ concludes that the FeO₅ layers create the CB (conduction band), the FeO₆-layers form the VB (valence band) and the FeO₄ layers create insulating channels, leading to anisotropic electrical properties that coincide with the experimentally observed 2D magnetic, electrical, and structural characteristics of GdBa₂Ca₂Fe₅O₁₃.

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1. Introduction

Research on materials oriented towards technological applications is one of the most valuable areas in solid state chemistry and materials science. In this context, metal oxides with perovskite-type crystal structures (ABO₃ stoichiometry) constitute prominent archetypes of functional materials, since they display an extraordinary wide range of electronic properties.¹ The cubic perovskite structure can be described as a corner-linked BO₆-octahedra network with the A-atoms located in 12-coordinated sites formed by four linked octahedra. Transition metal atoms are located at the B-sites while the A-sites are commonly occupied by alkaline-earth metals or lanthanides. The electronic properties of the perovskites are tuned by partial or total substitution of ions into the A and B-sites, sometimes combined with creation of defects such as cation or anion-vacancies. As a consequence, metals, insulators and materials with metal–insulator transitions, also presenting noticeable magnetic properties, are found among different perovskite-related families.^2–5 Perovskite-related oxides have gained a great deal of attention in the field of so-called energy-applications.^6–8 In this regard, numerous perovskites have been studied as electrodes for solid oxide cells, both for fuel cells and electrolyzers (SOFCs and SOECs).^9–14 Particularly interesting are the mixed-conducting non-stoichiometric layered-perovskites, as for instance those of the system LnBaCo₂O_6−δ (Ln = lanthanide metal), with ordering of Ln and Ba in alternated layers along the [001] direction of the structure.^14–19 Layered-type ordering of Ln and Ba in the crystal structure seems to be crucial to the location of the anion vacancies within the LnO_1−δ planes, creating a higher ionic conductivity than in perovskites with the anion vacancies placed at random in the structure.^20,21 The high demand of Co-materials for energy applications and the low thermal stability of the LnBaCo₂O_6−δ systems associated with their rapid oxygen release/up-taking makes it essential to develop new Co-free perovskites. Substitution of Co by other transition metals in the abovementioned layered-perovskites improves thermal stability, although in general decreases the electronic conductivity, while the ordering in both cationic and anionic sublattices can be modified.^22–24 The ordering of the A-site cations is not only assisted by differences between their ionic radii and oxidation states but also depends on the type and oxidation state of the B-site cations, which highly affect the oxygen content of the compound. Thus, layered-ordering of Gd and Ba in GdBaMnFeO_6−δ is only achieved by using a reducing atmosphere in the first step of the synthesis;^25,26 in a similar way, Pr/Ba ordering in PrBaFe₂O_6−δ requires synthesis under reducing conditions.²⁷

Coupling of cation and anion-vacancy orderings to form perovskite-based superstructures is clearly perceived in some complex Fe³⁺-perovskites, where the Fe³⁺ cations can adopt different oxygen coordination.^28,29 Thought provoking is the modulation of the nuclear and magnetic structures in the Ln_0.8−xBa_0.8Ca_0.4+xFe₂O_6−δ systems (Ln = Gd, Tb).^30,31 Three single-phase oxides have been isolated in the system, which are better formulated as Ln_1.2Ba_1.2Ca_0.6Fe₃O₈, Ln_2.2Ba_3.2Ca_2.5Fe₈O₂₁ and LnBa₂Ca₂Fe₅O₁₃ (corresponding to x = 0.00, 0.25 and 0.40 respectively) in agreement with their oxygen content; the compounds constitute the A_3m+5nFe_3m+5nO_8m+13n homologous series with m and n values 1, 0; 1, 1; and 0, 1, respectively. These three complex perovskite-related oxides present different layered-ordering of the A-site cations in combination with different oxygen-coordination environments around the Fe³⁺ atoms (octahedra, tetrahedra and squared pyramids) along the stacking c-axis. In the system, substitution of Ln³⁺ by Ca²⁺ decreases the oxygen content, which is accommodated by formation of FeO₅-squared pyramid layers intercalated with one A-layer containing mainly Ln³⁺ cations. Like in layered Co-perovskites,^20,21 the isolated Ln³⁺ layers might favor the anion conductivity through them, explaining that LnBa₂Ca₂Fe₅O₁₃, with a unit cell with the highest isolated Gd³⁺ layers rate in the series, also presents, in addition to the highest electronic conductivity, the best electrocatalytic properties.³⁰ The antiferromagnetic behavior of these Fe³⁺ containing oxides is also influenced by the ordering and type of oxygen-polyhedra around the Fe atoms, demonstrating, once again, the relationship between crystal and electronic properties.³¹ Interestingly, the variation of their electrical conductivity with temperature, typical for a semiconducting mechanism, shows a smooth transition in a relatively wide range of temperatures (between ∼600 K and ∼900 K). The electrical transition is coupled with a broad magnetic transition, in the same temperature range, associated with the evolution of 3D into 2D-magnetic behavior.^30,31

The studies on these complex layered perovskites are primarily focused on their physical properties in connection with their crystal structure, while the electronic structure is scarcely considered. Relatively recent advances in electronic structure theory, in particular density functional theory (DFT), in parallel with the improvements in computing power, allow explaining a great variety of materials properties and predict new compounds with certain thermodynamically stable crystal structures and behaviors. We present here a thorough computational study on the layered-perovskite GdBa₂Ca₂Fe₅O₁₃ (GBCFO), a potential oxygen electrode material for SOFCs. The work is focused on determining the electronic structure of the oxide to understand its functional properties and, therefore, to establish adequate relationships between its phase composition, crystal, and electronic characteristics. Among the oxides of the A_3m+5nFe_3m+5nO_8m+13n series, GdBa₂Ca₂Fe₅O₁₃ presents the highest electronic conductivity and oxygen reduction reaction activity.³⁰ In addition, the crystal structure of the studied oxide contains three types of oxygen-polyhedra around the Fe³⁺ cations (octahedra, tetrahedra and squared pyramids), which also seem to be associated with its superior properties. The results of the calculations also elucidate the previously observed 2D magnetic nature interplayed with the electrical behavior of the GdBa₂Ca₂Fe₅O₁₃, suggesting new approaches for further comprehensive experimental research in layered perovskites with 2D-behaviors.

2. Computational methods

All first-principles calculations performed in this work were executed in the VASP program package utilizing U-modified density functional theory (DFT+U).³² The wave functions of atomic valence electrons 2s²2p⁴ for O, 3s²3p⁶4s² for Ca, 5s²5p⁶6s² for Ba, 3p⁶3d⁷4s¹ for Fe and 5s²5p⁶6s²5d¹ for Gd were treated with the plane augmented wave (PAW) method; the corresponding states of core electrons and atomic nuclei were evaluated with the program-supplied pseudopotentials. Some calculations implied the use of Gd pseudo-potential that included 4f-states explicitly, i.e. the configuration of the valence electrons for Gd atom was 5s²5p⁶6s²5d¹4f⁷. The exchange–correlation functional used is the one proposed by Perdew–Burke–Ernzerhof (PBE) formulated within the general gradient approximation (GGA). In addition, the more recently formulated MetaGGA SCAN functional was also tested.³³ These functionals tend to give an underestimated band-gap energy while hybrid functionals more accurately deal with electron localization, in particular HSE06 ones, which are preferable for periodic solids.³⁴ However, hybrid functionals are computationally more expensive than semi local methods, and are also heavily dependent on the choice of the mixing parameter.³⁵ A common approach is the Hubbard correction that introduces a U_eff in the calculation of systems with strongly localized electrons of d and f types. The correction term in DFT+U was selected by following the simplified scheme proposed by Dudarev et al.³⁶ It should be stressed that the selection of the U_eff parameter for the 3d electronic states of Fe atoms often depends on the desired accuracy of a particular property simulation.³⁷ In this work, the respective value was fixed at 4.0 eV, which has previously provided correct Fe₂O₃ to Fe₃O₄ reduction enthalpies,³⁸ as well as reasonable electronic band gaps, crystal structure parameters and oxygen ion migration barriers for perovskite-like ferrites containing Fe³⁺ ions in octahedral, square pyramidal and tetrahedral sites.^37,39 More recently, the insulating character of the YSr₂Cu₂FeO₇ compound was well captured when electron correlations were treated by the GGA+U approximation using a U_eff value of 4.0 eV for both Fe and Cu.⁴⁰ Also, the value of U_eff = 4.1 eV was obtained self-consistently for Fe³⁺ cations with octahedral oxygen coordination.⁴¹ For comparative purposes a U_eff value equal to 6.0 eV, calculated for lanthanides,⁴² was also introduced for Gd 4f states in one of the calculations. Due to the presence of ions that possess pronounced magnetic properties (Gd, Fe) in the studied oxide, spin polarization was also considered. The cutoff energy for the plane waves was set to be 520 eV; self-consistency of the solution of Kohn–Sham equations was attained when the total energy difference between two consecutive iteration steps did not exceed 10⁻⁶ eV. The increased precision criterion (10⁻⁸ eV) was used for the calculation of the electronic spectra and related properties. All the crystal structures studied were preliminarily relaxed with respect to forces acting on every ion considered. The lattice was supposed to be converged, when the total energy difference between the two last steps of structural optimization was less than 10⁻³ eV. In addition to this, the energy-volume curve was calculated at 16 different volumes of the GBCFO unit cell; thermal dependence of the respective unit cell volume was assessed using a quasiharmonic Debye model.⁴³

The crystal structure of the GdBa₂Ca₂Fe₅O₁₃ oxide was modelled based on the √2a_p × √2a_p × 10a_p unit cell (a_p refers to the lattice parameter of the cubic perovskite), with orthorhombic symmetry (Imma space group), that contains 4 formula units and 92 atoms in total. In this work, the unit cell and atomic coordinates obtained from the structural refinement of the isostructural TbBa₂Ca₂Fe₅O₁₃ (TBCFO) and Y_0.9Ba_1.7Ca_2.4Fe₅O₁₃ (YBCFO) oxides using neutron diffraction data were taken as a reference.^{30,31,44–47} The graphic representation of the unit cell is depicted in Fig. 1;³⁰ the respective unit cell directions in Fig. 1 are given in terms of the symmetry of a simple cubic perovskite lattice.³⁰ Three different magnetic structures (with respect to Fe³⁺ 3d electrons) were considered for the calculations (Fig. 1): one ferromagnetic (FM) and two antiferromagnetic (AFM) orderings: A-AFM and G-AFM.

Fig. 1 Graphic representation of the RBa₂Ca₂Fe₅O₁₃ (R = Gd, Tb, Y) crystal structure along different projections and the corresponding magnetic orderings used in this work. Oxygen atoms are not depicted for simplicity. Instead, Fe³⁺–oxygen polyhedra are presented; the corresponding color-coding marks the specific crystalline environment of a particular polyhedron (for the magnetic orderings the color represents the spin state of the Fe³⁺ ion located inside; light blue represents spin-down states and light red represents spin-up states). The thick black solid line shows the borders of the unit cell, horizontal dashed lines designate the structural layers under consideration. Experimentally observed defects in the ordering of the A-cation sublattice have not been contemplated in the structure.

To assess the thermodynamic stability of various magnetic configurations of the GdBa₂Ca₂Fe₅O₁₃ oxide, the total electronic energies of the Gd₂O₃, Fe₂O₃, BaO and CaO oxides were calculated. Defect formation energies in GBCFO were computed for the 2a_p × 2a_p × 10a_p supercells by utilizing the approach reported earlier.⁴⁸ Oxygen interstitials were created by introducing an additional oxygen atom in the pristine 184-atomic supercell; in turn, oxygen vacancies were made by removing one of oxygen atoms from it. The scheme of defect allocations used is provided in the SI.

Total (DOS) and partial (pDOS) density of states spectra of the GdBa₂Ca₂Fe₅O₁₃ oxide were computed by utilizing the Gaussian smearing method, since the unit cell parameter c is almost an order of magnitude larger compared to other typical perovskite unit cell dimensions; accordingly, the k-point sampling of the Brillouin zone (BZ) was selected to be N × N × 1, where N is a natural number. During the optimization procedure N was taken to be 4, while the electronic properties were calculated assuming N = 8. The band centers of mass were computed like in BaFeO₃.⁴⁹ Intralayer band gaps were estimated based on the position of O 2p and Fe 3d orbitals of oxygen and Fe atoms located in the different layers. Band dispersion was evaluated for the preliminarily optimized GBCFO unit cell under the constraint of preserving orthorhombic symmetry and G-AFM magnetic ordering; the distances between two adjacent high-symmetry points in BZ were interpolated by 25 k-points. Optical spectra were computed via a common approach reported previously.⁵⁰ The pre- and post-processing of the results was accomplished using vaspKIT software.⁵¹ Additional calculations were made for estimating Fe–O bond strength using the LOBSTER package.⁵²

To study the influence of the lattice arrangement of different FeO_x polyhedra on the electronic properties of the resulting compounds, several alternative Fe³⁺-based crystal structures were investigated using the same GGA+U methodology as described above. Namely, orthorhombic GdFeO₃, tetragonal BaCaFe₂O₅ and hexagonal CaBaFe₄O₈ were utilized to represent the contribution solely from FeO₆ octahedra, FeO₅ square pyramids and FeO₄ tetrahedra, respectively. In addition, combinations of two different polyhedra (i.e. FeO₄ + FeO₅; FeO₄ + FeO₆ and FeO₅ + FeO₆) were assessed with the help of NaCa₂Fe₃O₇, GdBaCaFe₃O₈ and GdBa₂Fe₃O₈ orthorhombic crystal structures, which were based on the earlier reported structural data.⁴⁰ All of the respective calculation details are provided in the SI (Table S1 and Fig. S1).

The electronic conductivity of the GdBa₂Ca₂Fe₅O₁₃ compound was computed with the linearized Boltzmann's transport equation under the constant relaxation time (CRT) approximation using Boltztrap2 software.⁵³ The input data on band structure were obtained from the self-consistent calculation of G-AFM ordered GBCFO unit cell with the 14 × 14 × 2 k-point sampling of its BZ (the partial occupancies of orbitals were determined within the tetrahedron method using Blöchl corrections⁵⁴). The electronic conductivity tensor ([small sigma, Greek, circumflex]_CRT) was estimated using the following equation:

where k_B denotes Boltzmann's constant, ē is the elementary electrical charge, T is the absolute temperature, E is the energy, E_F is the Fermi level and Σ_αβ(E) is the so-called transport distribution function which is determined as a sum of the integrated tensor products of electron energy derivatives over all bands considered, multiplied by the relaxation time τ:with ħ denoting the reduced Planck's constant, E_ik the energy of i-th band at a specific k vector in the reciprocal unit cell and δ_K is the Kronecker's delta function. The summation in eqn (2) is carried out over all N_b bands in the system, while α and β indices designate two distinctive cartesian coordinates of the particular k vector. In this work the σ_CRT was estimated as a function of temperature and the doping level p_d; the latter one was implicitly accounted for in the actual value of the Fermi level inserted into eqn (1).

3. Results and discussion

3.1. Crystal structure and thermodynamic stability of GdBa₂Ca₂Fe₅O₁₃

According to the experimental data, the GBCFO oxide has a highly anisotropic orthorhombic lattice, which can be unambiguously indexed within the √2a_p × √2a_p × 10a_p unit cell with the Imma space group.³⁰ An important feature of the synthesized RBa₂Ca₂Fe₅O₁₃ (R = Gd, Tb, Y) compounds is the presence of anti-site defects in the A-positions, more noticeable in the case of Y and Ca in the YBCFO.^44,46 However, ab initio modeling of such defect compounds implies a complicated computational approach; accordingly, certain approximations allow achieving reasonable time-to-result ratios. In this work, the A-site disordering in GdBa₂Ca₂Fe₅O₁₃ has been modeled by mixing Gd and Ca atoms only in their respective layers. Considering this disordering, three different 92-atomic unit cells were generated with Gd atoms substituting Ca ones and vice versa; the corresponding graphic representations can be found in the SI (Fig. S2). Accordingly, to assess the relative influence of A-site mixing on thermodynamic stability of the j-th structural model of the GdBa₂Ca₂Fe₅O₁₃ compound, the ΔH_f(j) enthalpy change (i.e. the formation enthalpy) of the following chemical reaction has been computed:The term j in brackets underlines the particular structural model of the GBCFO used for the calculations. In turn, the ΔH_f(j) quantity can be expressed as:where E^tot_l denotes the total electronic energy of l-th component in reaction (3), p stands for pressure and ΔV_j is the overall molar volume change in the j-th reaction (3). One should note now that the results of using formula (4) may strongly depend on the type of magnetic ordering of the GBCFO (all other oxides involved in reaction (3) are assumed to be in their most stable magnetic states at 0 K). Therefore, the calculated formation enthalpies in the framework of the GGA+U method are given in Table 1 for both structural disorder and magnetic ordering models in GBCFO. As seen, all the estimated ΔH_f(j) parameters are strongly negative, which indicate the thermodynamic preference of the perovskite-like structure over the mixture of binary oxides. Also, one can acknowledge that the anti-site defects in the A-sublattice of GBCFO make little difference (less than 100 meV) for the resulting ΔH_f(j) value; it is probable that the small energy gain observed for perfectly ordered GBCFO is counterbalanced by a higher entropy of mixing, which eventually enables the stabilization of some Ca atoms within the Gd layer in the GBCFO material. On the contrary, a much more pronounced energy difference is observed for different magnetic orderings. For instance, the calculations revealed that FM and A-AFM structures result in higher ΔH_f(j) values than the G-AFM ordering; the corresponding subtraction gives 1.26 and 0.78 eV per f.u. differences between G-AFM/FM and G-AFM/A-AFM configurations of perfectly ordered GBCFO, respectively. Similar results are achieved for the unit cells with the Gd/Ca mixing (models I, II and III in Table 1). One should note that these outcomes appear to be in general agreement with the Goodenough–Kanamori rules⁵⁵ and experimental observations.³¹ Accordingly, the G-AFM magnetic structure is proved to be the most stable in the GdBa₂Ca₂Fe₅O₁₃ crystalline framework, disregarding the degree of Gd and Ca mixing; hence, further analysis will be performed only for this type of spin ordering.

Table 1 The GGA+U computed formation enthalpies of the GdBa₂Ca₂Fe₅O₁₃ oxide with different magnetic orderings and A-site mixing models

Magnetic ordering	Formation enthalpy, eV
	Non-mixed	Model I	Model II	Model III
G-AFM	−1.597	−1.510	−1.552	−1.520
A-AFM	−0.820	−0.729	−0.776	−0.745
FM	−0.339	−0.254	−0.291	−0.268

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Experimental data are a valuable tool for validating the results of U_eff corrected GGA (GGA+U) calculations. Indeed, the theoretical unit cell parameters of GdBa₂Ca₂Fe₅O₁₃ obtained during structural relaxation (within the G-AFM magnetic ordering) compared with those experimentally determined by X-ray diffraction in the synthesized oxide²³ are in reasonable agreement (Table 2). The very close matching of the GBCFO unit cell volumes estimated by GGA+U and Rietveld methods is especially remarkable; the respective relative difference is less than 0.01%. The computed energy-volume curve of the G-AFM ordered GBCFO additionally supports the computing results – the equilibrium volume at 0 K determined by fitting the respective points with the Birch–Murnaghan equation of state⁵⁶ is extremely close to that obtained by simple structural relaxation (see Fig. S3(a) in the SI). In addition, the extracted temperature dependence of GBCFO unit cell volume appears to be in good agreement with the experimental observations, Fig. S3(b);⁵⁷ moreover, the obtained Debye temperature (∼430 K at room temperature) coincides well with the known data for similar oxide systems.⁵⁸ In contrast, the use of the SCAN functional in conjunction with the U_eff term (SCAN+U) brings worse results – the difference between experimental and computed unit cell volumes of GBCFO is estimated to be ∼5%. Therefore, one may assume that the GGA+U approximation is more precise in terms of estimating the unit cell dimensions. Hence, GGA+U approximation was used in this study.

Table 2 Experimentally determined and DFT+U computed unit cell parameters of GdBa₂Ca₂Fe₅O₁₃. Experimental data were taken from elsewhere³⁰

Parameters	Method
	Experiment	GGA+U	SCAN+U
a, Å	5.5222	5.53056	5.49541
b, Å	5.5625	5.58393	5.44159
c, Å	38.309	38.106	37.375
α, deg	90	90	90
β, deg	90	89.991	90
γ, deg	90	90	90.006
V, Å^3	1176.75	1176.79	1117.66

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Despite the perfect coincidence between GGA+U computed and the experimental unit cell volume of the GBCFO, the respective values of individual unit cell parameters differ significantly although the inconsistencies lie within the typical range of ab initio errors.⁵⁹ In addition, the structural relaxation of the GBCFO lattice has led to symmetry breaking, thus transforming the initially orthorhombic unit cell with the Imma space group into the monoclinic P2₁/n one. The application of SCAN+U approximation has also resulted in the symmetry lowering of the GBCFO lattice, which suggests insufficient precision of structural models used in the calculations (i.e. one does not account for antisite-defects). However, the respective angle deviations from 90° were found to be less than 0.01 in both cases, which implies minor distortion of the GBCFO crystallographic planes.

Although estimation of unit cell dimensions is an important probing for the computational approach, it does not give information on the chemical bonds in the solid. Therefore, it is desirable to compare the calculated bond lengths with those derived from the Rietveld refinement of the structure from the corresponding diffraction data. Since these data are not available in the case of the GdBa₂Ca₂Fe₅O₁₃ in the literature, experimental bond lengths of the isostructural compounds Y_1.1Ba_1.6Ca_2.3Fe₅O₁₃ and TbBa₂Ca₂Fe₅O₁₃ will be used in this work.^31,45 The respective bond lengths and angle distributions calculated using the GGA+U method for GBCFO are given in Table 3 (the atomic coordinates obtained can be found in Table S2 in the SI). The overall agreement between the experimental bonding distances and angles, and the DFT+U computed values for the GBCFO is satisfactory. The most noticeable difference in the two datasets presented is linked with the geometry of FeO₄ tetrahedra: ab initio calculations predict more symmetric distribution of Fe–O bond lengths, both in the apical and equatorial positions of the oxygen-atoms. Contrary to that, experimental data of YBCFO and TBCFO show different equatorial Fe–O distances. Still, the degree of mismatch does not exceed 1% in most cases, thus suggesting an appropriate level of theoretical approximations used. In addition, it should be mentioned that the derived bulk modulus, B₀, of the GBCFO (Fig. S3(a)) agrees with the one measured for Fe³⁺ containing ferrites that simultaneously contain FeO₆ octahedra and FeO₄ tetrahedra in the structure with ∼1.8 Å and ∼2.0 Å Fe–O bond lengths respectively.⁶⁰ When only FeO₆ octahedra are present, the B₀ magnitude is usually higher.⁶¹ Finally, it must be stated that the in-plane Fe–O–Fe bond angles for the different oxygen-polyhedra around the Fe³⁺-cations in RBCFO are significantly different, which is well reproduced using the GGA+U method. This observation suggests that the degree of overlap between Fe-orbitals and O-orbitals differs depending on the coordination number of the Fe³⁺ cations. Hence, individual contributions from the respective 3d electronic states to DOS might vary depending on the type of Fe³⁺–O polyhedron, as has already been reported for the YBCFO.⁴⁵

Table 3 Computed interatomic distances and in-plane bond angles for G-AFM GBCFO in comparison with the corresponding experimental data determined for the oxides containing another element R (Y_1.1Ba_1.6Ca_2.3Fe₅O₁₃ and TbBa₂Ca₂Fe₅O₁₃).^31,45 GGA+U derived bond lengths in different Fe³⁺–O polyhedra were preliminarily averaged

Bond distances (Å) and angles (deg.)	Experimental		GGA+U
	R = Y	R = Tb	R = Gd
FeO6
d Fe–O apical	2.1089	2.1332	2.1187
	2.0824	2.032	2.1187
d Fe–O equatorial	1.9462	1.9559	1.9640
	1.9462	1.9559	1.9640
	1.9684	1.9667	1.9870
	1.9684	1.9667	1.9870
FeO5
d Fe–O apical	1.8718	1.9234	1.8610
d Fe–O equatorial	2.0127	1.9958	2.0201
	2.0127	1.9958	2.0201
	2.0031	1.9915	2.0170
	2.0031	1.9915	2.0170
FeO4
d Fe–O apical	1.8686	1.7532	1.8410
	1.8686	1.7532	1.8410
d Fe–O equatorial	1.8556	1.8572	1.9230
	1.9425	2.249	1.9230
d Fe–Fe (R layer), Å	3.7691	3.724	3.7692
ΘFe–O–Fe (FeO6)	168.94	166.77	171.66
ΘFe–O–Fe (FeO5)	149.34	155.58	151.41
ΘFe–O–Fe (FeO4)	125.06	117.88	123.96

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3.2. Electronic band structure & chemical bonding of GdBa₂Ca₂Fe₅O₁₃

Prior to discussing the electronic structure of the studied compound, it is instructive to note that Gd is a 4f-element. Therefore, the presence of 4f-electrons might render certain influence on the resulting electron density. However, it is often assumed that in the vicinity of the Fermi level, the electronic states are essentially independent of the respective 4f-states in the rare-earth containing crystals.⁶² This is also known as the frozen-core approximation.⁶³ With the aim to examine whether the Gd 4f electrons can indeed influence the DOS of the GBCFO, a separate calculation was made involving the previously studied GdFeO₃ oxide as a test compound. The actual calculations made for the G-AFM GdFeO₃ with and without frozen-core approximation have shown that the obtained E_g values are similar, corresponding to 2.421 and 2.412 eV respectively, see Fig. S4 in the SI. It should be noted that both results fall into the experimentally determined GdFeO₃ band gap range of 2.1–2.5 eV;⁶⁴ moreover, they are almost indistinguishable from each other. In addition, the computed distribution of Fe and O electronic states within the bands for both approaches used appears to be the same in the vicinity of the Fermi level (Fig. S4). The obtained result is reasonable, as the corresponding Gd 4f states were found to lie ∼5.5 eV beneath the top of the valence band (VB), (Fig. S4); accordingly, their influence on the Fe and/or O states should be insignificant. The reported numerical estimations of magnetic interactions fully support this conclusion.⁶⁴ Hence, the frozen core approximation can be accurately utilized for treating the electronic structure of the GBCFO oxide.

The computed DOS spectrum under the GGA+U approximation of the GBCFO oxide is presented in Fig. 2(a). The results indicate that the studied compound is a narrow-gap semiconductor with a band gap value of E_g = 0.53 eV. The corresponding estimation of partial contributions to DOS reveal typical particularities for Fe³⁺ containing perovskites – the severely spin-split Fe 3d band, part of which is forming the bottom of the conduction band (CB) and a wide O 2p band providing the top of the VB.^64–67 One can also acknowledge the overlying Gd 4d states (∼4 eV above the bottom of the CB) which are too high in energy to participate in the electron conduction process. Accordingly, the electronic structure of the GBCFO, in general, represents the common picture of localized electrons in a transition metal oxide.⁴⁹

Fig. 2 The GGA+U computed electronic spectra of GdBa₂Ca₂Fe₅O₁₃. (a) The total DOS (dotted line) and pDOS (filled areas/solid lines) curves evaluated for all Gd, Fe and O atoms in GBCFO (filled areas show filled states, solid lines above the Fermi level designate empty states). (b) pDOS of the Fe-sublattice in GBFCO (filled grey area/black solid line) and respective partial contributions from Fe³⁺ cations located in different oxygen polyhedra (filled colored areas); lighter colors correspond to empty states. (c) pDOS of the O-sublattice (filled grey area/black solid line) and respective partial contributions from oxygen ions located in layers belonging to different oxygen polyhedra around the Fe³⁺ ions (filled colored areas); non-colored areas correspond to empty states. Negative values designate spin down states. The vertical dashed line shows the position of the Fermi level. The type of polyhedron associated with a certain pDOS, is also marked on the figure.

However, the implication of the layered crystalline architecture causes some peculiarities to appear in the DOS spectrum. Namely, the distribution of electronic states for Fe and O atoms was shown to depend on the Fe³⁺-environment in the unit cell, in particular on the exact type of oxygen-polyhedron around the Fe³⁺ cations. The mentioned data are shown in more detail in Fig. 2(b) and (c) for Fe 3d and O 2p states, respectively. Considering the information presented in Fig. 2(b) one can notice the wide low-intensity Fe band ranging from −6 eV up to the Fermi level; these are the Fe 3d–O 2p hybridized states. At the same time, peak-shaped Fe 3d states located both in the VB and in the CB are split with the lower energy peak corresponding to pyramidally coordinated Fe atoms. In turn, higher energy states are provided by a combination of electrons bound to the Fe atoms located in the FeO₄ tetrahedra and FeO₆ octahedra. Given the large splitting energy (>1 eV), it is concluded that the FeO₅ layers are mainly responsible for the formation of the CB in GBCFO. On the other hand, the pDOS of the O 2p states in equatorial oxygen positions of the different polyhedron (i.e. FeO₄ tetrahedron, FeO₅ square pyramid or FeO₆ octahedron) shows that the top of the VB is also spatially specific (Fig. 2(c)). Namely, only those states that belong to equatorial oxygen atoms located in FeO₆ octahedra do contribute to the top of the VB. The states of oxygen ions in FeO₄ and FeO₅ polyhedra appear to be ∼1 eV lower in energy and hence their influence on electrical transport properties can be neglected. Accordingly, GBCFO may be considered as a 2D-semiconductor with FeO₅-layers acting as the CB, FeO₆-layers as the VB and FeO₄ ones as effectively insulating channels. This result seems to be controversial, as a previous report in the isostructural YBCFO suggested the top of VB is in the FeO₄-layer.⁴⁵ Still, the use of other functionals (i.e. SCAN) in this work gives similar distribution of the electronic states.

It is important to notice that the computed band gap value (0.53 eV) for the GBCFO is unusually small in comparison with the other Fe³⁺-perovskites.^60,68,69 Considering the relationship between electric bandgap and optical bandgap, we have calculated the optical spectrum of the G-AFM GBCFO, see Fig. S5 in the SI. The color of the solid can be estimated theoretically from its reflectance spectrum.⁷⁰ The one of the GBCFO (Fig. S5) shows that its reflectance is approximately constant in the visible range; the respectively estimated CIE-color⁷¹ is dark brown in agreement with the experimental color of the studied oxide,³⁰ although the determined band gap suggests the material should be totally black.⁷⁰ This counterintuitive result originates from the peculiar band dispersion in GBCFO – the calculated energy band diagram (Fig. S6 in the SI) indicates that the band gap in the studied compound is indirect in nature and hence the relative contribution of the electronic transition to the overall light absorption should be small. In contrast, the strongest impact on optical absorption is provided by direct transitions, which are higher in energy (above 2 eV). Consequently, the intense increase of optical absorbance in GBCFO is observed only at UV wavelengths, which explains the non-black color of the material considered.

Although the DOS spectra provide important information about the electronic structure, they do not allow to estimate the spatial localization of bands, which is of special importance for semiconductors. Fig. 3 displays the calculated electron charge densities in the CB and VB projected onto specific crystallographic planes of the GBCFO structure. The spatial distribution of charge within the CB and VB is significantly different. The charge distribution in the CB is revealed to be essentially localized on Fe atoms located in the FeO₅ square pyramids, while in the VB the charge is more delocalized with comparable contributions of the Fe atoms and equatorial oxygens of the FeO₆ octahedra. These findings fully coincide with the conclusions derived from the DOS/pDOS spectra (Fig. 2). Importantly, both bands show pronounced anisotropy of the charge distribution – they appear to be concentrated mainly in the quasi-2D layers perpendicular to the c-axis ([001]_p direction), see Fig. S7 in the SI, in agreement with the evaluated band dispersions (Fig. S6). The analysis of different projections of charge density in the CB, Fig. 3(b) and (c), indicates the conducting states are mainly formed by d_3z²−r² Fe orbitals. The VB shows the Fe 3d–O 2p hybridized behavior of the d_x²−y² Fe states, Fig. 3(d) and (e). It should be stated that the derived picture of charge distribution in conjunction with the observed band curvatures (Fig. S6) suggests that the electronic transport in the CB should be more localized than in the VB in agreement with earlier studies in other Fe³⁺-perovskites.^37,72,73

Fig. 3 Graphical representation of the crystal structure of GdBa₂Ca₂Fe₅O₁₃ (a); the corresponding representation omits the A-site cations and oxygen anions for simplicity – only oxygen polyhedra around Fe³⁺ cations are shown. The GGA+U computed charge densities of electrons in the conduction, (b) and (c), and valence, (d) and (e), bands of GBCFO are shown, projected onto specific crystallographic planes. The respective planes are schematically shown in panel (a) as rectangles that cross-section the unit cell of GBCFO. Panel (b) corresponds to the charge density projected onto plane 1, panel (c) to that projected onto plane 3, and panel (d) to that projected onto plane 2. Panel (e) represents the electron density projected onto plane 3. Panels (c) and (e) show the charge density projections along the [001]_p direction; the respective span range is marked on plane 3 by white solid lines (a). The presented charge density maps (b, c, d and e) also contain the positions of Fe atoms resting in FeO₆ octahedra (shown as grey spheres) and in FeO₅ pyramids (shown as black spheres). Oxygen sites are also marked as bright yellow spheres along with the respective Fe–O bonds (dashed white lines).

The unique coexistence of the three different types of Fe³⁺-environment in the crystal structure makes the RBCFO (R = Y, Gd, Tb) oxides an interesting playground for testing the influence of the oxygen coordination of Fe³⁺ cations on their complete electronic behavior. In this sense, the NPD experiments on TBCFO revealed different magnetic moments localized on Fe³⁺ ions (μ_Fe) depending on their oxygen coordination.³¹ The calculated data for GBCFO and the experimental values determined for TBCFO are compared in Table 4. Regarding the individual magnetic moments, GGA+U calculations give correct (within the 6% error limit) values for Fe atoms located in the FeO₅ square pyramids and FeO₆ octahedra but fail to predict the significant decrease of the μ_Fe in FeO₄ tetrahedra. It is important to notice that a similar issue was detected for the Ca₂Fe₂O₅ brownmillerite where the experimentally observed difference between the μ_Fe in FeO₄ and FeO₆ polyhedra⁷⁴ was not properly reproduced by DFT+U calculations.⁶⁶ Nevertheless, the detailed analysis of the pDOS spectra of GBCFO has shown that the electronic states in FeO₄ layers do not contribute to VB and CB edges and hence the improper μ_Fe values cannot be considered as substantial incorrectness of the calculation method.

Table 4 Calculated electronic parameters of the GBCFO oxide. U_dd is the magnitude of splitting of the Fe 3d states, Δ_CT is the charge transfer energy, and Δ_ox is the average amount of energy required to transfer electron from the O-2p band to the top of the VB. The presented data are associated to the oxygen environment of the Fe atoms. The numbers in brackets in the top row represent the experimental values obtained for TBCFO²⁴

Parameters (eV)	Crystallographic environment
	Tot	FeO6	FeO5	FeO4
a The value is estimated from the temperature dependence of the GBCFO conductivity.^23 b The band gaps are obtained for the compounds with only one type of FeOx polyhedron.
|μFe|, μB	0.0 (0.0)	4.09 (3.9)	4.04 (3.8)	4.05 (3.0)
E g	0.53 (0.38^a)	1.90 (2.4^b)	1.67 (1.35^b)	2.79 (3.02^b)
U dd	+9.05	+9.16	+8.7	+9.05
Δ CT	−4.08	−3.83	−3.73	−3.63
Δ ox	+2.93	+2.93	+3.41	+3.23

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As follows from the data presented in Table 4, the partial E_g values are significantly higher (if compared to the total E_g) for each polyhedron type but E_g is especially large in the case of the FeO₄ tetrahedra (∼2.8 eV), assuming these sites form insulating layers in the GBCFO lattice. Notably, a very close band gap is obtained for CaBaFe₄O₈ that contains entirely FeO₄ structural units in its lattice, see Fig. S8(a); this result is also supported by the respective optical measurements (Fig. S8(b)). In addition, the computed E_g values for GdFeO₃ and BaCaFe₂O₅ oxides (Fig. S9 in the SI) that contain only FeO₆ and FeO₅ structural units respectively, were found to be in qualitative coincidence with the data obtained for individual FeO₆ and FeO₅ layers in GBCFO, see Table 4. Considering all the above results, one can suppose that the combination of FeO₄, FeO₅ and FeO₆ polyhedra in one lattice provides a synergetic effect on the resulting electronic band structure.

To support this conclusion, it is important to study the implications in the band structure due to the elimination of one of the types of the FeO_x polyhedra. In this context, another compound of the A_3m+5nFe_3m+5nO_8m+13n series was considered, the Gd_1.2Ba_1.2Ca_0.6Fe₃O₈, which contains only FeO₄ tetrahedra and FeO₆ octahedra.³⁰ The calculation was made for the perfectly ordered GdBaCaFe₃O₈. The GGA+U computed DOS/pDOS spectra for the GdBaCaFe₃O₈ oxide are presented in Fig. 4(a) (the corresponding representation of the unit cell is given in the inset³⁰). In this case, the Fe-3d levels in the CB and the VB show similar positioning on the energy scale, disregarding the actual FeO_x polyhedron type, which differs from the spectrum of the GdBa₂Ca₂Fe₅O₁₃. In addition, the band gap of GdBaCaFe₃O₈ is estimated to be ∼2.2 eV, thus falling in the typical range of E_g values of Fe³⁺-perovskites.^68,69

Fig. 4 DOS spectra (dashed lines) of GdBaCaFe₃O₈ (a), GdBa₂Fe₃O₈ (b) and NaCa₂Fe₃O₇ (c) oxides calculated using the GGA+U method. Colored areas depict the pDOS of Fe 3d states computed for Fe atoms located in octahedra (red color), square pyramids (blue color) and tetrahedra (green color). The spin-down states are not shown as they are fully symmetrical to the spin-up ones. The vertical dashed lines show the position of the Fermi level. The insets depict graphic representations of the crystal structures of the respective compounds. The hatched rectangles show the width of the estimated band gap; the respective numerical values are also provided for each panel.

Another possibility is the combination of either FeO₅ square pyramids and FeO₆ polyhedra, or FeO₅ pyramids and FeO₄ tetrahedra. These combinations containing only Fe³⁺ ions can be modeled by GdBa₂Fe₃O₈ and NaCa₂Fe₃O₇ structures, respectively. Although such compounds have not been reported experimentally, the corresponding unit cells were built based on the known structures of YSr₂Cu₂FeO₈ and YSr₂Cu₂FeO₇,⁴⁰ which resemble in a certain sense the GBCFO lattice. The computed DOS/pDOS spectra of GdBa₂Fe₃O₈ and NaCa₂Fe₃O₇ are presented in Fig. 4(b) and (c), respectively. As shown, the copresence of FeO₆ and FeO₅ polyhedra results in the lowest band gap of ∼1.1 eV; importantly, this result corroborates with the noticeable splitting of Fe 3d states in the CB of GdBa₂Fe₃O₈ among the respective polyhedra, Fig. 4(b). In turn, the combination of FeO₄ tetrahedra and FeO₅ pyramids provides a band gap of ∼1.6 eV, Fig. 4(c), which is in between of E_g values obtained for GdBa₂Fe₃O₈ and GdBaCaFe₃O₈. Interestingly, bringing two different types of FeO_x polyhedra in one lattice allows for certain E_g reduction, if compared to the case of crystals that have only octahedra (E_g = 2.41 eV), only square pyramids (E_g = 1.35 eV) or only tetrahedra (E_g = 3.02 eV). Still, none of the double combinations provide as much band gap contraction as is observed in the GBCFO case. Therefore, it seems that the co-presence of all three FeO_x structural units discussed is indeed required to obtain a narrow band gap in Fe³⁺ containing perovskite-like compounds. The schematical depiction of the observed peculiarity is shown in Fig. 5.

Fig. 5 Graphic representation of the influence of different combinations of FeO_x polyhedra on the band gap of perovskite-related lattices.

The obtained DOS of GBCFO (Fig. 2) allows us to elucidate other important band structure parameters, also collected in Table 4: U_dd, the magnitude of splitting of the Fe 3d states; Δ_CT, the so-called charge transfer energy; and Δ_ox, the average amount of energy required to transfer electrons from the O-2p band to the top of the VB.⁴⁹ The magnitude of these parameters is schematically shown in Fig. 2(a). Determination of the respective band centers, which can be done using multiple ways, must be carried out to compute U_dd, Δ_CT and Δ_ox. In this work, it was assumed that the band center of the Fe-related states should not include those located in a wide Fe 3d–O 2p hybridized band; in turn, a similar approach was utilized for the O-2p states that contribute to Fe-3d peaks placed deep in the VB and CB. The calculated values suggest that the studied compound is a negative charge transfer oxide, which is characteristic of Fe³⁺-perovskite-type oxides.⁴⁹ The high magnitude of U_dd energy (>9 eV) provides strong evidence that under oxidation of GBCFO, the increased charge will be accumulated on oxygen sites.⁴⁷ In turn, the large Δ_ox parameter (∼3 eV) implies high energy required for oxygen vacancy formation in GBCFO, which is indeed experimentally observed.^30,44 Therefore, one can acknowledge the sufficient precision of GGA+U approximation utilized in this work, capable of reproducing the key experimental findings reported previously.

Considering the essential differences in the computed pDOS spectra between the alternating layers with different Fe-environments in GBCFO, individual estimation of U_dd, Δ_CT and Δ_ox for these layers makes certain sense. The respective data for FeO₄-, FeO₅- and FeO₆-containing layers are given in Table 4 (importantly, the discussed estimation of band structure parameters did not involve apical oxygen sites but only equatorial ones). The charge transfer energy remains essentially the same disregarding the polyhedron type, which means that the electrons transferred from any Fe site in GBCFO will be immediately replaced (with approximately the same speed) by other electrons from the O-2p formed VB. However, the magnitude of the U_dd energy is slightly lower for the Fe atoms in the FeO₅ square pyramids. Apparently, this observation may be responsible for the revealed particularity of CB formation in the GBCFO. In accord with that, the lowest Δ_ox value is calculated for FeO₆-containing layers, thus explaining why these d-orbitals form the top of the VB.

A certain alignment with the previously discussed data can be found in the LOBSTER computed partial Crystal Orbital Hamilton Population (pCOHP) curves, which basically show the bonding (if pCOHP < 0), non-bonding (pCOHP = 0) or antibonding (when pCOHP > 0) characteristic of the chemical interactions.⁵² The respectively estimated pCOHP curves for the Fe–O bonds in the different oxygen coordination polyhedra of GBCFO lattice are shown in Fig. 6. The results indicate that chemical interaction between Fe and oxygen atoms exhibits a strong bonding characteristic deep in the VB (<−6 eV) disregarding the type of polyhedron, apical or equatorial allocation of the oxygen ions and Fe³⁺ spin state. This conclusion agrees with previous findings in BaFeO₃.⁴⁹ Similarly, pCOHP values become positive in the CB of the GBCFO for all considered cases, which marks the anti-bonding characteristic of the interactions and hence the excited nature of these states. Interestingly, the bonding type between Fe-3d and O-2p electrons in the upper part of the VB (i.e. in the energy range from −6 to 0 eV) is found to be spin-dependent: spin-up electrons of Fe atoms form bonding type interactions, while the inverse process is observed for spin-down electrons. As discussed previously,⁴⁹ this observation comes from the inversion of the respective contributions of 3d and 2p orbitals to the Fe–O chemical bond depending on the spin state of the 3d electrons. In this sense, the type of oxygen polyhedron around the Fe atoms should not change that picture, which is indeed observed in the case of GBCFO. However, the shape and relative position on the energy scale of a certain pCOHP are shown to be highly dependent on the Fe-coordination. For instance, the anti-bonding Fe–O states in the CB, that belong to FeO₅ square pyramids are located much closer to the Fermi level in comparison with the FeO₄ or FeO₆ cases (Fig. 6(b)). The anti-bonding interactions between Fe and O atoms observed at the top of the VB correspond to the FeO₆ octahedra (Fig. 6(a)). Finally, in the case of FeO₄ tetrahedra, the characteristic of chemical interaction appears to be non-bonding in the vicinity of the Fermi level. All the above results suggest that the excited electronic states in the GBCFO tend to localize either in FeO₅ layers (when considering electrons in the CB) or in FeO₆ ones (when implying electron holes in VB). At the same time, those states located in FeO₄ layers can be considered as effectively insulating (Fig. 6(c)) in complete agreement with the conclusions derived from the DOS data.

Fig. 6 The LOBSTER computed pCOHP curves for Fe–O bonds in FeO₆ octahedra (a), FeO₅ square pyramids (b) and FeO₄ tetrahedra (c) of GBCFO. Solid lines show spin-up contributions and dotted lines spin-down ones. Colored lines correspond to Fe–O bonding of oxygen in equatorial sites. The Fermi level position is marked with a vertical dashed line. The respectively obtained IpCOHP values are presented in panel (d). The data on both spin channels were summed up for each bar. Filled bars show IpCOHP between apical oxygens and Fe, the hatched ones depict IpCOHP between equatorial oxygens and Fe atoms.

Essential information about the chemical bond strength can be obtained by the integration up to the Fermi level of the pCOHP curves (IpCOHP). These data quantitatively measure the energy of covalent bonds in eV units;⁵² it can often be used as a guideline for searching the most probable defect configurations in non-stoichiometric oxides.^49,75 The respective results for Fe–O bonds in GBCFO are presented in Fig. 6(d). All computed IpCOHP values are negative, which marks the essential chemical stability of Fe³⁺-perovskites. However, the magnitude of |IpCOHP| differs drastically depending on the Fe-coordination. In particular, the magnitude of IpCOHP for the apical oxygens and Fe atoms located in FeO₆ octahedra indicates low bond strength. Contrary to that, in the case of other FeO_x polyhedra, the respective quantity significantly increases, thus suggesting that apical oxygen ions are much more strongly bonded to FeO₄ tetrahedra and FeO₅ square pyramids. It should be noted that this conclusion nicely coincides with the GBCFO bond lengths presented in Table 3 – the shorter the bond, the stronger the bonding and vice versa. Accordingly, one can expect that apical oxygen sites of the FeO₅-units are unlikely to become vacant. In contrast, the equatorial positions have other distribution of IpCOHP energies with the highest one belonging to Fe–O bonds located in FeO₅ square pyramids, suggesting that these positions are the most probable sites for oxygen vacancy formation. However, one should consider that the largest (among other polyhedra) Δ_ox value in FeO₅ (see Table 4) implies higher energies to transfer electrons from the hybridized Fe 3d–O 2p band to the Fermi level. Hence, one may expect the decreased |IpCOHP| value in the case of FeO₅ pyramids as evidence of Fe–O bond extension induced by the absence of apical oxygen atoms in the Gd layer. Accordingly, the most probable place for oxygen vacancy formation in GBCFO is the equatorial O sites in FeO₆ octahedra.

3.3. Electronic conduction behavior of GdBa₂Ca₂Fe₅O₁₃

The studied compound has a crystal structure that induces a highly anisotropic magnetic behavior.²⁴ In this sense, the band structure of the GBCFO reveals pronounced 2D character of the electronic states’ distribution. This result implies that the electronic conductivity of the GBCFO should be dependent on the direction of the applied electric field propagation. Taking into account that the crystalline lattice of GBCFO has orthorhombic symmetry, and considering the crystallographic directions of the cubic perovskite structure, one can write its electronic conductivity tensor [small sigma, Greek, circumflex]_el as:where σ_el‖[110]_p, σ_el‖[−110]_p and σ_el‖[001]_p denote individual contributions, being parallel to the [110]_p, [−110]_p and [001]_p crystallographic directions respectively (see Fig. 1 for more details). Eqn (5) suggests that by measuring conductivity in different directions of a GBCFO single crystal, the exact values of each [small sigma, Greek, circumflex]_el tensor component can be determined. However, the available experimental data correspond to a polycrystalline sample, which makes a direct comparison with eqn (5) difficult. To overcome this problem one can assume that the GBCFO grains are small enough to be randomly oriented in the sample studied. Then, the experimental conductivity σ_exp can be expressed as a normalized trace of [small sigma, Greek, circumflex]_el which yields the following simple result:where angular brackets denote the averaging procedure. Accordingly, direct comparison between computed 〈[small sigma, Greek, circumflex]_CRT〉 and measured σ_exp³⁰ becomes possible; the respective data are given in Fig. 7(a). It is worth mentioning that the CRT approach used in this work does not allow us to properly estimate the magnitude of relaxation time τ. However, the previously published results suggest this parameter lies within the range of few dozens of femtoseconds (10⁻¹⁵ s).⁷⁶ Therefore, one can assume τ = 10 fs for the GBCFO, which agrees with the independently evaluated τ for the donor-doped SrTiO₃ perovskite.⁷³

Fig. 7 The calculated (lines) and experimentally measured (points) electronic conductivities of polycrystalline GBCFO vs. temperature (a). The numerical values of 〈[small sigma, Greek, circumflex]_CRT〉 are obtained assuming τ = 10 fs; the influence of different doping level p_d is also accounted for. Theoretically computed σ_CRT/τ functions being parallel to different directions of GBCFO lattice vs. temperature (b). The doping level for each curve is taken to be p_d = 10¹⁷ cm⁻³.

Prior to discussing the peculiarities of transport phenomena in the studied material it is instructive to provide a brief description of its defect structure; the respective information allows to elucidate the concentration of charge carriers in the VB and CB, which is of primary importance for electronic conductivity calculations. Given that GBCFO is a narrow band-gap semiconductor it is necessary to account for different extrinsic processes of defect formation, as well as for the intrinsic thermal ionization. For instance, one can suppose that oxygen incorporation can occur into the different structural vacancies in the GBCFO lattice, i.e. into FeO₅ or FeO₄ layers, thus producing interstitial sites O_i. Another possibility is that oxygen vacancies V_O can be formed – these can be located in all FeO_x polyhedra. The list of possible sites for hosting defects, as well as the corresponding defect formation reactions are given in the SI. The results of GGA+U calculations of the j-th defect formation enthalpies ΔH_df(j) in GBCFO oxide are presented in Table 5.

Table 5 Defect formation enthalpies in GBCFO oxide as obtained from GGA+U calculations

Defect formation process		ΔHdf(j), eV
VO formation	FeO6	3.057
	FeO5	3.287
	FeO4	3.253
Oi formation	FeO5	0.654
	FeO4	1.202

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As can be seen, oxygen vacancies are very unlikely to form with the respective enthalpies computed to be higher than 3 eV. Still, these results align well with the experimental data⁷⁷ and can be explained by high energy penalties to convert Fe³⁺ into Fe²⁺ in perovskite-oxides. In addition, the predicted trend of FeO₆ octahedra to the sites which are most susceptible to hosting vacancies (the Δ_ox parameter was found to be the lowest for FeO₆ polyhedra, see Table 4) seems to be supported by actual calculations, Table 5. On the other hand, the formation of interstitial oxygen appears to be intricated, although the relative fraction of O_i defects (especially of those formed in FeO₅ layers) will be higher than that of oxygen vacancies under oxidative conditions (i.e. in air atmosphere). Still, as follows from the calculated results, the overall deviation of oxygen content in GdBa₂Ca₂Fe₅O_13±δ from 13 should be small (|δ| will be less than 0.01 at 1000 K and under an air atmosphere). The available thermogravimetric data support this conclusion – the measured mass change of GBCFO when heating in air was found to be negligible.⁵⁷ What is even more interesting is that the positive sign of ΔH_df of the O_i formation process implies that the fraction of interstitial oxygen will increase with temperature. Accordingly, the number of holes in the VB will grow with heating. Given that ΔH_df(O_i) < ΔH_df(V_O) and ΔH_df(O_i) > 0, one can reasonably assume electron holes will be the main charge carriers in GBCFO under SOFC cathode conditions.

In agreement with the conclusions of the previous paragraph, the determination of the 〈[small sigma, Greek, circumflex]_CRT〉 value was made assuming acceptor doping level (p_d) to be of 10¹⁷–10¹⁹ cm⁻³ in magnitude, which roughly corresponds to δ ∈ [−10⁻⁵, −10⁻³] per GdBa₂Ca₂Fe₅O₁₃ formula unit. The estimated conductivity curves at p_d < 10¹⁸ cm⁻³ are shown in Fig. 7(a) and reveal the transition from the extrinsic to the intrinsic regime as the temperature increases. One can acknowledge the obtained 〈[small sigma, Greek, circumflex]_CRT〉 numerical values at p_d = 10¹⁷ cm⁻³ are close to those (σ_exp) measured in the experiment. Notably, when the doping amount is higher (i.e. p_d = 10¹⁹ cm⁻³), the shape of the conductivity plot becomes almost temperature independent. These results are in agreement with the experimental findings. Indeed, defect formation enthalpies suggest that the concentration of holes should increase with temperature, i.e. p_d = f(T). And hence, the experimentally observed σ = f(T) of GBCFO originates from the intrinsic regime at low temperatures (due to the narrow band gap) and hole-doped regime, which is invoked by the incorporation of interstitial oxygen into the crystal lattice, at higher temperatures. Importantly, this mechanism qualitatively explains the flattening of the σ_exp = f(T) curve at T > 800 K, see Fig. 7(a).

Accounting for the discussed crystalline anisotropy of GBCFO it is interesting to consider the direction-dependent components of 〈[small sigma, Greek, circumflex]_CRT〉. The respective data represented as σ_CRT/τ functions are given in Fig. 7(b). Those components of conductivity tensor that are parallel to either [110]_p or [−110]_p crystallographic directions (i.e. planes (001), also named planes ab) are 3–4 orders of magnitude higher than those parallel to the [001]_p direction. In addition, the σ_CRT‖[110]_p and σ_CRT‖[−110]_p values are almost similar. Therefore, both magnitudes will be referred to as σ_CRT(‖), while the quantity σ_CRT‖[001]_p will be designated as σ_CRT(⊥). This denomination underlines that the obtained result shows direct evidence of the anisotropy of the GBCFO electrical properties. However, the reason behind the uncovered phenomenon seems to be unclear. In this sense, the previous computational study for the YBCFO suggested that the absence of oxygen ions in the Y layer provided the main contribution to the 2D characteristic of electronic conduction.⁴⁴ In turn, the present study provides a more complicated picture where all polyhedral layers in the GBCFO lattice play a certain role in providing the anisotropic behavior. But indeed, the elimination of the FeO₅ square pyramids should result in a more isotropic 〈[small sigma, Greek, circumflex]_CRT〉; the respective evaluations made for GdBaCaFe₃O₈ and presented as a ratio between σ_CRT(‖) and σ_CRT(⊥) components of conductivity tensor clearly confirm this statement, see Fig. 8. In fact, the corresponding pDOS data suggest that the CB of GdBaCaFe₃O₈ is composed from electronic states of both FeO₆ and FeO₄ structural units (Fig. 4(a)). Therefore, it can be deduced that in the case of the acceptor doped GBCFO the main reason for the anisotropic behavior of 〈[small sigma, Greek, circumflex]_CRT〉 is the combination of FeO₆ layers forming the top of the VB and the large slabs composed of FeO₆₅ and FeO₄ polyhedra that provide effective isolation of the conductive octahedral layers from each other. Because of that the curvature of the VB along the [001]_p direction in GBCFO (b₂ direction in reciprocal space, as represented in Fig. S6) is almost zero, resulting in large effective masses for charge carriers and hence making the [001]_p direction highly unfavorable for electronic conduction in GBCFO. Similar behavior is observed for the acceptor doped GdBa₂Fe₃O₈ and NaCa₂Fe₃O₇ compounds (Fig. 8); the only difference from GBCFO is the nature of insulating layer - it is either FeO₄ tetrahedra (in NaCa₂Fe₃O₇) or FeO₅ pyramids (in GdBa₂Fe₃O₈).

Fig. 8 The ratio of σ_CRT(‖) to σ_CRT(⊥) components of the conductivity tensor as a function of temperature and acceptor doping level calculated for different Fe³⁺-containing oxides: GdBa₂Ca₂Fe₅O₁₃ (dark grey), GdBa₂Fe₃O₈ (red), GdBaCaFe₃O₈ (green) and NaCa₂Fe₃O₇ (dark blue).

4. Conclusions

DFT+U calculations on the GdBa₂Ca₂Fe₅O₁₃ perovskite-related oxide exposed a distinct band structure that reveals a 2D-electronic behavior. The computing methodology (GGA+U approximation) used in this work captures both the nuclear and magnetic structures of the compound. The crystal structure, which consists of layered Gd/Ba/Ca ordering in combination with the ordering of three different coordination-polyhedra around the Fe-atoms (FeO₆-octahedra, FeO₅-squared pyramids and FeO₄-tetrahedra), causes a peculiar contribution of the differently located Fe and O atoms to the resulting band structure. Both the valence and the conduction bands are shown to be spatially distributed in the lattice depending on the oxygen-coordination of the Fe atoms. In particular, the bottom of the CB is formed by 3d orbitals of Fe atoms possessing 3z² − r² symmetry, which are located within the FeO₅-squared pyramids. However, the electronic states near the Fermi level are mainly formed by 2p states of oxygen atoms occupying the equatorial positions of the FeO₆-octahedra. As a result, GdBa₂Ca₂Fe₅O₁₃ appears to be a narrow-gap 2D-semiconductor (the respective band gap E_g is found to be ∼0.5 eV) where the FeO₅-layers create the CB, the FeO₆-layers form the VB and the FeO₄-layers create insulating channels avoiding the intralayer transport of electrons. The existence of the three types of oxygen-coordination around the Fe-atoms seems to be essential for the narrow band gap, since other possible double combinations of oxygen-polyhedra forming the anion sublattice, as well as singular ones lead to higher E_g values. The computed pCOHP curves conclude a strongly bonding character of the chemical interaction between all the Fe atoms and the oxygens in the oxide and confirm the insulating character of the FeO₄ layers. Computed electronic conductivity tensors support the anisotropic electrical properties of the GdBa₂Ca₂Fe₅O₁₃, since the conductivity is established within the (ab)-planes (i.e. (001)_p planes), but it is about four orders of magnitude lower along the c-axis ([001]_p direction) of the structure. Overall, the band-structure of the GdBa₂Ca₂Fe₅O₁₃ oxide is a consequence of the existence and ordering of the three different types of FeO_x polyhedra within the crystal structure.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting this article have been included as part of the supplementary information (SI). Supplementary information: structural and magnetic data of various oxides used in the DFT+U calculations (Table S1); graphic representations of GdBaCaFe₃O₈, GdBa₂Fe₃O₈ and NaCa₂Fe₃O₇ crystal structures (Fig. S1); graphic representations of disordered GdBa₂Ca₂Fe₅O₁₃ crystal structures (Fig. S2); computed energy-volume curve for the GdBa₂Ca₂Fe₅O₁₃ oxide and the respectively derived Debye temperature and unit cell volume dependences on temperature (Fig. S3); optimized coordinates of the G-AFM GdBa₂Ca₂Fe₅O₁₃ unit cell (Table S2); comparison of GGA+U computed densities of states for GdFeO₃ ferrites with and without accounting for the 4f states of Gd atoms (Fig. S4); GGA+U computed optical spectra for the G-AFM GdBa₂Ca₂Fe₅O₁₃ oxide (Fig. S5); computed band dispersions of the orthorhombic GdBa₂Ca₂Fe₅O₁₃ unit cell with G-AFM magnetic ordering (Fig. S6); charge density projections of the CB and VB of the G-AFM GdBa₂Ca₂Fe₅O₁₃ along the [100]_p direction (Fig. S7); GGA+U computed band dispersion and experimentally measured optical absorption spectrum of CaBaFe₄O₈ oxide (Fig. S8); DOS/pDOS spectra of GdFeO₃, BaCaFe₂O₅ and CaBaFe₄O₈ as obtained within the GGA+U framework (Fig. S9); graphic representation of the allocation of different point defects in GdBa₂Ca₂Fe₅O₁₃ crystal lattice (Fig. S10). See DOI: https://doi.org/10.1039/d5cp01780g.

Acknowledgements

This research has been supported by the “Plan de Recuperación, Transformación y Resiliencia-C17.I1” with funds from the European Union – Next Generation EU, “European Regional Development Fund” as part of the action after the COVID-19 pandemic, and Comunidad de Madrid with the project “GREEN H2-CM”. Authors also thank MCIN/AEI/10.13039/501100011033 for funding the Project PID2022-139039OB-C22 and MCIN/AEI/10.13039/501100011033/“NextGenerationEU”/PRTR for funding the Project TED2021-130452B-C21. Special thanks to Mikhail Kalinkin from the Institute of Solid State Chemistry UB RAS for measuring optical spectra.

References

1. T. Ishihara, Inorganic Perovskite Oxides, in Springer Handbook of Electronic and Photonic Materials, ed. S. Kasap and P. Capper, Springer, Cham, 2017.
2. T. Wolfram and S. Ellialtioglu, Electronic and optical properties of d-band perovskites, Cambridge University Press, Cambridge, 2006.
3. D. I. Khomskii, Transition metal compounds, Cambridge University Press, Cambridge, 2014.
4. M. Nasir, I. Kim, K. Lee, S. I. Kim, K. H. Lee and H. J. Park, Phys. Chem. Chem. Phys., 2023, 25(5), 3942–3949.
5. J. Ding and X. Zhu, J. Mater. Chem. C, 2024, 12, 9510–9561.
6. J. Irvine, J. L. Rupp, G. Liu, X. Xu, S. Haile and X. Qian, et al. , JPhys Energy, 2021, 3(3), 031502.
7. G. R. Monama, K. E. Ramohlola, E. I. Iwuoha and K. D. Modibane, Results Chem., 2022, 4, 100321.
8. X. Liang, W. Yan, Y. Yu, K. Zhang, W. An and H. Chen, et al. , Angew. Chem., Int. Ed., 2023, 62(46), e202311606.
9. A. N. Zainon, M. R. Somalu, A. M. K. Bahrain, A. Muchtar, N. A. Baharuddin, S. A. Muhammed Ali, O. Nafisah, A. S. Abdullah, A. Abul Kalam and P. B. Nigel, et al. , Int. J. Hydrogen Energy, 2023, 48(53), 20441–20464.
10. S. E. Wolf, F. E. Winterhalder, V. Vibhu, L. B. de Haart, O. Guillon, R. A. Eichel and N. H. Menzler, J. Mater. Chem. A, 2023, 11(34), 17977–18028.
11. W. Zhang and Y. H. Hu, Catal. Today, 2023, 409, 71–86.
12. S. Baratov, E. Filonova, A. Ivanova, M. B. Hanif, M. Irshad and M. Z. Khan, et al. , J. Energy Chem., 2024, 94, 302–331.
13. E. Tezel, D. Guo, A. Whitten, G. Yarema, M. Freire and R. Denecke, et al. , J. Electrochem. Soc., 2022, 169(3), 034532.
14. A. I. Klyndyuk, E. A. Chizhova, D. S. Kharytonau and D. A. Medvedev, Materials, 2022, 15, 141.
15. J. A. Kilner and M. Burriel, Annu. Rev. Mater. Res., 2014, 44(1), 365–393.
16. J. H. Kim and A. Manthiram, J. Mater. Chem. A, 2015, 3(48), 24195–24210.
17. F. Fauth, E. Suard, V. Caignaert, B. Domenges, I. Mirebeau and L. Keller, Eur. Phys. J. B, 2001, 21, 163–174.
18. Z. Gao, L. V. Mogni, E. C. Miller, J. G. Railsback and S. A. Barnett, Energy Environ. Sci., 2016, 9(5), 1602–1644.
19. P. Winiarz, E. A. Sroczyk, A. Brzoza-Kos, P. Czaja, K. Kapusta and K. Świerczek, Acta Mater., 2024, 277, 120186.
20. A. A. Taskin, A. N. Lavrov and Y. Ando, Prog. Solid State Chem., 2007, 35(2–4), 481–490.
21. D. Parfitt, A. Chroneos, A. Tarancón and J. A. Kilner, J. Mater. Chem., 2011, 21(7), 2183–2186.
22. R. Pelosato, G. Cordaro, D. Stucchi, C. Cristiani and G. Dotelli, J. Power Sources, 2015, 298, 46–67.
23. D. Muñoz-Gil, D. Pérez-Coll, E. Urones-Garrote, U. Amador and S. García-Martín, J. Mater. Chem. A, 2017, 5(24), 12550–12556.
24. D. Muñoz-Gil, E. Urones-Garrote, D. Pérez-Coll, U. Amador and S. García-Martín, J. Mater. Chem. A, 2018, 6(13), 5452–5460.
25. D. Muñoz-Gil, D. Ávila-Brande, E. Urones-Garrote and S. García-Martín, Dalton Trans., 2015, 44(23), 10867–10874.
26. S. García-Martín, K. Manabe, E. Urones-Garrote, D. Ávila-Brande, N. Ichikawa and Y. Shimakawa, Inorg. Chem., 2017, 56(3), 1412–1417.
27. P. Karen, J. Solid State Chem., 2021, 299, 122147.
28. P. Karen, E. Suard and F. Fauth, Inorg. Chem., 2005, 44(23), 8170–8172.
29. J. M. Hudspeth, D. J. Goossens, A. J. Studer, R. L. Withers and L. Norén, J. Phys.: Condens. Matter, 2009, 21(12), 124206.
30. R. Marín-Gamero, X. Martinez de Irujo-Labalde, E. Urones-Garrote and S. García-Martín, Inorg. Chem., 2020, 59(8), 5529–5537.
31. X. M. de Irujo-Labalde, U. Amador, C. Ritter, M. Goto, M. A. Patino, Y. Shimakawa and S. García-Martín, Inorg. Chem., 2021, 60(11), 8027–8034.
32. G. Kresse and J. Furthmüller, Computational Materials Physics, Faculty of Physics, Universitat Wien, 2014.
33. J. Sun, A. Ruzsinszky and J. P. Perdew, Phys. Rev. Lett., 2015, 115(3), 036402.
34. H. D. Saßnick and C. Cocchi, Electron. Struct., 2021, 3(2), 027001.
35. J. He and C. Franchini, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 86(23), 235117.
36. S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys and A. P. Sutton, Phys. Rev. B: Condens. Matter Mater. Phys., 1998, 57(3), 1505.
37. J. K. Shenton, D. R. Bowler and W. L. Cheah, J. Phys.: Condens. Matter, 2017, 29(44), 445501.
38. L. Wang, T. Maxisch and G. Ceder, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73(19), 195107.
39. M. Alaydrus, I. Hamada and Y. Morikawa, Phys. Chem. Chem. Phys., 2021, 23(34), 18628–18639.
40. M. Gómez-Toledo, S. A. López-Paz, S. García-Martín and M. E. Arroyo-de Dompablo, Inorg. Chem., 2023, 62(8), 3445–3456.
41. M. Ferri, J. Elliott, S. Fabris and S. Piccinin, Phys. Rev. B, 2020, 101(15), 155201.
42. B.-L. Liu, Y.-C. Wang, Y. Liu, Y.-J. Xu, X. Chen, H.-Z. Song, Y. Bi, H.-F. Liu and H.-F. Song, J. Chem. Phys., 2023, 158, 084108.
43. A. Otero-De-La-Roza, D. Abbasi-Pérez and V. Luaña, Comput. Phys. Commun., 2011, 182(10), 2232–2248.
44. X. M. de Irujo-Labalde, M. Goto, E. Urones-Garrote, U. Amador, C. Ritter and M. Amano-Patino, et al. , Chem. Mater., 2024, 36(10), 5184–5191.
45. A. Demont, M. S. Dyer, R. Sayers, M. F. Thomas, M. Tsiamtsouri and H. N. Niu, et al. , Chem. Mater., 2010, 22(24), 6598–6615.
46. C. Tenailleau, M. Allix, J. B. Claridge, M. Hervieu, M. F. Thorpe and R. J. Cava, J. Solid State Chem., 2003, 176(2), 381–392.
47. X. M. de Irujo-Labalde, E. Urones-Garrote and S. García-Martín, Solid State Sci., 2024, 151, 107516.
48. O. A. Salawu, S. Boulfrad and U. Schwingenschlögl, J. Mater. Chem. A, 2016, 4(9), 3560–3564.
49. M. F. Hoedl, C. Ertural, R. Merkle, R. Dronskowski and J. Maier, J. Phys. Chem. C, 2022, 126(30), 12809–12819.
50. M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller and F. Bechstedt, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73(4), 045112.
51. V. Wang, N. Xu, J. C. Liu, G. Tang and W. T. Geng, Comput. Phys. Commun., 2021, 267, 108033.
52. R. Dronskowski and P. E. Blöchl, J. Phys. Chem., 1993, 97, 8617–8624.
53. G. K. Madsen, J. Carrete and M. J. Verstraete, Comput. Phys. Commun., 2018, 231, 140–145.
54. P. E. Blöchl, O. Jepsen and O. K. Andersen, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 49(23), 16223.
55. H. Weihe and H. U. Güdel, Inorg. Chem., 1997, 36(17), 3632–3639.
56. T. Katsura and Y. Tange, Minerals, 2019, 9(12), 745.
57. R. Marín Gamero, Óxidos metálicos con superestructura derivada de la perovskita y estudio de sus propiedades electrocatalíticas en la reducción de oxígeno, 2024.
58. A. Scrimshire, A. Lobera, A. M. T. Bell, A. H. Jones, I. Sterianou, S. D. Forder and P. A. Bingham, J. Phys.: Condens. Matter, 2018, 30(10), 105704.
59. S. Lutfalla, V. Shapovalov and A. T. Bell, J. Chem. Theory Comput., 2011, 7(7), 2218–2223.
60. P. Adler, U. Schwarz, K. Syassen, A. P. Milner, M. P. Pasternak and M. Hanfland, J. Solid State Chem., 2000, 155(2), 381–388.
61. M. Capone, C. J. Ridley, N. P. Funnell, M. Guthrie and C. L. Bull, Phys. Status Solidi B, 2021, 258(2), 2000413.
62. E. Marelli, J. Lyu, M. Morin, M. Leménager, T. Shang and N. S. Yüzbasi, et al. , EES Catal., 2024, 2(1), 335–350.
63. U. Von Barth and C. D. Gelatt, Phys. Rev. B: Condens. Matter Mater. Phys., 1980, 21(6), 2222.
64. X. H. Zhu, X. B. Xiao, X. R. Chen and B. G. Liu, RSC Adv., 2017, 7(7), 4054–4061.
65. V. R. Galakhov, E. Z. Kurmaev, K. Kuepper, M. Neumann, J. A. McLeod and A. Moewes, et al. , J. Phys. Chem. C, 2010, 114(11), 5154–5159.
66. X. Bo, D. Wang and X. G. Wan, Phys. Lett. A, 2021, 394, 127202.
67. B. Ouyang, T. Chakraborty, N. H. Perry, T. Mueller, N. R. Aluru and E. Ertekin, Chem. Mater., 2019, 31(9), 3144–3153.
68. M. D. Scafetta, A. M. Cordi, J. M. Rondinelli and S. J. May, J. Phys.: Condens. Matter, 2014, 26(50), 505502.
69. M. K. Warshi, V. Mishra, A. Sagdeo, V. Mishra, R. Kumar and P. R. Sagdeo, Ceram. Int., 2018, 44(7), 8344–8349.
70. Encyclopedia Britannica, https://www.britannica.com/science/color, (accessed April 2025).
71. Colorimetry: Understanding the CIE System, ed. J. Schanda, John Wiley & Sons, 2007.
72. Y. Shin, K. Y. Doh, S. H. Kim, J. H. Lee, H. Bae, S. J. Song and D. Lee, J. Mater. Chem. A, 2020, 8(9), 4784–4789.
73. A. D. Bamburov, A. A. Markov, M. V. Patrakeev and I. A. Leonidov, Solid State Ionics, 2019, 332, 86–92.
74. J. E. Auckett, G. J. McIntyre, M. Avdeev, H. De Bruyn, T. T. Tan, S. Li and C. D. Ling, J. Appl. Crystallogr., 2015, 48(1), 273–279.
75. I. Kim, H. Lee and M. Choi, J. Appl. Phys., 2022, 131(7), 075106.
76. V. P. Zhukov and E. V. Chulkov, Phys. Solid State, 2022, 64(4), 422–431.
77. I. A. Leonidov, A. A. Markov, M. A. Zavyalov, O. V. Merkulov, E. V. Shalaeva and S. S. Nikitin, et al. , Materials, 2022, 15(13), 4390.

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