Introduction

Materials with oxygen hyperstoichiometry have excellent electronic, magnetic, and oxygen storage properties and can be used in a wide variety of applications1,2,3,4,5. For example, La2CuO4+δ forms two phases depending on the oxygen content6: the phase with δ = 0 is semiconducting, while the other has a wide range of values (0.03 < δ < 0.18) and it becomes superconducting for δ = 0.08 with Tc ~ 38 K. The discovery of the series of 114 cobaltites ((Ln,Ca)1BaCo4O7) revealed the existence of closely related structures with various crystallographic symmetries and the possibility of oxygen nonstoichiometry in the range “O7”–“O8.5” in those systems, which opened up a new field for the investigation of strongly correlated electron systems. This change of oxygen stoichiometry, which induces the variation of the Co2+:Co3+ ratio in the system, is expected to influence the physical properties of these compounds considerably. This is the case of the oxygen-rich “114” cobaltites YBaCo4O8 and YbBaCo4O7.2, which were shown to be magnetically frustrated rather than magnetically ordered at low temperatures7,8. Therefore, oxygen hyperstoichiometry materials with useful functionalities are attractive subjects of research for chemists, physicists, or materials scientists.

In recent years, materials with oxygen hyperstoichiometry received great attention in the field of solid oxide fuel cells because of the low activation energy (Ea) of interstitial ion migration. Well known and studied examples are apatites La10-δ(MO4)6O3−1.5δ (M = Si, Ge)9, melilite La1+δSr1-δGa3O7+0.5δ10, layered perovskites La2NiO4+δ11, fluorite UO2+δ12, or scheelite La0.2Pb0.8WO4+δ13, Bi1−δSrδVO4−0.5δ14, and also the subject of this study, CeNbO4+δ. CeNbO4+δ is a mixed ionic and p-type electronic conductor with fast oxygen ion diffusion at moderate temperatures (total conductivity up to 0.030 S cm−1 at 850 °C; ion transference number up to 0.4; diffusion coefficient up to 8.3 × 10−8 cm2 s−1), making it a promising material for applications in energy generation and storage devices15,16,17,18,19,20,21,22,23. CeNbO4+δ was first reported by Cava et al.15 in 1970s. It has been identified as a family of compounds with variable oxygen contents, with distinct phases CeNbO4, CeNbO4.08, CeNbO4.25, and CeNbO4.33. Although Thompson et al.16 successfully indexed the unit cell of these four compounds by selected area electron diffraction (SAED) in 1999, no progress on the structure solution of these phases was made until in 2016 Pramana et al.17 solved the structure of CeNbO4.25 by single-crystal X-ray diffraction (SCXRD) and revealed by molecular dynamics (MD) simulations that the fast ion migration occurs within planes of the neighboring NbOn polyhedra. However, the atomic structures of CeNbO4.08 and CeNbO4.33 remained unknown, which hindered the full understanding of the oxide ion conduction behavior for the whole system of CeNbO4+δ.

In order to better understand the oxygen transport mechanism and to optimize its performance, atomic structures of CeNbO4.08 and CeNbO4.33 need to be understood. However, for CeNbO4.08 and CeNbO4.33, it was very difficult to grow large single crystal due to their specific syntheses. Furthermore, CeNbO4.08 appeared to be a (3 + 2)-dimensional incommensurately modulated structure with monoclinic symmetry and CeNbO4.33 a commensurately modulated structure with triclinic symmetry according to SAED16. The large unit cell parameters and complex diffraction patterns make it very difficult to determine their atomic structures by conventional powder X-ray diffraction (PXRD). Fortunately, nanocrystals and microcrystals can be treated as single crystals in electron microscopy. The recently developed 3D ED technique, continuous rotation ED24,25,26,27, can use single nanocrystals to obtain single-crystal diffraction data that can be used for structure determination by utilizing the software developed for SCXRD (ShelxT28, Superflip29,30).

In this study, we determine the incommensurately modulated structure of CeNbO4.08 and the superstructure of CeNbO4.33 by combining 3D ED, synchrotron X-ray powder diffraction (SPD), and neutron powder diffraction (NPD). Using the same methods, the superstructure of CeNbO4.25 was also re-determined. The structure models are sufficiently accurate to reveal the interstitial oxygen sites and allow to elucidate how the extra oxygen atoms change the structural connectivity. Combining the structural information with MD simulations, we describe the oxide ion migration mechanisms in CeNbO4+δ. The relationship between the structure and oxide ion migration for the whole series of CeNbO4+δ compounds here provides means to optimize the performance of these compounds and to develop better oxygen hyperstoichiometric materials for a wide variety of applications.

Results

A series of phases of CeNbO4+δ was obtained from CeO2 and Nb2O5 by the solid-state synthetic route in Supplementary Fig. 1. The most striking chemical feature of the studied cerium niobate phases is that they are progressively oxidized from one structure to another under relatively mild conditions. The oxygen contents of the phases were analyzed by thermogravimetric analysis (TGA), which yields formulae CeNbO4.11(2), CeNbO4.24(2), and CeNbO4.33(1) (Supplementary Fig. 2). Figure 1 shows the [010]p (p denotes parent material CeNbO4) zone axis SAED patterns of the CeNbO4+δ (δ = 0, 0.08, 0.25, 0.33) phases. In all the four SAED patterns, the strong reflections have the same pseudo-tetragonal symmetry as that of the parent structure. However, for CeNbO4.08, CeNbO4.25, and CeNbO4.33, many additional reflections can be observed.

Fig. 1: [010]p zone axis SAED of CeNbO4+δ.
figure 1

a CeNbO4.0. b CeNbO4.08. c CeNbO4.25. d CeNbO4.33. The strong reflections in b–d have the same pseudo-tetragonal symmetry as the reflections of the parent structure in a. (p denotes parent material CeNbO4).

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Incommensurately modulated structure of CeNbO4.08

In previous transmission electron microscopic study, CeNbO4.08 was identified as a monoclinic (3 + 2)D incommensurately modulated phase with average unit cell parameters close to the parent CeNbO4 and modulation vectors q1 ~ (0.138, 0, 0.344), q2 ~ (0.345, 0, −0.138) and centering I = (½, ½, ½, 0, 0) (Supplementary Fig. 3a)16. In the present study, analysis of the [010]p zone axis SAED of CeNbO4.08 indicated that the structure can be described as monoclinic (3 + 1)D incommensurately modulated with a single modulation vector q = 0.069a* + 0.175c* and a nonstandard centering X = (½, ½, ½, ½). However, satellites up to order 5 or even 7 are needed to describe the whole pattern (Supplementary Fig. 3b). The qualities of the LeBail fit of SPD are similar for the 2D modulation and 1D modulation, which further confirmed that the 1D modulation is the correct description (Supplementary Fig. 4). It is extremely unlikely that a single modulation vector would be able to describe two independent modulation vectors just by coincidence with such accuracy. Satellites up to order 7 in the synchrotron powder data are visible. However, the number of satellites is very large and the sensitivity of the SPD pattern to oxygen positions is relatively low, making it extremely challenging to solve the (3 + 1)D incommensurately modulated structure of CeNbO4.08 ab initio from the SPD pattern. Therefore, the 3D ED technique was applied to solve its incommensurately modulated structure.

Due to the limitation of processing continuous mode 3D ED data with (3 + 1)D superspace, we processed the 3D ED data with approximate supercell. The 3D reciprocal lattice reconstructed from the 3D ED data of CeNbO4.08 are shown in Fig. 2a–d. Reciprocal lattice along the [010]p direction can be cut from the reconstructed 3D reciprocal lattice (Supplementary Fig. 6b), in which the strong reflections show the similar pseudo-tetragonal symmetry as the [010]p zone axis SAED of CeNbO4.. Then we transformed the processed 3D ED data (hkl file based on the approximate supercell) to (3 + 1)D superspace. Reflection conditions indicated the superspace group of X2/c(α0γ)0s and satellites up to order 7 were required. The determination of the modulation functions turned out to be extremely difficult due to their complexity and large number of contributing harmonic waves. Finally, the (3 + 1)D model was constructed from the supercell structure. The structure contains one additional oxygen site (O3) compared with the parent structure of CeNbO4 (Fig. 3a, see Supplementary Information for details on the localization of this additional atoms). Detailed analysis of the interatomic distances and difference potential maps showed that the atom O1 has two positions—one is the main atomic domain while the other is occupied only when O1 coexists with O3. The modulation structure model was first refined against 3D ED with modulation functions up to order 7 and modulation of ADP parameters up to order 3 for Ce and 2 for Nb. The final modulated structure model of CeNbO4.08 was obtained by the Rietveld refinement of combined SPD data (Fig. 3b) and NPD data (Fig. 3c). Modulation functions up to order 7 were used together with isotropic atom displacement parameters. The model resulted in chemically reasonable bond distances (Supplementary Fig. 9) and modulation functions similar to the model obtained against 3D ED data (Supplementary Figs. 10 and 11).

Fig. 2: 3D ED data of CeNbO4.08 and CeNbO4.33.
figure 2

3D reciprocal lattice reconstructed from the 3D ED data of CeNbO4.08 (a) and CeNbO4.33 (e). b (0kl), c (hk0), and d (h0l) slices extracted from the reconstructed reciprocal lattice of CeNbO4.08, reflection conditions: 0klm: k + l = 2n. hk0m: h + k = 2n, h0lm: h + m = 2n, l + m = 2n, h00m: h + m = 2n, 0k00: k = 2n, 00 lm: l + m = 2n. f (0kl), g (hk0), and h (h0l) slices cut from the reconstructed reciprocal lattice of CeNbO4.33, reflection conditions: no special systematic reflections appear forbidden.

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Fig. 3: Structure characterization and analysis of CeNbO4.08.
figure 3

a Average structure of the (3 + 1)D incommensurately modulated structure of CeNbO4.08 along the [100]p direction. The interstitial O3 (blue atom) is located between Ce cations within the Ce cationic chain. b Final Rietveld refinement against the SPD data for CeNbO4.08 with (3 + 1)D superspace. Note: the unexplained peaks at 7.46, 7.77, and 8.86 degree (marked at inset pattern) belong to the impurity phase of CeNbO4.25. c Final Rietveld refinement against the NPD data for CeNbO4.08 with (3 + 1)D superspace. (Red multi (×) symbol: observed profile, green curve: simulated profile, blue curve: difference profile, black Bragg line: main reflections, pink Bragg line: satellite reflections).

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Supercell structure of CeNbO4.33

From the [010]p zone axis SAED, the satellites in CeNbO4.25 and CeNbO4.33 can be indexed with a single vector q = 1/12 [204]p and 1/3 [101]p, respectively. Thus the commensurately modulated structure of CeNbO4.25 and CeNbO4.33 can be described by supercell models. Again, 3D ED data were collected on CeNbO4.33 (Fig. 2e–h) and CeNbO4.25 (Supplementary Fig. 5e–h). The reflection conditions derived from the 3D ED datasets indicated that the possible space group of CeNbO4.33 is P\(\bar 1\). All the crystallographically unique Ce and Nb atoms as well as a part of oxygen atoms in the structure were located directly by using the software SHELXT28. The supercell structure model of CeNbO4.25 could also be obtained with 3D ED data. The final supercell structure model for CeNbO4.33 and CeNbO4.25 with interstitial oxygen sites were obtained by the Rietveld refinement of combined SPD and NPD data (Supplementary Figs. 12–15).

Identification of the interstitial sites Oi in the structure models

Comparing the three oxide supercell structures with the parent structure (Fig. 4 and Supplementary Fig. 16), the additional oxygen atoms incorporated into the CeNbO4 host lattice have relaxed the original oxide ion positions, which resulted in three different incommensurately or commensurately modulated structures. Figure 4 shows the structures of the parent CeNbO4 and the structures of CeNbO4.08, CeNbO4.25, and CeNbO4.33 projected along the principal axes of [010]p and [100]p. Significant displacement of oxygen atoms from the original positions in the parent structure is observed and additional oxygen sites appear in the cationic layers, close to the interstitial position O3 identified in the average structure of CeNbO4.08 (Fig. 3a).

Fig. 4: Structure model of CeNbO4+δ.
figure 4

CeNbO4 (a, b), oxidized CeNbO4.08 (c, d), CeNbO4.25 (e, f), and CeNbO4.33 (g, h) projected along the principal axes of [010]p and [100]p. Displacement of oxygen from the original position in the parent structure is observed. Note that c and d were 6a × 2b × 6c approximant superstructure in (3 + 1)D incommensurately modulated model along [010]p and [100]p.

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In the oxidized supercell structures, the excess oxygen atoms are located at one ((3 + 1)D incommensurately modulated model of CeNbO4.08), three (CeNbO4.25), and one (CeNbO4.33) fully occupied general Wyckoff sites, respectively. The interstitial sites are the blue atoms in Fig. 4 (O3 in CeNbO4.08 6a × 2b × 6c approximant superstructure structure, O4, O11, and O41 in CeNbO4.25 and O13 in CeNbO4.33). Bond valence sum (Supplementary Tables 2 and 3 and Supplementary Fig. 17) analysis shows that some of the Ce positions are Ce4+ sites. The change of the oxidation state of Ce is responsible for the shorter Ce-O bond lengths and allows that excess oxygen is incorporated into the host lattice at the interstitial sites between Ce cations (Fig. 4c–h). In CeNbO4.33, the amount of additional oxygen is so much that part of the oxygen atoms are pushed away from the original normal site into an interstitial site (O11) due to the relaxation of the structure. The interstitial site oxygens (O11, O13) between Ce cations are occupied in an ordered manner along the a and c axes in CeNbO4.33 (Supplementary Fig. 18). The ordered interstitial oxygen (O13) and the relaxation of the other oxygen site (O11) generate a zigzag-shaped [Nb6O26]∞ infinite chain comprising edge-sharing [NbO6]4 units bridged by edge-sharing [NbO6]2 units through vertex (Supplementary Fig. 18a).

Oxide ion migration

MD simulations based on interatomic potential method were performed to elucidate the dependence of oxide ion migration on the variation of oxygen hyperstoichiometry and structure in CeNbO4+δ compounds. The mean square displacement (MSD) values (Fig. 5a–d and Supplementary Fig. 19) and scatter plots (Fig. 5e–l and Supplementary Fig. 20) indicate that all atoms in the parent CeNbO4 show only lattice vibration without long-range migration as expected. With inclusion of the excess oxygen into the host lattice of CeNbO4, the oxide ions become mobile in CeNbO4.08 and CeNbO4.25 but hardly migrate in CeNbO4.33. The MD simulations show that CeNbO4.08 and CeNbO4.25 phases have similar oxide ion migration pathways.

Fig. 5: Molecular dynamics simulations study of CeNbO4+δ.
figure 5

Calculated MSD values of Ce3+/4+, Nb, and O atoms as a function of simulation time for CeNbO4 (a), CeNbO4.08 (b), CeNbO4.25 (c), and CeNbO4.33 (d) from the MD simulation at 1200 °C. Trajectory scatter plots for CeNbO4 (e, i), CeNbO4.08 (f, j), CeNbO4.25 (g, k), and CeNbO4.33 (h, l) at 1200 °C where cyan, blue, and red dots represent Ce3+/4+, Nb5+, and O2− atoms, respectively.

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Two kinds of oxide ion migration events contributing to the long-range migration can be identified in the MD simulations of CeNbO4.08 (Fig. 6a, b). The first one is oxide ion migration between NbOn polyhedra isolated by the Ce cationic chain by ~5.0–5.4 Å (labeled as path A). In this migration event, the oxide ion leaves one NbOn polyhedron and passes through the interstitial sites (referred to as Oi) within the chain of Ce cations along the a or c axis of the parent CeNbO4 structure to subsequently enter into the coordination environment of another NbOn polyhedron located at the other side of the Ce chain through a knock-on process. The second type of migration is the oxide ion migration between the neighboring NbOn polyhedra separated by ~3.6−4.2 Å (labeled as path B) through a chain knock-on process via the NbOn polyhedral rotation and deformation. This migration event also frequently involves the interstitial sites Oi (Fig. 6a, b). MD simulations indicate that the migration along path A is much less frequent than that along path B, which is consistent with the larger separation between the NbOn polyhedral units involved in path A than that in path B.

Fig. 6: The oxide ionic transport mechanism of CeNbO4+δ.
figure 6

a, b Trajectory scatter plots of Ce and O atoms showing the migration paths A (black dash dot line) and B (yellow dash dot line) with the NbOn polyhedral network embedded in, and c snapshots of long-range oxide ion (marked with the blue circle) migration involving the breaking and reforming of the Nb2O9 units (highlighted by dashed lines) in CeNbO4.08 from the MD simulations at 1200 °C. In order to illustrate the Nb2O9 breaking and reforming mechanism clearly, in b, c the weak Nb-O bonds >2.2 Å are not included in the NbOn polyhedral units. d Arrhenius plot of the oxygen diffusion coefficient calculated using the MSD values of oxygen atoms for CeNbO4+δ (δ = 0.08, 0.25, 0.33). The activation energies are labeled.

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As Nb atoms are usually found in 6 or 4 coordinate environments and the Nb–O bond lengths normally vary within 1.8–2.2 Å, the Nb–O bonds >2.2 Å were not included in the NbOn polyhedral units during a close examination of oxide ion migration, which simplifies the description of the polyhedral units and assists understanding the complex migration process. This representation leads to isolated tetrahedral NbO4 units and corner-sharing Nb2O9 units most of the time during simulation. The oxide ions were found to move between the polyhedral units in the long-range migration routes composed of paths A and B mainly through the synergic-cooperation mechanism involving the continuous breaking and reformation of the Nb2O9 units assisted by rotation and deformation of the NbOn polyhedra and a knock-on process between the oxygen atoms (Fig. 6b, c and Supplementary Video 1). Therefore, the interstitial sites within the Ce cationic chain and NbOn polyhedra define more like 3D pathways for the oxide ion migration in CeNbO4.08, which is consistent with the fact that inclusion of the extra oxygen atoms disturbs anisotropically the polyhedral linkage in CeNbO4.

The same oxide ion migration pathways are found also in CeNbO4.25 phase in our study. Oxide migration in CeNbO4.25 was analyzed also by Pramana et al.17. They identified anisotropic oxide ion diffusion paths within the linked NbO6 layers and pointed out that migration between the adjacent layers also took place when the migration paths along the linked NbO6 layers were blocked. The migration routes proposed by Pramana et al.17 are similar to those composed of paths B only identified in our study. However, the work of Pramana et al.17 did not identify the interstitial sites and as a consequence did not recognize migration paths going through the interstitial sites and the related dynamic process of the adaptation of the polyhedral units to the oxide ion migration. The proper description of the interstitial sites turns out to be crucial for the correct description of the full 3D long-range migration and also for the elucidation of the knock-on dynamic process of breaking and reformation of the Nb2O9 units during the oxide ion migration.

The calculated oxide diffusion coefficients of CeNbO4.08 are about one magnitude lower than that of CeNbO4.25 (Fig. 6d). This is ascribed to the larger concentration of the mobile charge carriers in CeNbO4.25. However, the oxide ions in CeNbO4.33 phase with more excess oxygen are essentially immobile at the same simulation temperatures compared with CeNbO4.08 and CeNbO4.25. The well-organized mixed corner/edge-sharing zigzag-shaped polyhedral [Nb6O26] chains in CeNbO4.33 phase resulting from the ordered distribution of the excess oxide ions have much less flexibility on the rotation and deformation. This rigidity blocks the transfer of the oxide ions between the polyhedral units. Also the high concentration of excess oxide ions in the oxygen sublattice seems to have a significant blocking effect to the oxide ion migration. The oxide ions in CeNbO4.33 phase get mobile only when the temperature is increased to above 2200 °C during the MD simulations (Supplementary Fig. 21). Even in that case, the migration path is confined only along the [Nb6O26] chain (Supplementary Fig. 22). The large Ea (~2.07 eV) calculated from the Arrhenius plot of oxygen diffusion coefficients confirm that the oxide ions are hardly mobile in CeNbO4.33 phase (Fig. 6d).

The MD simulations show an increase of Ea for the oxide ion migration with the temperature for both CeNbO4.08 and CeNbO4.25 phases (Fig. 6d), in accordance with the results by Pramana et al.17. The low-temperature Ea values (0.27–0.41 eV) are close to those from the conductivity data of CeNbO4+δ (δ = 0.08, 0.25, 0.33; Supplementary Fig. 23), which did not show apparent Ea changes when placed under N2 flow and below 600 °C to maintain their oxygen contents (Supplementary Fig. 2). However, CeNbO4+δ displays mixed electronic and oxide ion conduction and it is hard to quantify the electronic and ionic contributions to the conductivity. Therefore, the calculated Ea from the MD simulations at low temperatures here cannot be compared directly with the experimental values from these conductivity measurements. Owing to the instability of the CeNbO4+δ phases to lose the extra oxygen at temperatures above 600 °C even in the O2-rich atmosphere18,20, there is no accurate Ea in higher temperature region for direct comparison with the calculated high-temperature Ea value for each phase. However, the high-temperature Ea values for CeNbO4.08 (0.84 eV) and CeNbO4.25 (0.93 eV) are close to the experimental Ea (~0.99 eV) for oxide diffusion in CeNbO4+δ from the 18O tracer diffusion measurements by Packer et al.22, further validating our MD simulations. The increase of Ea with the temperature may be explained by the fact that the concentration of mobile oxide anions in CeNbO4+δ could depend on the temperature. At low temperatures, the loosely bonded oxide ion could be the major mobile oxide ion, while at higher temperatures the more strongly bonded oxide ions start to move and contribute to the oxide ion migration, which requires extra energy and therefore increases the total Ea.

In nonstoichiometric CeNbO4+δ, the incorporation of extra oxygen atoms is coupled by the oxidation of Ce3+ to smaller Ce4+ and the interstitial oxide ion migrations in the scheelite-based CeNbO4.08 and CeNbO4.25 take place through a synergic mechanism of breaking and reformation of the Nb2O9 dimers, which is possible thanks to the spatial proximity of the tetrahedral units. This is akin to the continuous breaking and reformation of the tetrahedral dimers in the vacancy-mediated oxide ion conductors based on isolated tetrahedral anion structures, e.g., La1–xBa1+xGaO4-0.5x31 and Bi1–xSrxVO4–0.5x14. However, in these two oxygen-vacancy conducting materials, the tetrahedral-dimer-assisted oxide ion migration does not modify the coordination number, while in the case of CeNbO4+δ the coordination number in the polyhedral units involved in the oxide ion migration changes. The oxide ion migration mechanism in CeNbO4+δ revealed here thus emphasizes the key roles of the coordination-number-variable cations as well as rotation and deformation flexibility of 4/5-coordinated polyhedral units for the oxide ion migration. This is generally consistent with the previous findings in interstitial oxide ion conducting apatite31 and melilite-type materials10. Therefore, the structures containing cations forming polyhedral units with rotation, deformation, and coordination flexibility, e.g., Ga, Ti, V, Nb, Mo, and W, are promising candidates for new oxide ion conductors if, at the same time, flexible oxidation state or donor substitution with smaller cations allows the introduction of excess oxide ions.

Discussion

In summary, atomic structures of three oxygen hyperstoichiometric materials (CeNbO4.08, CeNbO4.25, CeNbO4.33) were determined by combining data from 3D ED, SPD, and NPD. The superstructure of CeNbO4.33 and the (3 + 1)D incommensurately modulated structure of CeNbO4.08 were obtained for the first time, to the best of our knowledge. The interstitial sites Oi were identified in all the three compounds and the structure analysis elucidates how the structure adapts for the oxygen hyperstoichiometry change, advancing our understanding of the complex CeNbO4+δ system. Cationic size contraction of Ce upon the oxidation allows not only the incorporation of excess oxygen into the host lattice of CeNbO4 but also the relaxation of the NbOn polyhedra and their interconnection through mixed corner/edge-sharing in three dimensions. MD simulations show that, with the inclusion of the excess oxygen into the host lattice of CeNbO4, the oxide ions become mobile in CeNbO4.08 and CeNbO4.25 with coordination-number-variable network but hardly migrate in the CeNbO4.33 phase owing to the ordered distribution of the excess oxide ions and the 6-coordinated [Nb6O26] polyhedral chain network with constrained deformation and rotation. Two kinds of oxide ion migration events are identified in CeNbO4.08 and CeNbO4.25 involving the interstitial Oi sites: (i) migration between the NbOn polyhedra isolated by Ce cations; (ii) migration between neighboring NbOn polyhedra. These two processes together form a long-range 3D network of migration pathways through which the oxygen ions migrate via a synergic-cooperation knock-on mechanism involving the continuous breaking and reformation of the Nb2O9 units assisted by the polyhedral rotation and deformation. The relationship between the structure and oxide ion migration for the whole series of CeNbO4+δ compounds here provides means to optimize the performance of these compounds and to develop better oxygen hyperstoichiometric materials for a wide variety of applications.

Methods

Materials

The parent material CeNbO4 was prepared by traditional solid-state reaction. The powders of CeO2 and Nb2O5 were mixed and homogenized through grinding with an agate mortar and a pestle. The mixtures were annealed at 1623 K for 10 h and then sintered at 1073 K for 10 h in flowing argon gas atmosphere. The yellow colored powder, pure phase of CeNbO4, was finally obtained. The oxidized phases CeNbO4.08, CeNbO4.25, and CeNbO4.33 were synthesized as described in Supplementary Fig. 1. In short, the CeNbO4.08 phase was obtained by heating CeNbO4 in air at 1123 K for 15 min, then quenching in the furnace to 948 K, holding at 948 K for another 20 min, and finally quenching to room temperature in air. CeNbO4.25 was prepared by heating CeNbO4 at 873 K in air for 24 h and then cooling in furnace to room temperature. The CeNbO4.33 was prepared by heating CeNbO4 at 673 K in flowing oxygen atmosphere for 72 h and then cooling to room temperature under oxygen atmosphere. The powder was reground and then reheated under the same conditions until the powder was in a single crystalline phase (Schematic representation of the synthesis of CeNbO4+δ is shown in Supplementary Fig. 1).

Characterizations

3D ED data was collected on a 200 kV JEOL JEM-2100 transmission electron microscope. The goniometer was continuously rotated while SAED patterns were simultaneously captured from crystals with the sizes ranging from 100 to 500 nm using the quad hybrid pixel detector (Timepix). The 3D ED datasets were then processed using the X-ray Detector Software package32, which can export the results as an hkl list. TGA was performed on a TGA-Q500 from room temperature to 1000 °C with a heating rate of 10 °C min−1 under flowing N2 atmosphere. The AC impedance spectroscopy (IS) measurements were performed with a Solartron 1260 frequency response analyzer over the 10−1–10−7 Hz frequency range within the 100–700 °C temperature range under N2 atmosphere flows. Prior to the IS measurements, the platinum paste was coated on the opposite faces of the pellets to form electrodes. Conventional PXRD pattern for parent phase CeNbO4 was collected at room temperature on a PANalytical X’Pert Pro diffractometer in Debye–Scherrer geometry with Cu Kα1 radiation with a minimum full width half maximum of 0.028°. NPD data were collected on CeNbO4+δ at ambient temperature over the 10–120° 2θ range at 2θ intervals of 0.05° on the 3T2 diffractometer at Laboratoire Leon Brillouin (France) using wavelength λ = 1.22997 Å (CeNbO4.08 and CeNbO4.33) and λ = 1.54 Å (CeNbO4.25). High-intensity and high-resolution SPD data were recorded on CeNbO4+δ (δ = 0.08, 0.25 and 0.33) on the 11BM diffractometer at the Advanced Photon Source, Argonne National Laboratory. SPD data were collected over the 0.5–40° 2θ range with a 0.001° step size at room temperature with 0.3 mm sample capillary using λ = 0.4130370 Å. The structure models derived from the 3D ED data were used as initial models for Rietveld refinements by combining SPD with NPD using Jana 200633 for (3 + 1)D incommensurately modulated structure of CeNbO4.08 and the software Topas Academic Version 534 for CeNbO4.25 and CeNbO4.33.

MD simulations

The oxide ion migration in CeNbO4+δ was investigated through MD atomistic simulations based on interatomic potential approach with the DL_POLY code35,36. The Buckingham potential function37 was used to model interactions between ions and the shell model38 to describe the electronic polarizability. The interatomic potential parameters, which were used in the previous atomistic simulations of CeNbO4 and CeNbO4.25 by Pramana et al.17, were slightly modified (Supplementary Table 4), especially regarding the Ce4+-O2− potential parameters39 for better reproduction of all experimental CeNbO4+δ structures (Supplementary Table 5) by the General Utility Lattice Program (GULP)39,40. The lattice parameters and most of the bond lengths were reproduced within ±6% error except for 2–4 bond lengths in each oxidized phase showing relatively large discrepancies (±10–20%) from the experimental values. The MD simulations were performed for the whole series of CeNbO4+δ using the updated interatomic potential parameters. The simulation box consisted of a 6 × 3 × 6 supercell containing 2592 atoms for the parent CeNbO4, a 1 × 3 × 1 supercell containing 2112 atoms for CeNbO4.08, a 3 × 2 × 4 supercell containing 7200 atoms for the CeNbO4.25, and a 5 × 5 × 3 supercell containing 2850 atoms for CeNbO4.33. The systems were equilibrated first under a constant pressure of 1 atm at specific temperatures within 1273–2673 K for 105 time steps with a time step of 0.1 fs before carrying out the main MD simulation for 200 ps with 2 × 106 time steps in the NVT ensemble. The Visual Molecular Dynamics package41 was used to perform MD data analysis and the MSDs were calculated with the nMoldyn3 code42. Oxygen diffusion coefficients were calculated from the slope of the MSD plots as a function of simulation time.