Introduction

Perovskite oxides (ABO3) can accommodate different types of cations on A- or B-sites, thereby having great tunability in oxygen vacancy formation1,2, which can benefit various applications such as solar thermochemical hydrogen generation3,4,5, solid oxide fuel cells6, and lithium-ion batteries7. In the solar thermochemical hydrogen (STCH) process, hydrogen can be produced by the two-step thermochemical redox cycle as follows8,9,10:

$$TR:AB{O}_{3-{\delta }_{GS}}\to AB{O}_{3-{\delta }_{TR}}+\frac{\Delta \delta }{2}{O}_{2}(g)$$
(1)
$$GS:AB{O}_{3-{\delta }_{TR}}+\Delta \delta {H}_{2}O(g)\to AB{O}_{3-{\delta }_{GS}}+\Delta \delta {H}_{2}(g)$$
(2)

where δTR and δGS are the oxygen non-stoichiometry at the thermal reduction (TR) condition and the gas splitting/oxidation (GS) condition in ABO3−δ, respectively. The Δδ (=δTR − δGS) is the difference in δ between the TR and GS conditions. Thermodynamically, it sets the oxygen exchange capacity of the material11 and identifies whether water splitting is possible or not12. For a positive Δδ, it can represent the ideal molar amount of H2 produced per mole of perovskite oxide and water13. For this thermochemical reaction, thermodynamic properties are more important than kinetic properties at high temperatures and operational conditions8,14,15,16 and the kinetic factors become dominant when the water splitting temperature is reduced16. Thus, even though H2 yield is controlled by both thermodynamics and kinetics, an accurate estimation of δ is required for computational material design.

Perovskite oxides with mixed cations provide the opportunity to optimize the STCH process by tuning the oxygen vacancy formation17,18. Previous studies for STCH applications have been mostly limited to the binary mixing of cations on the A- and B-sites1,19,20,21 or on either the A- or B-site individually22,23,24. Recent advancements in high-entropy perovskite oxides, which contain five or more principal elements on either the A-site or the B-site, have introduced more tunability for various applications25,26,27,28. Assuming equimolar mixing, the cation mixing entropy increases from 0.7kB (kB is Boltzmann constant) to 1.4kB when the number of elements increases from two to four types. This contribution will help to stabilize the mixed structure against phase segregation, especially at higher temperatures during STCH, however, its impact on δ and H2 generation has not been addressed. High entropy mixing will certainly impose structural distortion and variation on the charge states of multivalent lattice ions that determine the oxygen non-stoichiometry. Thus, it is worthwhile to systematically investigate how structural and oxygen non-stoichiometry change with increasing types of cations on one site, in order to take advantage of the tunability.

In this paper, we systematically compared equimolar binary and quaternary A-site mixed AFeO3 systems. Namely, three binary mixed A-site perovskite oxides (Nd0.5La0.5FeO3 (NLFO), Sr0.5La0.5FeO3 (SLFO), and Ba0.5La0.5FeO3 (BLFO)) and one quaternary mixed A-site (Nd0.25Sr0.25Ba0.25La0.25FeO3 (NSBLFO)) perovskite oxides were used as model systems. NLFO consists of two different A3+ cations while SLFO, BLFO, and NSBLFO represent the mixing of 50% of A2+ and 50% of A3+ cations. Many previous density functional theory (DFT) calculations on the oxygen vacancy formation energy (\({E}_{V}^{f}\)) often used ordered arrangements of cations in the mixed A-/B-site perovskite oxides21,22,29,30,31, although randomly distributed cations have been experimentally found in the mixed cation perovskite oxides, especially at high-temperature32,33. Furthermore, Energy Dispersive X-Ray Spectroscopy typically revealed compositional homogeneity in high entropy oxides28,34,35,36,37,38. For systematical comparison, randomly distributed, or disordered, A-site cations in these systems will be modeled. The changes of \({E}_{V}^{f}\), which is associated with the atomic structures, were compared by increasing the number of elements (from 2 to 4) on the A-site while keeping the ratio of A2+ to A3+ the same. This systematical comparison, especially the structural and property changes caused by the increasing number of elements on the A-site, will serve as guidance for investigating oxygen vacancies in high entropy oxides with even more cation elements (e.g., 5 or 6).

Modeling the STCH performance requires the accurate prediction of δ under different operating conditions based on DFT computed \({E}_{V}^{f}\), which faces several specific challenges in high entropy oxides. First, mixing cations with different sizes and charge states will cause local bond distortions. Since oxygen vacancy formation is sensitive to the local environment, the randomly mixed cations on the A-site would likely cause a spread of \({E}_{V}^{f}\) values, as opposed to a single value in a simple lattice, where all the oxygen vacancy sites are equivalent. Therefore, all individual oxygen vacancy sites need to be sampled in DFT calculations and the statistical distribution \({E}_{V}^{f}\) will be investigated. While most of the previous models to predict δ based on DFT computed \({E}_{V}^{f}\) assumed equivalent oxygen sites and utilized a constant \({E}_{V}^{f}\) with no spreading1,21,22,39,40,41, a statistic model is required for high entropy materials. For high-entropy alloys, recent studies have started to use configuration statistics. Daigle and Brenner42 applied traditional Boltzmann statistics to each site configuration and summed all the configurations to calculate the total vacancy concentration42. Recently, Chae et al.43 simplified this approach and only used the configurations of the six nearest cations around an oxygen vacancy for high-entropy metal oxides with a rock-salt structure43. This method considered the non-equivalent oxygen vacancy formation energy due to local cation arrangement but may be too short range. In this study, a statistic model was derived focusing on the oxygen vacancy formation energy statistics in mixed cation systems. We will refer to this model as the high entropy (HE) model and the traditional equal oxygen site model as the low entropy (LE) model. Here the entropy refers to the cation mixing entropy and its impact on the \({E}_{V}^{f}\) distribution and ultimately the δ, instead of the oxygen vacancy formation entropy (See Methods Section).

Secondly, a high oxygen vacancy concentration at the TR condition (high temperature and low oxygen partial pressure) is desired for better STCH performance. This may lead to oxygen vacancy interaction, characterized by higher oxygen vacancy formation energy with increasing oxygen nonstoichiometry, such as in SrFe0.75Mo0.25O3−δ44, which is more complicated due to B site mixing and competition of different d-electron configurations45. For simpler A-site mixed ferrite, the oxygen vacancy interaction has also been investigated previously in LaxSr1−xFeO3−δ (x=0, 0.5, and 1)31,46. The oxygen vacancy repulsion is due to the excess electrons left on the lattice upon oxygen vacancy formation. The excess electron distribution can be either localized at the first two nearest neighbor Fe atoms (e.g., LaFeO331,45) or delocalized (e.g., in La0.5Sr0.5FeO3 and in SrFeO346), depending on the cations in the systems. The excess electrons and the local bond distortion associated with an oxygen vacancy are defined as oxygen vacancy polaron. Das et al. have previously correlated the onset of the oxygen vacancy interaction with the vacancy distance approaching the polaron size. Specifically, the oxygen vacancy polaron size could be as large as ~ 8.3 Å in ferrite perovskites depending on the type of A-site cations and the crystal structures, and that the oxygen vacancies start to interact when δ > 0.0531,46. Thus, the non-interacting and dilute limit approximation, often used in past modeling efforts1,22,40,41, could lead to inaccurate vacancy concentrations at the TR condition, where the δ can readily exceed the dilute limit. Recently, several methods were proposed to predict δ under non-dilute limitations by using concentration-dependent vacancy formation free energy46, large sampling47, and shifting the Fermi level due to defect concentrations48. In the current work, we simplified the vacancy interaction by a linear interaction model to capture the concentration-dependent oxygen vacancy formation energy, \({E}_{V}^{f}(\delta )={E}_{V}^{f,d}+\alpha \cdot \delta\), where \({E}_{V}^{f,d}\) is the vacancy formation for the dilute system, and α is a constant. To obtain \({E}_{V}^{f,d}\), we choose a simulation cell dimension of 11 × 11 × 16 Å (shown in Fig. 1a), which is beyond the vacancy interaction distance in all three directions. The simulation cell contains 160 atoms, so the oxygen vacancy concentration is about 1% and the corresponding δ is 0.03, well below the dilute limit (according to previous studies in ordered LaxSr1−xFeO3). The interacting coefficient α was obtained from fitting DFT results, it can be compared with experimental results16. This linear approach was used to predict high temperature δ for SrFeO3−δ46. Overall, four models (LE-Dil, LE-Int, HE-Dil, and HE-Int), listed in Table 1, will be compared with experimentally measured δ and Δδ values to calibrate the validity of these models and their assumptions (See “Methods” section for details).

Fig. 1: Structural characterization.
figure 1

a Representative of the simulation cell at dimensions of ~11 × 11 × 16 Å (lattice parameters listed in Supplementary Table 4)b cell distortion factor Dc for different compositions, c representative of Fe–O bond length (l), O–Fe–O bond angle (θO−Fe−O), and Fe–O–Fe bond angle (θFe–O–Fe), d distribution of the octahedral distortion (Do), e distribution of bond angle variance of θO–Fe–O (Σ2), and f distribution of θFe–O–Fe. In d, e, and f, the interquartile range (difference between the top and bottom of the box) and the max-min range (difference between maximum and minimum of the extended vertical lines) are used as a measure of distribution to show how much spread out the data set is except for outliers (hollow circles). The red line and the blue triangle indicate the median and mean, respectively. More details about the boxplot are in Supplementary Fig. 4.

Full size image
Table 1 A comparison of four vacancy concentration models based on oxygen vacancy formation enthalpy (\(\Delta {H}_{V}^{f}\)) which was obtained from DFT-computed vacancy formation energies in dilute systems (\({E}_{V}^{f}\)) and the thermodynamic correction for oxygen.
Full size table

Results

Structural characterization of randomly mixed perovskites

The effects of mixed A-site cations on structural properties were first investigated. Figure 1a shows a representative simulation cell with 32 FeO6 octahedra highlighted in brown (Fe) with red corners (O), the four colored sites are A-sites. The mixing can induce different behaviors of global and local structural distortions49. The global distortion was described by the cell distortion factor Dc, which quantifies the lattice distortion from a cubic perovskite: \({D}_{c}=\frac{{(a/\sqrt{2}-{a}_{p})}^{2}+{(b/\sqrt{2}-{a}_{p})}^{2}+{(c/2-{a}_{p})}^{2}}{3{a}_{p}^{2}}\times 1{0}^{4}\) where ap is a pseudocubic parameter, calculated as \({a}_{p}=(a/\sqrt{2}+b/\sqrt{2}+c/2)/3\)50, where a, b, c are relaxed cell lengths (listed in Supplementary Table 4). The local distortions were estimated by the distortion of one FeO6 octahedron and the tilting between two octahedrons, as shown in Fig. 1c. Three local octahedral distortion parameters were defined. The octahedral bond distortion (Do) was calculated as \({D}_{o}=1/6\Sigma {\{({l}_{i}-{l}_{avg})/{l}_{avg}\}}^{2}\) where li is an individual Fe–O bond length in FeO6 and lavg is the average bond length in an individual FeO6 octahedron51,52. The octahedral angle distortion was analyzed by the bond angle variance (Σ2) using θO−Fe−O: \({\Sigma }^{2}=\mathop{\sum }\nolimits_{i = 1}^{12}{({\theta }_{O-Fe-O,i}-90)}^{2}/11\) where θO−Fe−O,i is the individual O–Fe–O angle in each octahedron53. Also, the local inter-octahedral tilting was evaluated using the θFe−O−Fe and the deviation of θFe−O−Fe from the ideal cubic (180 °) was analyzed (180° − θFe−O−Fe).

Figure 1b shows a comparison of Dc (Dc = 0 for the ideal cubic structure) for single component, binary, and quaternary mixing systems. For the single component systems (Supplementary Table 2), it has been known that LFO54 and NFO55 have orthorhombic structures while SFO56 and BFO57 have cubic structures as listed in Supplementary Table 2. Accordingly, Dc of the orthorhombic LFO (Dc = 0.23) and NFO (Dc = 2.12) deviate from 0 while the cubic SFO and BFO have Dc = 0. For all systems with mixed A-site cations, Dc deviates from cubic. In binary mixed systems, Dc is close to the compositional average. Specifically, the lattice of NLFO shows 1.25 of Dc, which is slightly greater than a simple compositional average (1.18). The Dc for SLFO and BLFO are 0.16 and 0.15, respectively, indicating that they cannot maintain the perfect ideal cubic structure, and their Dc values are also slightly greater than their compositional averages (0.13 for both). For the quaternary mixing system, the cell distortion factor is further reduced from the binary systems to Dc = 0.10 which is much lower than the compositional average (0.59) and is very close to the Dc of the ideal cubic structure. There is a significant change by mixing four different cations on the A-site. The global structural distortion could be minimized and the structure of NSBLFO becomes closer to a cubic structure which is known to be the most stable perovskite structure. This result is aligned with the experimental observation showing high-entropy mixing stabilized the cubic phase in fluorite oxides58.

The local distortion, especially its distribution is visualized by boxplots, in Fig. 1d–f. The detailed description of a typical boxplot is in Supplementary Fig. 4. More specifically, the interquartile range (difference between upper and lower quartile) and the max-min range (difference between maximum and minimum) are used as a measure of distribution to show how spread out the data set is except for outliers. In Fig. 1d–f, the boxplots also show the median (red line), mean (blue triangle), and the outliers (empty circles) of a distribution. For the binary systems, the magnitude of the mixing effect on the local distortion varies by the type of mixing cations, especially their charge states and radius. For the NLFO, the mixing of La and Nd on the A-site makes a trivial effect on the local environments in the NLFO since Nd and La are alike. The radii of Nd3+ and La3+ are comparable (Nd3+ is 6.6% smaller than La3+ as listed in Supplementary Table 1) and their single component perovskite (LFO and NFO) has an orthorhombic structure. The average Do and Σ2 are respectively 0.5 × 10−9 and 1.8, which means most Fe cations are placed at the center of the FeO6 octahedron and the octahedrons are almost undistorted. But the inter-octahedral tilting (180° − θFe−O−Fe) is maximized by mixing La and Nd cations, which agrees with the fact that the NLFO has the largest cell distortion factor Dc due to its orthorhombic nature. The distributions of all three parameters are narrower than those of other systems, implying the octahedron-to-octahedron variation is smaller. On the contrary, the BLFO shows the widest distribution among the binary systems in all three distortion parameters. That is because Ba and La are dissimilar elements in terms of size and charge state. Ba2+ is 18.4% larger than La3+ and the charge state on Fe can be either 3+ or 4+ whose radius has a 10% difference due to the different charge state on the A-site (2+ vs. 3+) (Supplementary Table 1). Also, BFO and LFO have different crystal structures. Because of these dissimilarities between Ba and La, the local environments have a large variation shown as the wide distribution. The large Do and Σ2 show that the FeO6 octahedra in BLFO are considerably distorted. However, the inter-octahedral tilting of BLFO is minimal and close to that of the ideal cubic structure possibly because the large Ba2+ ions expand the overall lattice and give room to the FeO6 octahedra so that they could be well aligned. For the SLFO, the mixing effect is intermediate. The SLFO shows similar but less local distortion than the BLFO because Sr2+ is only 5.9% larger than La3+.

For the quaternary NSBLFO system, the distributions of all local distortion parameters are wider than the binary systems indicating each FeO6 octahedron has its own distinct local environment and the local distortion shows more variation in the quaternary system. The large average value and the wide distribution of the Do and Σ2 support the fact that the four different cations on the A-site disturb both bond length and bond angles of the FeO6 octahedra more randomly and largely. The average inter-octahedral tilting angle of NSBLFO is slightly more away from the ideal cubic than that of BLFO but less away compared to the LFO and NFO systems. The FeO6 octahedra are relatively well aligned and the inter-octahedral tilting angles have the largest variation in the NSBLFO due to the mixing of four elements. Also, the ranges of distributions of BLFO and NSBLFO are similar. This implies the effect of Ba mixing regardless of the amount of Ba is significant on the local distortion compared to the mixing of Nd or Sr due to the large Ba cation size.

Overall, the quaternary mixed NSBLFO shows the most cubic-like structure on the global scale and the largest variance of local distortions. Typical high entropy alloys often emphasize more severe lattice distortions, probably due to their simple lattice structures. Oxides have multiple sublattices for cations and anions. The effects of size mismatch and different charge states of the four elements on the A-site sublattice cause significant local distortions, but in turn, the wide range of local distortions eventually relax the cell lattice over long range. At the macroscopic scale, the NSBLFO system has the most cubic-like lattice structure, a similar trend that has been seen in some fluorite oxides as well58. The local distortion will impact local properties, such as oxygen vacancy formation or some catalytic behaviors, while the global cubic structure may contribute to the overall stability of the structure; highlighting the benefits of high entropy mixing.

Oxygen vacancy formation energy and its distribution

For systems with mixed cations, a first-order approximation is to take a compositional average of the DFT computed vacancy formation energies (\({E}_{V}^{f}\)) of single component systems. Therefore, \({E}_{V}^{f}\) for the constituent single component systems: AFeO3 (A: Nd, La, Sr, and Ba) was first calculated as shown in Fig. 2a. AFeO3 with trivalent A-site cations (A being Nd or La) shows 3 eV larger \({E}_{V}^{f}\) compared to that with divalent cations (A being Sr or Ba), but the variation within the same type of valency is not pronounced (<1 eV). This indicates that the \({E}_{V}^{f}\) of AFeO3 is mainly controlled by the charge state on A-site cations, not by the type or by the size of the cations because the charge state on A-site determines the charge state on Fe and its electron configuration. The \({E}_{V}^{f}\) for a dilute system is mainly affected by the reducibility of the Fe cation in AFeO3, which is determined by its electron configuration. In a simplified picture with classical charges, Fe3+ (3d5 electron configuration) is harder to be reduced owing to its half-filled d-orbitals, thereby showing the larger \({E}_{V}^{f}\). In contrast, Fe4+ (3d4 electron configuration) is highly reducible to the effect that AFeO3 with divalent cations has a lower \({E}_{V}^{f}\). The full picture of the DFT computed oxygen polaron size and shape for all the A cations are similar to those in the LaFeO331,45 and in SrFeO346, while others can be interpreted similarly to LaxSr1−xFeO346. \({E}_{V}^{f}\) values from sampling structures with randomly mixed A-site cations are visualized in histogram form (Fig. 2b, c, e and f). The mean and distribution of \({E}_{V}^{f}\) values obtained from sampling are also depicted as a box plot in Fig. 2b, in comparison with the simple compositional average. The simple compositional average of four systems falls within the max-min range of boxplots (the range between the upper and lower whisker ends). However, the simple compositional average and the median value (the central line in box plots) show discrepancies that can be as large as 0.72 eV. Thus, the simple compositional average \({E}_{V}^{f}\) from the single component systems does not necessarily represent the average \({E}_{V}^{f}\) of the mixed systems, and using it could lead to an inaccurate estimate of related properties (e.g, δ). The box plot in Fig. 2b also shows a broader distribution of \({E}_{V}^{f}\) when moving from binary SLFO and BLFO systems to quaternary NSBLFO, consistent with the broader local structure variation.

Fig. 2: Oxygen formation energy.
figure 2

a Single neutral oxygen vacancy formation energy (\({E}_{V}^{f}\)) in the constituent single component AFeO3 systems and d the composition weighted average \({E}_{V}^{f}\) for the binary/quaternary mixed systems in circle markers and the spread of individual \({E}_{V}^{f}\) in the typical box plot (see Supplementary Fig. 4). The histogram plots of oxygen vacancy formation energies for b NLFO, c BLFO, e SLFO, and f NSBLFO along with the fitted Gaussian distribution (shown in a red curve).

Full size image

The histogram distribution of \({E}_{V}^{f}\) was fit with a Gaussian distribution function as shown in red lines in Fig. 2, from which the average (\(\Delta {\bar{E}}_{V}^{f}\)) and standard deviation (σ) were obtained as annotated in Fig. 2b–f. Here, SLFO, BLFO, and NSBLFO systems will be compared due to the difference between them and the NLFO system. For SLFO, BLFO, and NSBLFO, the ratio of divalent and trivalent cations are the same, but they showed different average \({E}_{V}^{f}\) and distributions. Also, the cell structure in these systems is similar to cubic structures. However, NLFO consists of two types of A3+ cations and its cell structure is similar to the orthorhombic structure (Fig. 1), thus it cannot be directly compared to the other three systems. For SLFO, BLFO, and NSBLFO, the larger variation in local distortion (Fig. 1d–f) results in a broader distribution (larger σ values). As the degree of local distortion spreads out, oxygen sites become diverse in terms of \({E}_{V}^{f}\), and σ increases from binary SLFO and BLFO systems to quaternary NSBLFO.

Oxygen vacancy formation energy and local structures

Since Fig. 2a shows \({E}_{V}^{f}\) strongly depends on the A-site charge state in a single component AFeO3, agreeing on numerous previous studies on these single or binary A-site mixed systems, such as LaFeO331,45, SrFeO346, and LaxSr1−xFeO346, it is natural to explore if the local chemical composition and bond lengths will affect the distribution of \({E}_{V}^{f}\). The distribution of \({E}_{V}^{f}\) is likely to be correct, as to be shown in Section 2.4, they lead to predicted δ and Δδ values agreeing with experiments. Each O site bridges two FeO6 octahedrons, in which each Fe cation occupies the body center of 8 A-sites; thus, the O sites can be affected by the neighboring 12 A-site cations. Figure 3a–d shows that \({E}_{V}^{f}\) distribution is not strongly correlated with the number of divalent cations among those neighboring 12 A sites. For the systems having divalent cations (SLFO, BLFO, and NSBLFO), \({E}_{V}^{f}\) values spread out regardless of the number of divalent cations near the oxygen vacancy site. The boxplots in Fig. 3b–d show many outliers (black circles), implying \({E}_{V}^{f}\) values are not strongly tied to the number of neighboring divalent cations. For NLFO, two types of trivalent cations are mixed and its orthorhombic-like structure made the spreading of \({E}_{V}^{f}\) even though it does not include divalent cations.

Fig. 3: Correlation of oxygen vacancy formation energy with local structures.
figure 3

Effect of mixing A-site cations on \({E}_{V}^{f}\) for a NLFO, b BLFO, c SLFO, and d NSBLFO. The black squares indicate \({E}_{V}^{f}\) in the single component system and the boxplots show the minimum, lower quartile, median (red line), upper quartile, maximum, outlier (empty circle), and mean (blue triangle) of a distribution. The oxygen vacancy formation energy versus two Fe-O bond lengths associated with each oxygen vacancy site for e NLFO, f BLFO, g SLFO, and h NSBLFO.

Full size image

Additionally, the analysis of Fe–O bond lengths was conducted for all oxygen sites in the simulation cells since it has been found that the \({E}_{V}^{f}\) can be affected by strain in oxides59,60. In the mixed A-site perovskites, two Fe–O bond lengths connecting to the Vo site do not correlate with \({E}_{V}^{f}\). Figure 3e–h shows the \({E}_{V}^{f}\) value of all oxygen sites and the two corresponding Fe–O bond lengths. The color scheme indicates the \({E}_{V}^{f}\) values, and \({E}_{V}^{f}\) values do not show any strong correlation with the Fe–O bond lengths; rather, they are randomly distributed. In the systems with randomly mixed A-site cations, our results suggest that there is no preference for \({E}_{V}^{f}\) on the local chemical composition and bond lengths. It indicates that the random mixing of A-site cations leads to the spreading of \({E}_{V}^{f}\) into a distribution function.

Prediction of δ and Δδ

To validate the four models listed in Table 1, we compared their predicted δ (Fig. 4) and Δδ (Fig. 5) with experimental dry TGA (no hydrogen or water vapor in testing) measurements first. In the dry TGA experiment, the TR condition was 1350 °C with \({P}_{{O}_{2}}=1{0}^{-6}\) atm and the GS condition was \({P}_{{O}_{2}}=0.2\) atm with varying temperatures. It is noted that the experimentally measured Δδdry should be more accurate than the absolute δ.

Fig. 4: Model prediction of oxygen vacancy nonstoichiometry.
figure 4

Comparison of \({\delta }_{GS}^{dry}\) and δTR between models (LE-Int and HE-Int) and experiments. The TR condition is 1350 °C with \({P}_{{O}_{2}}=1{0}^{-6}\) atm and the dry GS condition is \({P}_{{O}_{2}}=0.2\) atm with varying temperatures. a–d Show δTR at the TR condition and \({\delta }_{GS}^{dry}\) under a range of TGS. The inserted figure in a is a zoomed-in view.

Full size image
Fig. 5: Model prediction of oxygen nonstoichiometry change in STCH cycles.
figure 5

a–d Shows a comparison of Δδdry between models (LE-Int and HE-Int) and experiments in each system. The Δδdry is the δ difference between the TR and dry GS condition (\(\Delta {\delta }_{dry}={\delta }_{TR}-{\delta }_{GS}^{dry}\)). The TR condition is 1350 °C with \({P}_{{O}_{2}}=1{0}^{-6}\) atm and the dry GS condition is \({P}_{{O}_{2}}=0.2\) atm with varying temperatures. The inserted figure in a is a zoomed-in view.

Full size image

Figure 4 shows the predicted δ at the TR condition and different GS temperatures in comparison with experimental data. At GS conditions in the temperature range of 600 to 1200 °C, the HE-Int model predicted δ values that are closer to experiments for all three cases, especially for SLFO. The discrepancy in the HE-Int model could arise from the value of α, which was obtained using ordered La0.5Sr0.5FeO3−δ structures assuming uniform vacancy distance31. The degree of vacancy interaction varies by materials because the size of the oxygen vacancy polaron varies. The local distortion may lead to different polaron sizes and thus the variation of oxygen vacancy interactions and α. Thus, using a material-specific α would reduce the discrepancy in BLFO and NSBLFO. In addition, the HE-Int model overestimated δ in all three cases at the TR condition, while agreeing better with the experiment for NLFO. This overestimation originates from the assumption that an oxygen vacancy occurrence at each state is considered to be an independent event, in other words, the functional form of \(G({E}_{V,i}^{f})\) is independent of δ. However, the probability of finding an oxygen site with \({E}_{V,i}^{f}\) in a subsystem would decrease at higher δ due to vacancy interactions. This mechanism has not been implemented in the current HE-Int model and this effect can be exaggerated especially under very reducing conditions (low \({P}_{{O}_{2}}\) and high temperature) when the experimental δ is above 0.2 or the computed \({E}_{V}^{f}\) distribution has many low \({E}_{V}^{f}\) states.

The difference in δ between TR and dry GS condition (\({\delta }_{TR}-{\delta }_{GS}^{dry}\)) is expressed as Δδdry, which is one of the parameters to evaluate water splitting performance in STCH. Based on the accuracy in predicting δ, our recently developed statistical model (HE-Int) showed improved agreement with experimental observations, especially the temperature dependence, in terms of Δδdry. When the \({E}_{V}^{f}\) is high and oxygen vacancy concentration is low (~0.05), the differences between the four models are not obvious in NLFO (Supplementary Fig. 5(a)). The non-interacting vacancy models (LE-Dil and HE-Dil) all showed unrealistically high Δδdry (larger than 1.5) in SLFO, BLFO, and NSBLFO (Supplementary Fig. 5), suggesting vacancy interactions must be considered for STCH applications. Therefore, only the two interacting vacancy models (LE-Int and HE-Int) are shown in Fig. 5. For the HE-Int model, both the direct sampling and Gaussian fitting methods are shown to be comparable. The calculated Δδdry of the LE-Int and HE-Int models at various GS temperatures and experimental values are displayed in Fig. 5b–d. The LE-Int model can be higher or lower than the experimental data and showed almost no temperature dependence, thus it is not a reliable predictive model. The HE-Int model consistently showed higher Δδdry values than experiments and captured a similar temperature dependence trend. Therefore, the HE-Int model is more accurate and can be used to screen materials for STCH computationally.

Effects of distribution of \({E}_{V}^{f}\) on Δδ

The distribution of \({E}_{V}^{f}\) (therefore, \(\Delta {H}_{V}^{f}\)) showed several impacts on the oxygen vacancy generation and filling in STCH cycles. If all of the oxygen sites have the uniform \({E}_{V}^{f}\) value, oxygen atoms would be released without any sequence. However, the presence of the distribution of \({E}_{V}^{f}\) implies that the oxygen site with the lowest \({E}_{V}^{f}\) would be vacant first in the TR process and filled last in the appropriate GS process. Thus, the spreading of \({E}_{V}^{f}\) will enable some of the oxygen sites to be more easily utilized while the uniform \({E}_{V}^{f}\) provides the equal probability of vacancy formation for all. Therefore, the impacts of \({E}_{V}^{f}\) distribution on Δδ were investigated.

First, the HE-Int model with Gaussian distribution was used for the Δδdry analysis to systematically show the effect of σ (Fig. 6) since this model could predict the trend of Δδdry with a constant overestimation. The LE-Int model was used to show how Δδdry changes with \({E}_{V}^{f}\) and TGS with no spreading which is the same as σ = 0 in the HE-Int model with Gaussian (Fig. 6a, b). Figure 6b shows the change of Δδdry with \({E}_{V}^{f}\) at TGS = 850 °C annotated as a horizontal dashed line in Fig. 6a. As expected, the Δδdry (between the given dry TR and GS conditions) monotonically decreases as \({E}_{V}^{f}\) increases. The Δδdry shows the most tunability when \({E}_{V}^{f}\) is in the range of 1.5–3.5 eV, suggesting a target for perovskite composition design.

Fig. 6: Improving STCH performance by tuning the distribution of oxygen vacancy formation energy.
figure 6

LE-Int model: a Δδdry as a function of \({E}_{V}^{f}\) and TGS. A solid line indicates a positive contour line and the dashed horizontal line indicates TGS = 850 °C, and b the dashed curve represents Δδdry as a function of \({E}_{V}^{f}\) at TGS = 850 °C. HE-Int model: (c) Δδdry as a function of average \({E}_{V}^{f}\) and σ. A solid line indicates a positive contour line and the dashed vertical line indicates 2.60 eV of average \({E}_{V}^{f}\), d the dashed curve shows Δδdry as a function of σ at \({E}_{V}^{f}\) = 2.60 eV under the dry GS condition (\({P}_{{O}_{2}}\) = 0.2 atm), and e trends of Δδ as a function of σ at \({E}_{V}^{f}\) = 2.60 eV under wet condition at TGS = 850 °C with different \(\Delta {\mu }_{O}^{(GS,85{0}^{\circ }C)}\). f Comparison of Δδ between the HE and LE models in NSBLFO (average \({E}_{V}^{f}\) = 2.99 eV and σ = 0.64 eV) with varying \(\Delta {\mu }_{O}^{(GS,85{0}^{\circ }C)}\) and the vertical lines denotes H2 yield at different ΔμO.

Full size image

When the distribution of \({E}_{V}^{f}\) is taken into account, the HE-Int model with Gaussian shows that the Δδdry is simultaneously affected by the average \({E}_{V}^{f}\) and the degree of spreading of the distribution (σ) in Fig. 6c. Figure 6d shows how the Δδdry changes with varying σ when the average \({E}_{V}^{f}\) is set to 2.60 eV. The Δδdry first increases when increasing spreading (σ) and is maximized when σ is 0.68. In other words, in addition to tuning the average \({E}_{V}^{f}\) via cation mixing, the spreading of \({E}_{V}^{f}\) distribution (σ) further increases Δδdry and can be optimized for a maximum Δδdry. This can be viewed as one benefit provided by high entropy oxide design.

To further explore the benefits of increasing cation mixing entropy in STCH, Δδ at different ΔμO, which can be correlated with different equilibrium GS conditions, were plotted in Fig. 6e. The Δδ values are computed at different ΔμO with varying σ when the average \({E}_{V}^{f}\) is 2.60 eV. The ΔμO = −1.33 eV is the dry TGA condition (with \({P}_{{O}_{2}}\) = 0.2 atm). At an ideal wet GS condition of \({P}_{{H}_{2}O}\) = 0.4 atm and \({P}_{{H}_{2}}\) = 0.1 atm (leading to 25% H2 yield), the calculated ΔμO is −3.03 eV. It is smaller than −2.86 eV of \(\Delta {\mu }_{O}^{TR}\) at the TR condition, making water splitting thermodynamically challenging61,62,63, as indicated by the negative Δδ values (implying water splitting is not possible). Figure 6e shows that the gain in Δδ by the optimized spreading (σ) drops with decreasing ΔμO. The Δδ gain due to the spreading of the \({E}_{V}^{f}\), as shown in Fig. 6e, started to diminish at ΔμO = −2.26 eV, which corresponds to \({P}_{{H}_{2}O}\) = 0.4 atm and \({P}_{{H}_{2}}\approx 3.5\times 1{0}^{-5}\) atm (0.0088% yield). This transition point of ΔμO varies with both the average \({E}_{V}^{f}\) and the spreading (σ) of a distribution. For instance, when the average \({E}_{V}^{f}\) is 2.99 eV as seen in the case of NSBLFO, the transition from Δδ gain to no gain occurs at ΔμO = −2.63 eV, corresponding to \({P}_{{H}_{2}O}\) = 0.4 atm and \({P}_{{H}_{2}}\approx\) 0.0016 atm, meaning 0.4% yield (Supplementary Fig. 8). Figure 6f exemplifies NSBLFO to compare the HE-Int model with average \({E}_{V}^{f}\) of 2.99 eV and σ of 0.64 eV, and the LE-Int model assuming the same average \({E}_{V}^{f}\) as the HE-Int model but no spreading. The gain in Δδ due to the \({E}_{V}^{f}\) distribution reduces as ΔμO decreases until the crossover point around ΔμO ≈ -2.46 eV (\({P}_{{H}_{2}O}\) = 0.4 atm and \({P}_{{H}_{2}}\approx\) 0.00027 atm (0.068% yield)). Therefore, both the average \({E}_{V}^{f}\) and the σ should be designed to maximize the gain in Δδ for different water splitting conditions.

Discussion

STCH process is challenging because the oxygen chemical potential at the TR condition is larger than that at the GS condition if a significant H2 yield is considered61,62,63. If the oxygen vacancy formation enthalpy is considered as an average value, a positive oxygen vacancy formation entropy of considerable magnitude is required for successful STCH materials. However, the vibrational and magnetic contributions to the vacancy formation entropy are typically small and often neglected by modeling64,65,66. Lany suggested the electronic entropy contribution from charged defect formation can be significant and should be considered63. Here, we showed another mechanism that the distribution of oxygen formation energy (therefore enthalpy) can lead to some H2 production gain, especially when \({P}_{{H}_{2}}\)/\({P}_{{H}_{2}O}\) ratio is relatively low.

Our results showed the distribution refers to both the average and the spreading of the oxygen vacancy formation energy, which can be tuned by mixing more types of divalent and trivalent cations, and possibly at different ratios. As the number of A-site cations increases from the single component AFeO3 to the quarternary mixed component {A}FeO3, local bond distortion increases and the \({E}_{V}^{f}\) distribution (fitted Gaussian) showed different average values and a larger standard deviation (single: 0; binary: 0.51 and 0.56 eV; quaternary: 0.64 eV), meaning a broader distribution.

For predicting the Δδ, four different models were compared. Our results suggest that the \({E}_{V}^{f}\) distribution as well as the interactions of oxygen vacancies must be considered to increase the accuracy of Δδdry prediction. Our HE-Int model with direct sampling from DFT accurately predicts the experimental observations in terms of the Δδdry and the δ values under the dry and wet STCH conditions. Furthermore, the HE-Int was used to explore the dependence of Δδ at both dry and wet STCH cycles on the distribution of \({E}_{V}^{f}\). Mixing of divalent to trivalent cations can vary the average \({E}_{V}^{f}\), generally inducing larger Δδ. The spreading of \({E}_{V}^{f}\) by mixing more types of divalent and trivalent cations on A-site further increases Δδ, at a range of STCH conditions, especially when \({P}_{{H}_{2}}\)/\({P}_{{H}_{2}O}\) ratio is low.

We also have investigated the effects of mixing on the structural distortion and spreading of \({E}_{V}^{f}\) in perovskites with mixed A-site cations. As the number of A-site cations increases from the single component AFeO3 to the quarternary mixed component {A}FeO3, the global lattice distortion becomes smaller and more similar to a cubic structure while the local distortions tend to vary more. However, the \({E}_{V}^{f}\) distribution does not show any strong correlation with the local A-site charge distribution or the structural distortions. Rather, \({E}_{V}^{f}\) values show a random distribution.

By extrapolating these results to high entropy perovskites with A-site mixing, we can expect a globally more cubic but locally more distorted structure, with a broader tunability of \({E}_{V}^{f}\) distribution that can be optimized to achieve larger Δδ in the STCH process. Going beyond STCH applications, since the local distortions can also lead to enhanced reaction activities67, we can envision that high entropy perovskites show even higher catalytic reactivity in oxygen reduction reaction67 and solid oxide fuel cell applications38.

Methods

Randomly mixed structures

Random A-site cation distribution was generated by the special quasi-random structures (SQS) approach with the Alloy Theoretic Automated Toolkit (ATAT)68,69,70 for binary and quaternary mixed A-site cations in simulation cells with 160 atoms. This simulation cell dimension (~11 × 11 × 16 Å) is large enough71 to accommodate the oxygen vacancy polaron (with a maximum diameter of 8.3 Å in La0.5Sr0.5FeO346) and prevent their interactions. The pair correlation functions were identical to those of the theoretically random structures up to the nearest atomic shell with a distance of ~ 4 Å, indicating that A-sites in the periodic simulation cell structures are occupied by cations randomly. Due to a large possible configurational space of A-cation arrangements in binary/quaternary mixed systems and a large number of possible SQS simulation cells, three different A-cation arrangements were generated using the SQS method. To reveal the structural randomness, the A-site cation arrangements around Fe or O site were analyzed to ensure perfect random mixing of A2+ and A3+ cations in a longer range than the ~ 4 Å of distance used for the SQS. It was performed based on the hypothesis that the oxygen vacancy formation energy can be affected by the ratio of the number of nearby A2+ and A3+ cations and by the charge state of neighboring Fe sites which can be determined by nearby A cations. One Fe site (or FeO6 octahedron) is coordinated by eight A sites, and one O site is surrounded by 12 neighboring A sites. The number of A2+ cations centered by O or Fe sites was analyzed in Supplementary Fig. 3, which shows the distribution follows the Gaussian distribution. It confirms the cations were randomly displaced on A-sites in the selected structures.

These three configurations were applied to both the cubic and the orthorhombic initial structures which were chosen based on the experimentally known structure of the constituent single component ABO3: LFO54 and NFO55 being orthorhombic, and SFO56 and BFO57 being cubic. In the binary systems (NLFO, SLFO, and BLFO), three different La configurations were commonly used while varying the rest A-site cations for a fair comparison. Supplementary Fig. 1 shows all six configurations tested for BLFO, and Ba was replaced with either Nd or Sr for NLFO and SLFO, respectively. For the quarternary system, three different A-cation arrangements were used for cubic and orthorhombic structures, as shown in Supplementary Figure 2. After both ionic and cell relaxation, the energy difference (ΔE) among the six different initial structures is listed in Supplementary Table 3. The small energy difference (much less than the proposed criterion of ΔE < 40 meV atom−172 or <100 meV atom−173 for ABO3 compounds) suggests that the six different initial random structures for each system are stable and thermodynamically comparable. Therefore, the structure with the minimum energy after full relaxation was selected as the representative structure of each system for structure and vacancy analysis.

DFT combined with thermodynamics

In STCH processes, perovskite oxides are exposed to high temperatures and oxygen vacancy formation enthalpy \(\Delta {H}_{V}^{f}\) relies on the chemical potential of oxygen (ΔμO). Thus, we combined DFT and thermodynamic modeling. All DFT calculations for the 0 K ground state were performed using the Vienna Ab initio Simulation Package74 with the spin-polarized generalized gradient approximation along with Perdew, Burke, and Ernzerhof (GGA-PBE) and the GGA+U method. The PAW potentials used in this study are La (5s2 5p6 6s2 5d1 of valence electronic configuration), Sr_sv (4s2 4p6 5s2), Ba_sv (5s2 5p6 6s2), Nd_3 (4f1 5s2 5p6 6s2), Fe_pv (3p6 3d7 4s1), and O_s (2s2 2p4). Due to the large data sampling, the soft “O_s” pseudopotential was used as previously used in perovskite oxides75,76,77 with the cutoff energy of 380 eV that is ~ 34% higher than the default cutoff energy78,79. It leads to less than 0.1 ~ 0.2 eV difference in the similar oxygen vacancy formation energy when the simulation cell consists of two and 32 formula units, compared to our previous work on LaxSr1−xFeO3 with the standard potential of O and Fe and higher cutoff energy of 500 eV31,46. The total energy and ionic force convergence criteria are 10−6 eV and 0.02 eV Å−1 respectively. A Gamma-centered k-point mesh was set to 2 × 2 × 1 for the simulation cell, to ensure a k-spacing of 0.3–0.4 Å−1. Ueff = 3 was chosen based on our previous extensive U parameter tests31. This single U parameter was able to capture the magnetic moments for Fe at a broad range of charge states, which can vary between 2+ and 4+ due to the mixing of A2+ and A3+ cations and oxygen vacancies, and gave the correct band gap and phases for various LaxSr1−xFeO3 compositions and phases31.

The DFT computed formation energy of a single neutral vacancy (\({E}_{V}^{f}\)) is defined as \({E}_{V}^{f}={E}_{defective}-{E}_{bulk}+{\mu }_{O}^{FERE}\) where Edefective and Ebulk are the DFT calculated energy of a simulation cell with and without a vacancy (VO), respectively. All possible VO structures were sampled using Python Materials Genomics (pymatgen)80 and individual \({E}_{V}^{f}\) was calculated for all. \({\mu }_{O}^{FERE}\) is the fitted elemental-phase reference energy (FERE) of oxygen (-4.76 eV for an oxygen atom) which was developed for accurate prediction of enthalpies of formation81. The free energy of oxygen vacancy formation (\(\Delta {G}_{V}^{f}(T,{P}_{{O}_{2}})\)) governs the vacancy concentration and it is expressed as \(\Delta {G}_{V}^{f}(T,{P}_{{O}_{2}})=\Delta {H}_{V}^{f}(T,{P}_{{O}_{2}})-T\Delta {S}_{V}^{f}(T,{P}_{{O}_{2}})\) where \(\Delta {H}_{V}^{f}(T,{P}_{{O}_{2}})\) is the oxygen vacancy formation enthalpy and the \(\Delta {S}_{V}^{f}(T,{P}_{{O}_{2}})\) is the oxygen vacancy formation entropy. The vibrational entropic contributions to \(\Delta {S}_{V}^{f}(T,{P}_{{O}_{2}})\) are relatively small and neglected64,65,66. The dominating contribution to \(\Delta {S}_{V}^{f}(T,{P}_{{O}_{2}})\) becomes oxygen gas82, \(\Delta {S}_{V}^{f}(T,{P}_{{O}_{2}})\approx \frac{1}{2}\Delta {S}_{{O}_{2}}(T,{P}_{{O}_{2}})\).

At a testing condition, finite temperature (T) and oxygen partial pressure \({P}_{{O}_{2}}\), \(\Delta {\mu }_{O}(T,{P}_{{O}_{2}})\) were used as the thermodynamic correction for an isolated oxygen molecule, describing the chemical potential change from 0 K to T at \({P}_{{O}_{2}}\) including the entropy of oxygen gas (\(\Delta {S}_{{O}_{2}}(T,{P}_{{O}_{2}})\)) as explicitly discussed in Ref. 46. The oxygen vacancy formation enthalpy \(\Delta {H}_{V}^{f}(T,{P}_{{O}_{2}})\) was calculated as follows:

$$\begin{array}{rcl}\Delta {G}_{V}^{f}\left(T,{P}_{{O}_{2}}\right)&=&\Delta {H}_{V}^{f}\left(T,{P}_{{O}_{2}}\right)-T\left(\frac{1}{2}\Delta {S}_{{O}_{2}}\left(T,{P}_{{O}_{2}}\right)\right)\\ &=&{E}_{defective}\left(T,{P}_{{O}_{2}}\right)-{E}_{bulk}\left(T,{P}_{{O}_{2}}\right)+\frac{1}{2}{\mu }_{{O}_{2}}\left(T,{P}_{{O}_{2}}\right)-T\left(\frac{1}{2}\Delta {S}_{{O}_{2}}\left(T,{P}_{{O}_{2}}\right)\right)\\ &=&{E}_{defective}-{E}_{bulk}+{\mu }_{O}^{FERE}+\Delta {\mu }_{O}\left(T,{P}_{{O}_{2}}\right)\\ &=&{E}_{V}^{f}+\Delta {\mu }_{O}\left(T,{P}_{{O}_{2}}\right)\\ \end{array}$$
(3)

For systems with mixed cations, an average vacancy formation energy \({\bar{E}}_{V}^{f}\) was used for this calculation. In experiments, various testing conditions have been used for STCH performance analysis. When water steam is not involved in the GS reaction, \(\Delta {\mu }_{O}(T,{P}_{{O}_{2}})\) is solely determined by temperature and \({P}_{{O}_{2}}\), which will be referred to as the dry condition. For a dry condition, \(\Delta {\mu }_{O}(T,{P}_{{O}_{2}})=\frac{1}{2}(\Delta {\mu }_{{O}_{2}}^{0}(T)+{k}_{{{{\rm{B}}}}}T\ln {P}_{{O}_{2}})\) is determined by the change in chemical potential of oxygen from 0 K to T at a given \({P}_{{O}_{2}}\), and \(\Delta {\mu }_{{O}_{2}}^{0}(T)\) is calculated with the specific heat of oxygen gas at temperature T and the entropy of oxygen gas at the standard state46. In a wet GS condition where water steam is injected into a reactor, \({P}_{{O}_{2}}\) can be calculated by using \({P}_{{H}_{2}O}\) and \({P}_{{H}_{2}}\) following equation:

$$\begin{array}{rcl}\Delta {\mu }_{O}&=&\Delta {H}_{{H}_{2}O}^{f}+\Delta {\mu }_{{H}_{2}O}-2\Delta {\mu }_{H}\\ \Delta {\mu }_{{H}_{2}O}&=&\Delta {\mu }_{{H}_{2}O}^{0}(T)+{k}_{{{{\rm{B}}}}}T\ln {P}_{{H}_{2}O}\\ \Delta {\mu }_{H}&=&\frac{1}{2}\left(\Delta {\mu }_{{H}_{2}}^{0}(T)+{k}_{{{{\rm{B}}}}}T\ln {P}_{{H}_{2}}\right),\end{array}$$
(4)

where \(\Delta {H}_{{H}_{2}O}^{f}\) is the formation enthalpy for water. Note that \(\Delta {H}_{{H}_{2}O}^{f}\) was used in place of the free energy of formation (\(\Delta {G}_{{H}_{2}O}^{f}\)) since entropic contributions are relatively small.

In this paper, three different testing conditions for \(\Delta {\mu }_{O}(T,{P}_{{O}_{2}})\) were used: TR, dry GS, and wet GS conditions. At the TR condition, temperature was set to 1350 °C and \({P}_{{O}_{2}}\) was 0.2 atm, which gives \(\Delta {\mu }_{O}^{TR}\) = –2.86 eV. For the dry GS condition, a wide range of temperatures (600-1200 °C) were examined at \({P}_{{O}_{2}}\) = 0.2 atm to analyze how δGS changes with temperature. For the wet GS condition, temperature and \({P}_{{H}_{2}O}\) were set to 850 °C and 0.4 atm, respectively, and multiple \({P}_{{H}_{2}}\) values were examined. The \({P}_{{H}_{2}O}\) and \({P}_{{H}_{2}}\) at equilibrium GS condition was also used to calculate H2 yield as \({P}_{{H}_{2}}\)/\({P}_{{H}_{2}O}\) ratio. All GS conditions could be incorporated as \(\Delta {\mu }_{O}^{GS}\) and Table 2 lists the GS conditions used in this paper. With the calculated \({E}_{V}^{f}\) and ΔμO, \(\Delta {H}_{V}^{f}\) was determined to calculate the δ in {A}FeO3 under various conditions. The Δδ under dry GS condition is referred to as \(\Delta {\delta }_{dry}(={\delta }_{TR}-{\delta }_{GS}^{dry})\). In the actual STCH processes, water steam is involved in the GS reaction; thus, Δδ is described as \({\delta }_{TR}-{\delta }_{GS}^{wet}\).

Table 2 \(\Delta {\mu }_{O}^{GS}\) values at different \({P}_{{O}_{2}}\), \({P}_{{H}_{2}O}\), and \({P}_{{H}_{2}}\) conditions at 850 °C.
Full size table

DFT combined with statistics

In contrast to single component ABO3−δ systems, multiple elements mixing on A- or B-sites leads to a spread of \({E}_{V}^{f}\) and the corresponding \(\Delta {H}_{V}^{f}\); therefore, we employed a statistical approach to predict the δ using the distribution of \({E}_{V}^{f}\) values. To elucidate the effects of the distribution of \({E}_{V}^{f}\) along with the effects of the vacancy interaction, we compared four different models that either consider or ignore the statistical distribution of oxygen vacancy formation energy and/or oxygen vacancy interactions. The most commonly used non-interacting dilute model, referred to as LE-Dil, assumes that vacancies are non-interacting and the oxygen formation energy/enthalpy is a single value83. The vacancy concentration is given by \(\delta /(3-\delta )=\exp (-\Delta {H}_{V}^{f}/({k}_{{{{\rm{B}}}}}T))\) where kB is the Boltzmann constant. Thus, the δ in ABO3−δ can be expressed as equation (5):

$$\delta =3\cdot \frac{\exp \left(\frac{-\Delta {H}_{V}^{f}}{{k}_{{{{\rm{B}}}}}T}\right)}{1+\exp \left(\frac{-\Delta {H}_{V}^{f}}{{k}_{{{{\rm{B}}}}}T}\right)}=3\cdot {P}_{s}\left(\Delta {H}_{V}^{f}\right)$$
(5)

where \({P}_{s}\left(\Delta {H}_{V}^{f}\right)\) indicates the site fraction of oxygen vacancies. This model has been used in ABO3−δ oxygen vacancy calculations83,84 despite the fact that oxygen vacancy interactions do arise as δ increases31,39,46,48,85,86.

Considering the interaction between vacancies, referred to as the LE-Int model, the \({E}_{V}^{f}\) becomes a function of δ. A simple linear relationship was used to describe increasing \({E}_{V}^{f}\) with increasing δ by a factor of an interacting coefficient, α, which can be material-dependent. In this study, the α value was set to 2.6 (eV) obtained by a linear fitting of the DFT-calculated \({E}_{V}^{f}\) values from our previous La0.5Sr0.5FeO3−δ study in Ref. 31, where an ordered arrangement of La and Sr on A-site was used.

In a system with randomly mixed cations, \({E}_{V}^{f}\) becomes a distribution function, \(G({E}_{V,i}^{f})\), which can be discretized into total k numbers of \({E}_{V,i}^{f}\) states. In this study, two approaches for \(G({E}_{V,i}^{f})\) were used: the direct sampling method used \({E}_{V}^{f}\) distribution from DFT calculation as \(G({E}_{V,i}^{f})\) and the Gaussian distribution fitting method utilized the fitted Gaussian distribution function for \(G({E}_{V,i}^{f})\). If an oxygen vacancy occurrence at each state is considered to be an independent event, the functional form for each subsystem that holds the vacancy state will be independent on δ. Each subsystem still takes the form of a two-level system (empty or filled sites) and the energetic preference of having a vacancy at the i state is the Boltzmann probability with a partition function \({Z}_{i}=1+\exp \left(\frac{-\Delta {H}_{V,i}^{f}}{{k}_{{{{\rm{B}}}}}T}\right)\)87. Then the probability of finding vacant oxygen sites at an energy level is \(G({E}_{V,i}^{f})\cdot {P}_{s}(\Delta {H}_{V,i}^{f})\). Summing all the states leads to the total number of oxygen vacancies, δ.

$$\delta =3\mathop{\sum }\limits_{i=1}^{k}G\left({E}_{V,i}^{f}\right)\cdot {P}_{s}\left(\Delta {H}_{V,i}^{f}\right)$$
(6)

The statistical distribution of \({E}_{V}^{f}\) was considered in both HE-Dil and HE-Int models. For the HE-Dil model, the vacancy formation enthalpy at i state, \(\Delta {H}_{V,i}^{f}\), is calculated as \({E}_{V,i}^{f}+\Delta \mu (T,{P}_{{O}_{2}})\); but for the HE-Int model, the interaction of vacancies was implemented using the interacting coefficient α: \(\Delta {H}_{V,i}^{f}={E}_{V,i}^{f}+\Delta \mu (T,{P}_{{O}_{2}})+\alpha \cdot \delta\).

Experiments

All samples were synthesized through a conventional solid-state method. The precursor powder La2O3 (99.99%), SrCO3 (99.9%), BaCO3 (99.9%), Nd2O3 (99.99%) and Fe2O3 (99.99%) were purchased from Alfa Aesar. The precursor powder was weighted and put in a poly(methyl methacrylate) high-energy ball mill (HEBM) vial with endcaps and milling balls made by tungsten carbide. The samples were dry milled for 100 min and further annealed in air at 1300 °C for 10 hours. The perovskite powder was ground by pestle and mortar for further tests.

Thermogravimetric analysis (TGA) was conducted using a simultaneous thermal analyzer (NETZSCH STA 449 F3 Jupiter). For each test, 30 mg of sample was placed on an alumina crucible. The sample was heated up to 1350 °C with a 25 °C min−1 ramping rate and was held isothermally for 45 min under ultra-high-purity (UHP) Ar. Then, the sample was cooled down to 1000 °C with the same ramp rate, and 21% O2 balanced with Ar was introduced to re-oxidize the sample. The cycle was repeated for six different oxidation temperatures (1000, 950, 900, 800, 750, 650 °C). The mass loss curves of samples were calibrated by a blank crucible run. The mass loss was converted into oxygen non-stoichiometry based on the following equation:

$$\delta =\frac{\Delta m\times {M}_{s}}{{m}_{s}\times {M}_{o}}$$
(7)

where δ is the extent of reduction, Δm stands for the mass loss measured by the TGA, ms is the initial mass of the specimen, and Ms and MO represent the molar mass of the specimen and oxygen, respectively.