High-temperature reduction thermochemistry of SrVO3−δ

Cubic SrVO3 perovskite oxide is an attractive candidate for high-temperature energy applications due to its favorable features such as multiple oxidation state cations, high structural and thermal stabilities, ability to accommodate a large number of oxygen vacancies, and cost-effectiveness. Herein, the temperature-dependent reduction properties of SrVO3 have been studied using accurate first-principles calculations to reveal the effects of oxygen vacancies and temperature on the reduction potential of SrVO3−δ , δ = 0–0.125. The reduction potential of SrVO3−δ was found to be significantly impacted by increasing oxygen vacancy concentration and temperature. Analysis of the electronic and vibrational properties of SrVO3−δ for differing δ revealed the origin of this reduction behavior. The electronic structure analysis shows that the reduction of SrVO3−δ upon oxygen vacancy formation is highly localized to the neighboring V4+ t2g states in the vicinity of the oxygen defect, irrespective of δ. A comparison of the vibrational density of states of defect-free and reduced SrVO3 demonstrated that the ionic contributions to the phonon density of states, and hence to the thermal contributions to the SrVO3−δ lattices, were significantly altered by the introduction of oxygen vacancies, which ultimately impacted the temperature-dependent reduction behavior of SrVO3−δ .


Introduction
ABO 3 perovskite oxides permit cations with a range of oxidation states located at A-and B-sites.This allows the physical properties to be tuned, drawing widespread interest in various applications involving high-temperature thermochemical energy conversion processes, e.g.thermochemical water splitting (TWS) and carbon dioxide splitting (CDS) [1][2][3][4][5], solid-oxide fuel cells (SOFCs) [6][7][8][9], solid state gas sensors (SSGSs) [10,11], thermochemical air separation (TAS) [12,13], partial oxidation of methane (POM) [14,15], and thermochemical heat storage (THS) [16,17].High-temperature routes for energy conversion with ABO 3 perovskite oxides are of particular interest due to their structural and thermal stabilities [18,19], ability to accommodate oxygen vacancies [20][21][22], efficient redox activities [23,24] and kinetics [25,26] to drive chemical reactions, and high earth-abundance of component elements [27,28].Typically, high-temperature processes start with the formation of mobile oxygen vacancies in the perovskite oxide during a redox reaction.The role of oxygen vacancies in ABO 3 perovskite oxides is critical in the reduction process, which leads to the formation of nonstoichiometric ABO 3−δ via an endothermic reaction [19], Furthermore, the oxygen vacancy formation energy and defect thermochemistry of ABO 3 perovskite oxides critically depend on the identity of their A-and B-site cations, and cations of different oxidation states are found to play an important role in determining these properties [29].Therefore, assessing the temperature-dependent reduction properties of stoichiometric and nonstoichiometric ABO 3 perovskite oxides can reveal their suitability for high-temperature energy applications.
The perovskite strontium vanadate, SrVO 3 , is an interesting material due to its nonequivalent oxidation state cations at the A and B sites.It has a divalent Sr 2+ alkaline-earth cation and a tetravalent V 4+ metal cation that has a 3d 1 electronic configuration with a single electron in the degenerate t 2g band.In the cubic phase of SrVO 3 , the larger Sr 2+ cation is positioned in the center of the unit cell with corner-shared VO 6 octahedra, where V 4+ is located at the center of octahedral oxygen cages (figure 1(a)) [30].Identified as a cubic perovskite in 1957 [31], SrVO 3 has been the subject of experimental [30,[32][33][34][35] and theoretical [36][37][38][39][40][41] studies for many low-and high-temperature applications.For instance, Hui and Petric experimentally investigated the conductivity and stability of SrVO 3 , reporting excellent electronic conductivity (∼1000 S cm −1 at 1073 K and p(O 2 ) ∼ 10 −20 atm) and concluding that its derivatives remain stable for p O 2 ∼ 10 −14 -10 −20 atm [32].Maekawa et al studied the mechanical, thermal and electrical properties of SrVO 3−δ as a function of temperature and reported it as a potential n-type oxide thermoelectric material [33].Peng et al synthesized the related compound (Sr, La)VO 3 and identified it as a promising anode material for SOFCs that exhibits a high tolerance to the presence of H 2 S in the fuels [34].Boileau et al examined the temperature-dependent structural, electrical and optical properties of SrVO 3 thin films grown onto different perovskite substrates, and reported enhanced performance compared to the bulk material [35].Li et al characterized SrVO 3 and reported it as a high-performance anode material with excellent electronic and ionic conductivities [30].
Complementary theoretical studies have been performed to assess the structural, electronic and thermodynamic properties of cubic SrVO 3 .Karolak et al employed the local density approximation (LDA) within density functional theory (DFT) to analyze the electronic structures of cubic SrVO 3 [36].The density of states and band structure analysis of SrVO 3 show that the degenerate t 2g states of V form three isolated bands at the Fermi level with a band width of 2.5 eV.The bands below the V d bands are mainly composed of O 2p and hybridized V d and O 2p states.Jacobs et al applied the HSE03 exchange-correlation (XC) functional within DFT to understand the work function of SrVO 3 and identified this material as stable and highly conductive with a low work function that makes it a potential candidate for thermionic energy conversion technologies [37].Parveen and Gaur employed a modified rigid ion model to investigate the elastic and thermodynamic properties of this perovskite over a wide range of temperature and show reasonable agreement with experimental data [38].Koçer et al performed LDA and a combined LDA + DMFT (dynamical mean-field theory) calculations to study the lattice dynamics of SrVO 3 and confirm the dynamical stability of this perovskite structure [39].A recent report revealed exceptional imaginary LDA phonon frequencies at the R point for cubic SrVO 3 , attributed to a phase transition of this material [40].However, Berry et al confirm that, similar to TiO 2 [42], the origin of these imaginary phonon frequencies is not due to a phase transition but are an artifact of insufficient k-point mesh sampling in DFT calculations [41].
Although cubic SrVO 3 has received a lot of attention for its promising structural, electronic and phonon properties, its high-temperature thermochemistry has not yet been characterized in the literature.The characterization of temperature-dependent reduction properties is crucial to elucidate the viability of SrVO 3 in high-temperature applications such as TWS, CDS, THS, TAS, SOFCs, SSGS, and POM processes.In the present study, the structural, electronic and high-temperature reduction properties of cubic SrVO 3 (Pm 3m) have been characterized using first-principles DFT.The temperature-dependent reduction properties of SrVO 3−δ (δ = 0-0.125)have been determined over the range 0-2000 K at various p O2 (10 −5 -10 −20 atm).DFT calculations revealed non-linear changes in the reduction free energies of SrVO 3−δ as δ changes, demonstrating an intrinsic reducibility of SrVO 3−δ at higher δ values.The collective effects of oxygen vacancies and temperature on the reduction-free energies significantly impacted the reduction temperature of SrVO 3−δ .Calculations with the TPSS functional predicted SrVO 3−δ to undergo a spontaneous reduction reaction at 1400, 1600 and 1700 K, respectively, for δ values of 0.016, 0.037 and 0.125, at p O2 of 10 −15 atm.
The TPSS-predicted structural properties of cubic SrVO 3 are in agreement with available experimental data.Comparison of the predicted partial densities of states (PDOS) of pristine and oxygen-deficient SrVO 3 revealed that the excess electrons resulting from oxygen vacancies are redistributed to the neighboring V 4+ t 2g states, a conclusion consistent with Bader and charge density difference (CDD) analysis.Phonon dispersions and vibrational density of states (VDOS) analysis confirmed the dynamic stability of cubic SrVO 3 , which agrees with previous experimental and theoretical observations [39,41].Comparison of the vibrational properties of pristine and defective SrVO 3 suggests that ionic contributions to the VDOS are strongly affected by the oxygen vacancies, and thereby the temperature-dependent reduction properties of SrVO 3−δ .

Electronic structure calculations
The experimental lattice structure of cubic SrVO 3 (Pm 3m) [43] was obtained from the Inorganic Crystal Structure Database [44] for all the DFT calculations reported in this study.The Vienna ab initio simulation program (VASP) was used to perform the spin-polarized DFT calculations [45].The PBEsol generalized gradient approximations (GGA), TPSS, SCAN and SCAN + U (with optimal U value adopted from Long et al [46]) meta-GGA and HSE06 hybrid XC-functionals were used with the projector augmented wave (PAW) method to investigate any potential dependency of SrVO 3 structural properties on the choice of XC-functional [47].Notably, the SCAN + U and HSE06 functionals have been proven to outperform LDA and GGA functionals in the prediction of electronic properties for periodic systems [48].However, they can still underestimate the electronic band gap of solid materials because they comprise only short-range nonlocal exchange components of the Coulomb potential.Alternatively, computationally cheaper options for accurate electronic bandgap predictions of periodic systems have been reported in the literature [49,50].
In this study, PAW potentials were employed for the following valence configurations: 4s 2 4p 6 5s 2 (Sr), 3p 6 3d 4 4s 1 (V) and 2s 2 2p 4 (O).All calculations employed a 520 eV plane wave cutoff and an accurate 'precision' mode in VASP.Gaussian smearing of band occupations with a width of 0.05 eV was used to alleviate the density convergence issues [51,52].The Brillouin zone (BZ) of the unit cell was sampled using Γ-point-centered Monkhorst-Pack k-point meshes with spacings between k-points of approximately 0.025 Å −1 or less.The total DFT energies were fully optimized with respect to ionic positions, cell shape and cell volume of the unit cell, using the conjugate gradient algorithm and the ionic and electronic convergence criteria of ⩽|0.01| eV Å −1 and 1.0 × 10 −8 eV, respectively.

Vibrational property calculations
The finite displacement approach was used to calculate the vibrational properties of cubic SrVO 3 [53,54].First, 2 × 2 × 2, 3 × 3 × 3 and 4 × 4 × 4 supercells were created from each of the optimized unit cells for calculations with PBEsol, TPSS, SCAN, and SCAN + U functionals.The number of k points used for each calculation was adjusted accordingly.HSE06 was discarded from the vibrational property calculations due to its computational expense.The self-consistent crystallographic supercell approach was adopted into VASP to calculate the force constants in terms of the Hellmann-Feynman forces induced by the atomic displacements of 0.01 Å in the supercells [55].These force constants were used to define the dynamical matrix, and diagonalization of the dynamical matrix yielded the phonon frequencies (ω) using the Monkhorst-Pack grid of 48 × 48 × 48 q points for the phonon wave vectors via Phonopy [56].The resulting phonon frequencies were then used to generate the phonon dispersion relation of SrVO 3 over the BZ and the integration over frequencies produced the projected VDOS of SrVO 3 .These phonon frequencies were also used to predict the temperature-dependent entropy (S), heat capacity at constant volume (C v ), Helmholtz free energy (A) within the harmonic approximation and the heat capacity at constant pressure (C p ).The supporting information (SI) provides comprehensive information on this approach.Notably, a comparison of the generated phonon dispersion relations of SrVO 3 using the PBEsol, TPSS, SCAN and SCAN + U functionals revealed that only PBEsol and TPSS calculations produced stable phonon frequencies along high-symmetry BZ paths, whereas SCAN and SCAN + U phonons failed to converge the phonon structures within 2 × 2 × 2, 3 × 3 × 3 and 4 × 4 × 4 supercells, as evident in figures S1-S4 in the SI.Therefore, considering the hierarchy of functionals on 'Jacob's ladder' , TPSS-predicted electronic and vibrational properties of cubic SrVO 3 (Pm 3m) are discussed below, and selected PBEsol-predicted properties are presented in the SI for comparison.

Electronic structure of SrVO 3
The lattice constant (a/Å) and volume (V/Å 3 ) of cubic SrVO 3 (Pm 3m) predicted by PBEsol GGA, TPSS, SCAN and SCAN + U meta-GGA, and HSE06 hybrid functionals are compared with experimental values in table 1. SCAN + U produced the exact experimental lattice constant, whereas the prediction by SCAN without the U value is just 0.01 Å less than that.On the other hand, PBEsol and HSE06 underestimated the experimental lattice constant by 0.02 Å, while TPSS overestimated this value by 0.02 Å.Of necessity, the differences between experimental and optimized lattice volumes have a similar degree of variation.For instance, PBEsol and HSE06 underestimated the experimental lattice volume by 1.6%, whereas the TPSS-predicted volume is 1.6% higher than the experimental one.Overall, the theoretical lattice constants have less than 1% deviation from the experimental values, while the maximum deviation between the experimental and theoretical lattice volume is only 1.6%.These results suggest that SCAN + U is highly competitive in predicting the experimental lattice parameters of Pm 3m cubic SrVO 3 (although we note that no geometric zero-point vibrational energy or thermal expansion effects are included [57]), and all other functionals are in reasonable agreement with experiment.Note that, for the rest of the section, only the TPSS-predicted electronic and vibrational properties were reported, as mentioned earlier in § section 2.3.

The free energy of the reduction of SrVO 3−δ and the effect of vibrations
The enthalpy of the thermal reduction reaction in thermochemical redox cycles of metal oxides critically depends on the energy required to form oxygen vacancies.Furthermore, the oxygen vacancy formation energy of metal oxides primarily depends on the metal-oxygen bond dissociation upon vacancy formation, which may reduce to its neighboring cations.The oxygen vacancy formation energies (E v ) of SrVO 3−δ at 0 K as a function of oxygen vacancy concentration (δ) were calculated by using equation (S1), and the TPSS-predicted values are shown in figure 2(a).For a single oxygen defect, the E v of SrVO 3−δ varied with the defect concentration.For instance, E v was found to be ∼409 kJ mol −1 for a δ value of 0.016, whereas E v for δ values of 0.037 and 0.125 were 444 and 469 kJ mol −1 , respectively.This implies that with 2.3 times increase in δ (i.e.δ between 0 → 0.016 and 0 → 0.037), E v increased by 35 kJ mol −1 , whereas the required E v is only 60 kJ mol −1 higher with respect to 7.8 times increment in δ (i.e.δ between 0 → 0.016 and 0 → 0.125).
A further comparison of E v for similar values of δ by PBEsol showed that E v is 12 and 21 kJ mol −1 higher than the TPPS prediction for δ of 0.125 and 0.016, respectively (figure S5(a)).On the other hand, this difference is moderately higher (∼64 kJ mol −1 ) for δ of 0.037 between TPSS and PBEsol.An exceptional previously reported E v of cubic SrVO 3−δ by Wexler et al [21] for a δ of 0.037 was found to be nearly 223 kJ mol −1 using SCAN + U functional, which significantly underestimated the TPSS and PBEsol-predicted values.However, they also reported a higher E v for both cubic CaVO 3−δ (∼345 kJ mol −1 ) and BaVO 3−δ (∼371 kJ mol −1 ) with similar defect concentrations, but the E v of cubic AVO 3−δ (A = Ca, Sr, Ba) is not expected to differ significantly due to the identical V−O bond in their crystal structures   S6 for a δ value of 0.016], showed that the reduction temperature of SrVO 3−δ deviated with the concentration of oxygen vacancies in the supercells.For instance, SrVO 3−δ is reduced at 1400 K (i.e.∆G RR < 0) with a δ of 0.016, whereas the reduction temperature was shifted to somewhat higher values of 1600 and 1700 K for δ values of 0.037 and 0.125, respectively, as predicted by TPSS (figure 2(b)).PBEsol predicts the reduction of SrVO 3−δ at 1500 K for the defect concentration corresponding to 0.016, whereas for δ of 0.037 and 0.125, SrVO 3−δ is reduced at 1800 K (figure S5(b)).
The deviation of ∆G RR for SrVO 3−δ with respect to δ consistently increased with increasing temperature.For instance, at 298.15 K, the difference between the ∆G RR of SrVO 3−δ for δ of 0 → 0.016 and 0 → 0.037, was 37 kJ mol −1 , while this value gradually increased with the increasing temperature and reached 50 kJ mol −1 at 2000 K. On the other hand, the gap between the ∆G RR of SrVO 3−δ for δ of 0 → 0.016 and 0 → 0.125, was 66 kJ mol −1 at room temperature, which increased to 95 kJ mol −1 at 2,000 K. Consistent with the trends of E v with respect to different δ, ∆G RR as a function of (δ, T) implied that for a 2.3 times increment of δ (i.e.δ between 0 → 0.016 and 0 → 0.037), ∆G RR increased by 13 kJ mol −1 between 298-2000 K.At higher δ for the same temperature range this value rose by only 29 kJ mol −1 with the 7.8 times increase of δ (i.e.δ between 0 → 0.016 and 0 → 0.125).The smaller increase in ∆G RR as a function of (δ, T) for defective SrVO 3−δ favors the spontaneous reduction of SrVO 3−δ (∆G RR < 0), which means that the reduction of SrVO 3 is thermodynamically favorable at high temperatures with oxygen vacancy concentrations increasing as the temperature is increased substantially above 1000 K.The changes to ∆G RR as a function of (δ, T) are driven by vibrational contributions of the defective SrVO 3−δ structures as detailed in § section 3.5.

Effects of oxygen vacancy on the electronic properties of SrVO 3−δ
In cubic SrVO 3, V 4+ has a 3d 1 electronic configuration, where the octahedral splitting of the crystal field splits the V 3d orbitals into low-energy triply degenerate t 2g orbitals (composed of d xy , d yz , d xz ) and  3) lies in the conduction bands where the V d-t 2g states are dominant, TPSS identified SrVO 3 as having metallic character in agreement with experimental observation [61].The V d-t 2g and V d-e g bands were well separated in the conduction bands of SrVO 3 , with V d-e g bands being located above the upper band edge of V d-t 2g .The presence of a single oxygen defect in SrVO 3−δ shifted the bottom of the V d-t 2g to a lower energy level relative to the pristine structure, and the magnitude of the V d-t 2g PDOS increased for all δ of 0.016, 0.037 and 0.125 (figures 3(a)-(c), right).These results indicate that the formation of oxygen vacancies in SrVO 3 leads to the redistribution of electrons from O 2p to V d-t 2g states.Therefore, V 4+ ions are reduced to various degrees with respect to the oxygen vacancy concentration, which is reflected in the magnitude of PDOS in the V d-t 2g states of defective SrVO 3−δ .For instance, the PDOS of V d-t 2g states for a δ of 0.037 was found to be more pronounced than for the other δ of 0.016 and 0.125.
The reduction of V 4+ was further confirmed by charge transfer analysis using Bader methods [62,63].The removal of an oxygen atom from a pristine supercell left two electrons previously associated with the removed oxygen atom in the vicinity of the vacancy in defective SrVO 3−δ .Bader charge analysis shows that these excess electron densities are primarily localized on V ions, with small proportions transferred to the Sr ions (figure 4).The electron density redistribution varies with the oxygen vacancy concentration.For  instance, for a δ of 0.016, roughly 0.76 e − of electron density was redistributed to each of the nearby V ions, which increased to 0.84 e − and 0.82 e − to the V ions for δ of 0.037 and 0.125, respectively.These results imply that the V ions in the defect SrVO 3−δ with a δ of 0.037 can be effectively reduced to a higher degree compared to the other two δ of 0.016 and 0.125.
The CDD analysis confirmed the relative reduction degree (figures 5(a)-(c)).The CDD analysis further showed that the reduction of the SrVO 3−δ remained highly localized around the two neighboring V ions, reducing them from V 4+ to an effective V 3+ state for all δ of 0.016, 0.037 and 0.125, while a smaller contribution was redistributed to the two neighboring Sr 2+ ions.

Phonon dispersions and thermodynamic properties of cubic SrVO 3
The TPSS-predicted phonon dispersion relations of cubic SrVO 3 (Pm 3m) are presented along the high-symmetry BZ paths of Γ-M-R-X-Γ-R in figure 6.The convergence of the phonon dispersion relation with the size of the supercell is shown in figure S2.Although a 3 × 3 × 3 expansion of the unit cell shifted the highest energy phonon modes slightly compared to the predictions by 2 × 2 × 2 and 4 × 4 × 4 expansions, both 2 × 2 × 2 and 4 × 4 × 4 supercells produced practically identical phonon dispersion relations at lowand high-energy levels.Unsurprisingly, no soft modes were observed in any of the BZ paths considered here.This is in line with other theoretical treatments [39,41], although we note that Hossain et al [40] did find soft modes, which we suspect to be artefactual.
In general, five threefold degenerate phonon modes were observed at the Γ point.Interestingly, TPPS predicts the highest optical modes at slightly lower energies compared to that from PBEsol (figure S7), while the predictions of acoustic modes by these two functionals remain identical.This implies that the acoustic modes dominated by the vibrations of heavy Sr 2+ ions moving against the rigid VO 6 octahedron are not affected by the used XC-functionals, while the highest-energy optical modes involving the motion of light O 2− against V 4+ are slightly impacted by the applied XC-functional.It is worth noting that no attempt was made to model LO-TO splitting in these phonon structures.We do not expect any such (neglected) splitting to have a significant impact on the calculated thermodynamic properties [42,64].
The TPSS-predicted thermodynamic properties of cubic SrVO 3 (Pm 3m) over the temperature range 0−2,000 K is shown in figures 7(a)-(c).The predicted thermodynamic properties are effectively invariant with the size of the supercells of cubic SrVO 3 (figures S8(a)-(c)).Furthermore, the TPSS-predicted thermodynamic properties are nearly identical to the PBEsol predictions (figure S9).Our calculated specific heat at constant pressure (C p ) of cubic SrVO 3 shows reasonable agreement with available experimental data from room temperature up to 1073 K (figure 7(a)).For instance, the largest deviation between this study and experimental [33] C p is 8 J K −1 mol −1 at 280 K, reducing to 2 J K −1 mol −1 at 1073 K.The variation of C p with temperature obeys the Debye's T 3 law [65,66] at low temperatures (0-300 K), but C p departs from the Dulong-Petit's law [67,68] above 500 K due to quasi-harmonic contributions.On the other hand, at room temperature the TPSS-derived C p was just 1 J K −1 mol −1 higher than the PBEsol value (∼98 J K −1 mol −1 ), while this difference is reduced to a fractional level (∼0.1 J K −1 mol −1 ) at 2000 K, with a TPSS value of 133.4 J K −1 mol −1 (figure S9(a)).
The calculated vibrational entropy (S) of cubic SrVO 3 at room temperature was 100 J K −1 mol −1 for TPSS (figure 7(b)), while this value was found to be 1 J K −1 mol −1 less for PBEsol (figure S9(b)).The small deviation of the calculated S between TPSS and PBEsol is maintained up to the temperature range 0-500 K, whereas the difference is only 2 J K −1 mol −1 between 500-2000 K.Both TPSS and PBEsol showed an exponential increase in S between 0-2000 K.At 2000 K, the entropies were calculated to be 323 and 321 J K −1 mol −1 , respectively, for TPSS and PBEsol.In addition, the calculated relative molar enthalpies

Effects of oxygen vacancies on the vibrational properties of SrVO 3−δ
The influence of δ on the vibrational properties of SrVO 3−δ in terms of Sr 2+ , V 4+ and O 2-ions is shown in figures 8(a)-(c).Within the low-frequency regime (0-200 cm −1 ) the movement of heavier Sr 2+ ions against the rigid VO 6 octahedron dominates the TPSS-derived VDOS for both pristine and defective SrVO 3−δ (figure 8(a)).The principal vibrational mode of Sr 2+ ions was observed at 114 cm −1 for pristine SrVO 3 .The introduction of an oxygen defect in SrVO 3 slightly impacts the peak amplitude of the VDOS of Sr 2+ , shifting the principal vibrational mode to 123 cm −1 for δ of 0.016.This trend holds true for the other two δ of 0.037 and 0.125 with a progressive decrease in the amplitude of the VDOS of Sr 2+ , while the principal vibrational mode splits into two weaker peaks at 107 and 130 cm −1 for δ of 0.125.On the other hand, the VDOS related to the V 4+ mainly contributes to the total VDOS between 150 and 600 cm −1 for both pristine and defective SrVO 3−δ (figure 8(b)).In pristine SrVO 3 , V 4+ has a VDOS peak at 342 cm −1 with a secondary peak at 194 cm −1 .However, in defective SrVO 3−δ , the magnitude of the peak in the VDOS of the V ions decreases and shifts to a slightly higher energy level ∼348 cm −1 for a δ of 0.016, with a secondary peak at 200 cm −1 .Similarly, the primary and secondary peaks of the VDOS of V ions in the defective SrVO 3−δ were observed at 351 and 203 cm −1 , respectively, for a δ of 0.037, and at 373 and 201 cm −1 , respectively, when δ was 0.125.Overall, the VDOS of phonons related to the V ions in the defective SrVO 3−δ was progressively shifted from 540 to 600 cm −1 with increasing δ.The most noticeable change in the VDOS of O 2− ions was that the two peaks observed in pristine SrVO 3 VDOS were significantly disrupted by the oxygen vacancy in the defect structures.The VDOS associated with the O 2− vibrations were between 100 and 740 cm −1 (figure 8(c)).The dominant contributions of O 2− ions in pristine SrVO 3 lay in the frequency range 320-740 cm −1 , with two peaks at 342 and 550 cm −1 , while for the defective SrVO 3−δ these two peaks in the VDOS were noticeable at 348 and 556 cm −1 , with a reduced intensity, for a δ of 0.016.Further shifts of these two peaks were evident

Conclusion
The temperature-dependent reduction properties of SrVO 3−δ have been reported at δ of 0.016, 0.037 and 0.125.DFT calculations demonstrated that the reduction free energies of SrVO 3−δ increased with increasing δ.However, the increment was not proportional to the rise in δ values, i.e. the reducibility of SrVO 3−δ is increased for higher δ.Furthermore, the thermochemical contributions to the reduction free energies significantly impacted the reduction temperature of SrVO 3−δ , and the mutual effects of oxygen vacancies and temperature to the defective SrVO 3−δ thermodynamically favored this structure to undergo spontaneous reduction reaction at 1400 K with δ values of 0.016, and at 1600 K and 1700 K, respectively, for δ values of 0.037 and 0.125, with a p O2 of 10 −15 atm for all three δ values.The PDOS, Bader analysis and CDD analysis confirmed that the reduction of SrVO 3−δ was highly localized to the neighboring V 4+ t 2g states of the defect sites, regardless of the defect concentration.The predicted phonon dispersion relation of cubic SrVO 3 (Pm 3m) identified it as a stable structure, and the calculated temperature-dependent thermodynamic properties such as C p show reasonable agreement with available experimental data.The corresponding VDOS analysis of pristine and defective SrVO 3−δ reflected that the vibrational contributions of Sr 2+ , V 4+ and O 2− ions were significantly impacted by oxygen vacancy formation.However, the nature of phonon modes in the defective lattices was unaffected by δ.Since SrVO 3−δ is reduced at high temperature and low oxygen partial pressure, this material is widely compatible with relevant technologies, such as superconductor coatings and aerospace applications, along with the proven high-performance anode materials [30].

[ 5 ,
29,[58][59][60].Nevertheless, in this study, the predicted E v with TPSS and PBEsol as a function of δ indicated that SrVO 3−δ exhibited facile reduction for a smaller concentration of oxygen defects and required higher energies for increasing δ, but the increment of E v decreased with increasing δ.Thermal contributions to the predicted E v of SrVO 3−δ were incorporated by equation (S2) as a function of temperature T and pressure p, using TPSS and PBEsol XC-functionals, and are respectively shown in figures 2(b) and S5(b).The TPSS and PBEsol-predicted Gibbs free energies of reduction (∆G RR ) of SrVO 3−δ between 0-2000 K at an oxygen partial pressure of p O2 = 10 −15 atm [∆G RR values between 0-2000 K at various p O2 (10 −5 -10 −20 atm) by TPSS are shown in figure

Figure 4 .
Figure 4. TPSS-predicted Bader analysis of electron density redistribution (e − ) from the vacancy site to the neighboring Sr 2+ and V 4+ cations in SrVO 3−δ with respect to δ.
[H(T) − H(298.15K)] of cubic SrVO 3 between 0-2000 K showed a linear increase with increasing temperature.At 300 K, the TPSS-predicted [H(T) − H(298.15K)] was ∼0.183 kJ mol −1 (figure 7(c)) and the PBEsol value was ∼0.181 kJ mol −1 (figure S9(c)).This negligible deviation within the [H(T) − H(298.15K)] between TPSS and PBEsol was carried throughout the temperature range considered here.For example, at 2000 K, [H(T) − H(298.15K)] was 204.61 and 204.34 kJ mol −1 , respectively, for TPSS and PBEsol.In summary, the predicted C p of cubic SrVO 3 showed consistency with the available experimental data, and the agreement between the TPSS and PBEsol-predicted C p , S and [H(T) − H(298.15K)] reflected that the thermodynamic properties of cubic SrVO 3 are effectively invariant to the choice of XC-functional, which is consistent with the behavior of other perovskite oxides and solid materials [42, 64].

Table 1 .
Comparison of optimized lattice parameters (a), and volumes (V) of cubic SrVO3 (Pm 3m) obtained by GGA, meta-GGA and hybrid functionals with experimental values.