Charge transfer drives anomalous phase transition in ceria

Ceria has conventionally been thought to have a cubic fluorite structure with stable geometric and electronic properties over a wide temperature range. Here we report a reversible tetragonal (P42/nmc) to cubic (Fm-3m) phase transition in nanosized ceria, which triggers negative thermal expansion in the temperature range of −25 °C–75 °C. Local structure investigations using neutron pair distribution function and Raman scatterings reveal that the tetragonal phase involves a continuous displacement of O2− anions along the fourfold axis, while the first-principles calculations clearly show oxygen vacancies play a pivotal role in stabilizing the tetragonal ceria. Further experiments provide evidence of a charge transfer between oxygen vacancies and 4f orbitals in ceria, which is inferred to be the mechanism behind this anomalous phase transition.

It is unclear whether the reported materials actually adopt a single phase, or whether a 2-phase mixture is formed. The authors clearly state that data up to 75 C require use of the tetragonal phase to get a good fit to the PDF data, but do not discuss whether all data can be accounted for exclusively by the tetragonal phase (e.g., the X-ray data should also be fittable with this phase!). Figure 1a shows lattice constants for a cubic phase extracted over the entire temperature range, and so does Table S1, implying that the cubic phase is observed at all temperatures. Table S2 seems to imply that the tetragonal phase is present at all temperatures up to 225 C. If both are correct/meaningful, then this should mean 2-phase coexistence. Of course, application of both models to the same data would then be necessary to determine relative amounts. Figure 2c shows tetragonal lattice constants at low temperature, and I presume cubic ones at higher T (this should be indicated somewhere!) -but the values for the tetragonal a parameters do not agree with the numbers in Table S2. Figure S8 claims to be displaying the a-parameter -yet once again, it is unclear whether this is the cubic or tetragonal parameter. Similarly, is Figure S6 displaying both cubic and tetragonal volumes? This is rather confusing, and makes it hard to truly evaluate the results presented. The authors need to clarify which phase is present at what temperature and refine/fit ALL data accordingly. E.g., if the material is not cubic at low T -then plotting a cubic lattice parameter as f(T) at low T is not meaningful. If the material forms a 2phase mixture, obviously major revisions will be necessary to adequately present the results.
corrections in the manuscript are highlighted in blue.

Responses to Reviewer #2
Q1: At least a line on the computational details is required for the main text. This information already allows to calibrate the reliability of the computational approach. A1: Thank you for your suggestion. We have added more computational details in the main text.

Correction in the manuscript:
In order to investigate the lattice dynamics of nanosized ceria, the first-principle calculations, as implemented in Vienna ab initio simulation package (VASP), are performed using Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) and the projector augmented wave (PAW) potential with the plane cut-off of 500 eV and 4×4×4 Monkhorst-Pack k mesh (see the technical details in Methods in Supplementary Information).

Q2:
The DFT calculations seem the minimum ad-hoc work. I do not understand why the authors do not provide the effect of vacancy formation from a computational point of view. This would help to understand if the origin of the tetragonal distortion is due to some particular arrangement of the generated vacancies. This study is a must to proceed further with the analysis. What is the role of the vacancies in the phonon dispersion?
A2: This comment leads us to take a significant step forward to the tetragonal-stabilizing mechanism using the DFT + U methodology 1 . First, we removed one oxygen atom (i.e. 12.5 % vacancies) from the Fm-3m conventional cell of CeO 2 , and performed the structural optimization. No oxygen atom displacement was observed in this fully relaxed structure (marked as C vac structure). However, when we induced a small perturbation along c direction to one of the oxygen atoms, this nonstoichiometric structure will be spontaneously relaxed to the P4 2 /nmc phase (marked as T vac structure), with a slight stretch of c-axis and a small displacement of O along c-axis (the relaxed displacement pattern is the same as the experimental pattern). In addition, the total energy of the T vac is lower than that of C vac (Table R1).
Hence, our DFT calculations show that the tetragonal structure becomes more stable when inducing oxygen vacancies, which is consistent with our experimental results. This calculation along with the related discussion is supplemented in the main text.
The phonon dispersions for both C vac and T vac structure are calculated with VASP and Phonopy 2 (Fig. R1). Although the T vac structure is more stable than C vac , the distinction of their phonon dispersions is insignificant. We agree with the reviewer that the specific arrangement of the vacancies might make a greater difference to the phonon. However, this also leads to a tremendous computing workload (at least 36 configurations for 2 × 2 × 2 supercell of CeO 1.75 ), which obviously lies outside the scope of the present study. We believe our experimental results and the phonon softening model, as well as the above DFT calculations with oxygen vacancy could give a clear picture of the tetragonal-cubic phase transition in nanosized ceria. Figure R1. The calculated phonon dispersions for the C vac and T vac unit cells. Supplemented in the manuscript: From the above, we propose that the formation of oxygen vacancies plays a key role in stabilizing the tetragonal ceria, which has been also investigated by the first-principle calculations. The Fm-3m unit cell with one oxygen vacancy (CeO 1.75 ) was employed for the structural optimization. When a small perturbation along c direction was applied to one of the oxygen atoms, this nonstoichiometric structure will be spontaneously relaxed to the P4 2 /nmc phase with 3.5meV/atom lower than cubic CeO 1.75 structure (Fig. S13). Consequently, the tetragonal structure becomes more stable when inducing oxygen vacancies, which is consistent with our experimental results at low temperature. In such case, the conventional cubic structure could become a metastable state, which turns into the tetragonal phase with a slight atomic-site fluctuation (perturbation). For the nanosized ceria, such fluctuation is frequently observed due to the short-range coherence of the lattice, giving rise to the tetragonal structure at low temperature.
Added in Supplementary Materials: Figure S13. The schematic diagram of the oxygen displacement for the relaxed tetragonal structure containing one oxygen vacancy.

Q3:
The authors claim in page 8 that the electrons can be localized at the vacancy for the tetragonal ceria. This can actually be calculated through DFT with some degree of accuracy. As it is know it is quite speculative and the results seem to preliminary to be published in the present form.
A3: Thank you for your suggestion. The paramagnetic defects in the tetragonal ceria have been confirmed by the DFT calculation of spin charge density 3 . We found excess spin charge in the vacancy of tetragonal unit cell, evidencing the existence of the unpaired electrons localized at the vacancy (Fig. S14a). When the tetragonal ceria transforms to cubic ceria, the trapped electrons transfer to Ce 4f orbitals (Fig. S14b), which is consistent with the EPR result.

Correction in the manuscript:
During the phase transition upon heating, a charge transfer occurs from the ( ) of tetragonal ceria to the Ce f orbitals of cubic ceria (Fig. 4b), which has been also verified by our DFT calculation of spin charge density (Fig. S14).

Added in Supplementary Materials:
The spin charge density was calculated based on the previous research for the tetragonal unit cell (using the experimental PDF results at -150 o C) and the cubic unit cell (using the experiment PDF results at 200 o C). One oxygen atom was removed for both tetragonal and cubic structure in conventional cells. Figure S14. The spin charge density of (a) tetragonal phase and (b) cubic phase with one oxygen vacancy in their unit cells.

Q4:
What is the origin of the critical size observed for the transition?
A4: On the one hand, the defect concentration, which is highly correlated with the dimension of the nanoparticles, is the origin of the critical size for the phase transition.
Our XPS results ( Figure S12) demonstrated that the amount of oxygen vacancies increases dramatically when reducing the particle size. The inhomogeneous dispersion of the abundant vacancies gives rise to surface stress 4,5 , which softens the phonon mode to induce the phase transition. For the ceria with larger particle size, less significant stress is induced due to the declined defect concentration. Consequently, there could be a critical size (critical defect concentration) for ceria, above which the induced stress is not strong enough to soften the phonon, and thereby the charge transfer and the phase transition is inhibited.
On the other hand, bulk non-stoichiometric CeO 2-δ , which possesses high concentration of oxygen vacancies, has been found to adopt irreversible structural reconstruction instead of reversible phase transition 6 . Consequently, there exists another key factor, apart from the defect concentration, to trigger the phonon softening and charge transfer, giving a critical size for the phase transition. We suppose that the unique phonon dispersion and electronic structure, as well as the atomic-site fluctuation on the surface of the nanoparticle might play key roles, since the proportion of surface atoms increases with decreasing the particle size.

Correction in the manuscript:
On the contrary, for the ceria with larger particle size, the induced stress is less significant to soften the phonon due to the declined defect concentration, and thereby the phase transition is inhibited.

Q5:
The conductivity can also be analyzed in terms of the electronic structure of the defective ceria.

A5:
We have supplemented DFT calculation of spin charge density to verify the charged defects in the tetragonal ceria. Such localized electrons are reported to serve as mobile charge carrier, which could significantly increase the conductivity 7 .

Q6:
The relevance of Figure 1c needs to be clarified.

A6:
The thermal expansion curves ( Fig. 1a-1b) clearly show distinct steps in the temperature range of -25 -75 o C, indicating a structural break for nanosized ceria as temperature rises. In addition, our specific heat capacity measurement also indicates a phase transition that could be relevant to a local structural change. These results motivate us to directly study the atomic arrangement of the nanoparticles using STEM with annular bright-field (ABF) detector. The ABF image in the present study is a key evidence of the dispersively oxygen vacancies in nanosized ceria, which plays a pivotal role in the phonon softening and charge transfer. In order to strengthen the relevance of ABF-STEM image, we have changed the sequence of Fig. 1c and Fig. 1d, and also revised the related descriptions in the manuscript.

Correction in the manuscript:
In order to provide a direct view of the local structure, scanning transmission electron microscopy with annular bright-field detector (ABF-STEM) was carried out for 5 nm ceria (Fig. 1d). Along the [001] projection, individual cerium-and oxygen-atom columns appear alternately. Faint dark dots could be recognized at the oxygen sites, but the contrast of these dots fluctuates, which is firmly demonstrated by the inconsistent peak valleys of the oxygen intensity profile (Fig. 1d inset). This result indicates the dispersive oxygen vacancies distributed in the nanoparticles, which are expected to play key roles in the NTE and the phase transition observed in nanosized ceria.

Q7:
The inset in Figure 1d needs to be redrawn to improve readability.

A7:
We have redrawn the inset of Fig. 1d (Fig. 1c in the revised version), and an explanation has been also added in the figure caption.

Correction in the manuscript:
The entropy (∆S) of this phase transition was estimated by integrating (C p -C fit )/T, where C fit is the background extracting with a polynomial function to C p (T) between -150 o C to 150 o C (Fig. 1c inset). Figure 1: Q8: An extensive revision of the English is required.

A8:
We have carefully revised many parts of the manuscript, and checked it in multiple proof readings. The English editing has been highlighted in blue.

Responses to Reviewer #3
Q1: The English is readable, however, it should be corrected before publication. As several authors reside in the United States, this should not be too difficult to achieve.

A1:
Thank you for your suggestion. The whole manuscript has been polished by an English native speaker, and the English editing has been presented by blue highlights.

Q2:
The supplemental information should contain some more details. E.g., what exactly is the configuration for the "low temperature attachment" for the PXRD instrument? Is this a flat plate setup? If so -then variable temperature data ought to be collected with an internal standard to correct for sample height changes as a function of temperature. Note that this is especially important when dealing with small changes in lattice constants as in the current paper! The PDF data clearly show a shortening of nearest-neighbor O-O distances, but the absolute numbers could be affected considerably.
A2: Thank you for your constructive comment. The low temperature attachment of PW 3040-X'Pert Pro diffractometer (PANalytical) is a flat plate setup. The exact configuration has been supplemented in Fig. S4. To ensure the accuracy and stability of temperature controlling, we had taken a series measures. The heat conducting grease was smeared between heating (cooling) stage and sample cell. All the XRD measurements were carried out under vacuum to reduce thermal fluctuation. Besides, the thermocouple is made of Pt100, which is considered reliable in the conducted temperature range. On the basis of the reviewer's advice, we have revisited our variable temperature data of 5 nm ceria involving quartz (SiO 2 ) internal standard to calibrate the lattice constants of nanosized ceria. The Rietveld refinements were carried out by fixing the SiO 2 lattice constants, obtained from linear regression of the quartz thermal expansion 8 (Fig. R2a). Remarkably, we found that the difference of the lattice constants extracted with or without the internal standard is subtle ( Fig. R2b and Table R2), which doesn't affect the final conclusion of the NTE and the phase transition. This is because the systematic errors associated with the thermal expansion of the cryostat could be maximally eliminated through correcting zero-shift. Taken together, we believe that the lattice parameters obtained from XRD in the present study are reliable. Some supplements have been added in Experimental Section in Supplementary Materials.   (7) 250 5.41583 (5) 5.41708 (7) 300 5.41816 (5) 5.41930 (7) 350 5.41986 (5) 5.42120 (7) Added in Supplementary Materials: The configuration of the low temperature attachment is shown in Fig. S4. The heating conducting grease was smeared between the stage and sample cell, and the thermocouple is made of Pt100.
The reliability of the obtained lattice constants were examined by an internal standard of SiO 2 . The difference of the lattice constants extracted with or without the internal standard is subtle, which could be due to the zero-shift correction that maximally eliminating the systematic errors. Figure S4. The configuration of the low temperature attachment of PW 3040-X'Pert Pro diffractometer.

Q3:
It is unclear whether the reported materials actually adopt a single phase, or whether a 2-phase mixture is formed.

A3:
Thank you for pointing out the unclear description in the manuscript. Before we response the comments concerning the nanostructure of ceria, two key issues needs to be clarified: First, the XRD method was used to determine the long-range structure of ceria in the present study. For the bulk material, the crystal structure can be determined by this conventional Bragg diffraction experiments with high precision. However, such method cannot give the short-range nanostructure in the case of nanomaterial, because of the broadening and overlapping of the Bragg peaks at high angle 9 . Only the average lattice constants and a distinct NTE could be surely determined for the present 5 nm ceria. Consequently, neutron PDF technique, which is emerging as a powerful tool for studying the nanostructure 10 , has been carried out to reveal the local structure (especially the oxygen displacement) of nanosized ceria.
Second, the tetragonal structure (P4 2 /nmc) observed in nano-ceria is a subgroup of the cubic structure (Fm-3m). For the tetragonal phase, three structural parameters are variable, i.e., the lattice constants (a t , c t ) and the atomic coordinate of O along c-axis . When the conditions of c t =√2 a t and O z = 0.75 are satisfied, the tetragonal structure turns into the cubic-fluorite structure (see the tetragonal-cubic relation in Fig.   S7). Consequently, for both XRD and PDF refinements, the data that is fittable with the Fm-3m model could be certainly described well by the P4 2 /nmc model, since the latter contains more variables than the former. In such case, the structure with higher symmetry should be used. Analogously, if the two-phase model, which contains more repeating variables, for XRD/PDF patterns cannot improve the refinements significantly, the structure should be regarded as single phase within the resolution of the structural-characterization methods.
In the present study, the PDF data of 5 nm ceria can be fitted well with the single-phase cubic model above 25 o C and the two-phase model cannot give a better description of the data (see the response in Q5). In addition, the PDF data below -75 o C can be well fitted with the single-phase tetragonal model but cannot be well fitted by the cubic model. Based on the above rules, the nanosized ceria possesses single cubic structure at high temperature, and single tetragonal structure at low temperature.
The tetragonal-cubic phase transition occurs in the intermediate temperature range from -25 o C to 75 o C.

Q4:
The authors clearly state that data up to 75 C require use of the tetragonal phase to get a good fit to the PDF data, but do not discuss whether all data can be accounted for exclusively by the tetragonal phase (e.g., the X-ray data should also be fittable with this phase!).

A4:
We appreciate the question mentioned in this comment. In the present study, all the XRD patterns could be fitted very well with the cubic model, as indicated by the low values of Chi 2 (R p , R wp and Chi 2 have been supplemented in Fig S5). We attempted to refine the XRD pattern of 5 nm ceria (collected at -100 o C) with the P4 2 /nmc structure. The refined patterns and the obtained results using cubic and tetragonal model are presented in Fig. R3 and Table R3, respectively. We found that the tetragonal model didn't improve the refinement significantly, and the c t is approximately equal to √2 a t (here, √ = 1.0003, while √ = 1.006 for the tetragonal structure from the corresponding PDF result), which means the obtained tetragonal structure is very close the cubic structure. This is because the peak broadening, which is generally observed for nanosized materials, conceals the peak splitting. In addition, the atomic coordinate cannot be exactly extracted for the very small nanoparticles through XRD, on account of the overlapped peaks especially at high angle 11,12 . In short, the tetragonal phase is difficult to be distinguished by XRD method due to its limited resolution, and only an NTE could be surely determined for the nanosized ceria. In the manuscript, all the XRD results (Fig. 1a, Fig. S5, Fig. S9 and Table S1) are based on the cubic-fluorite model.
The neutron PDF data can be fitted well with the cubic-fluorite model above 75 o C, but this cubic structure cannot be well reconciled below 75 o C, especially below -25 o C (see the increased agree-factor, R w , as temperature decreases in Fig. S6). The tetragonal structure (P4 2 /nmc) was found from the subgroups of the fluorite structure to give the best description of the data at low temperature. Of course, as mentioned above, the tetragonal subgroup with lower symmetry could also be suitable for the data above 75 o C. Actually, all the PDF results over the entire temperature range are given within tetragonal model in the manuscript, in order to obtain continuous structural evolution. When c t ≈√2 a t (see Fig. 2c) and O z ≈ 0.75 ( Fig. S8a and Table   S2), the nanosized ceria could be regarded as the cubic phase. On the contrary, the tetragonal structure could be recognized from the conditions of c t > √2 a t and O z > 0.75. From the overall neutron PDF results, the conclusion could be drawn that the nanosized ceria adopts cubic structure at high temperature, and possesses tetragonal structure at low temperature.

Corrections in the manuscript:
The cubic-fluorite model was initially utilized for the Rietveld refinements of X-ray diffraction (XRD) to determine the thermal lattice evolution (Fig. S4-5).
The observed NTE indicates a structural break for the nanosized ceria, which, however, cannot be further determined by XRD. This is due to the broadening and overlapping of the Bragg peaks that significantly reduces the XRD resolution.
The low-r neutron PDF data (from 1.5 to 15 Å) of 5 nm ceria was initially adopted with the cubic-fluorite model. This fluorite model fits well to the data above 75 o C, but fails to describe the data below 75 o C, especially below -25 o C (see the increased agree-factor, R w , as the temperature decreases from 75 o C to -25 o C, Fig. S6).
The PDF fitting was carried out with P4 2 /nmc model over the entire temperature range to obtain continuous structural evolution (Table S2).
Question 5: Figure 1a shows lattice constants for a cubic phase extracted over the entire temperature range, and so does Table S1, implying that the cubic phase is observed at all temperatures. Table S2 seems to imply that the tetragonal phase is present at all temperatures up to 225 C. If both are correct/meaningful, then this should mean 2-phase coexistence. Of course, application of both models to the same data would then be necessary to determine relative amounts.

A5:
As mentioned in Q4, all the XRD data were refined with the cubic model (Table   S1), and all the PDF data were fitted with the tetragonal model (Table S2). Such distinction doesn't imply a two-phase coexistence.
By analyzing the PDF data, the 5 nm ceria could be cautiously regarded as a single tetragonal phase at low temperature. The two-phase fitting with both cubic and tetragonal models were carried out for PDF data at -150 o C (

Correction in the manuscript:
It is worth mentioning that the PDF refinement didn't show a significant improvement with two-phase fitting with both cubic and tetragonal models. As a consequence, the 5 nm ceria should be regarded as single tetragonal structure at low temperature.
Question 6: Figure 2c shows tetragonal lattice constants at low temperature, and I presume cubic ones at higher T (this should be indicated somewhere!) -but the values for the tetragonal a parameters do not agree with the numbers in Table S2.

A6:
Thank you for pointing out the misleading part. The lattice constants in Fig. 2c was extracted from the PDF data with the tetragonal model. The unit cell of this tetragonal structure is smaller than that of the cubic structure (Fig. S7). In order to keep the coherence of the thermal evolution along a-axis, √2 a t (a t is the a-parameter in Table S2) was utilized in Fig. 2c. We have made the descriptions more explicit in the main text, and a column of √2 a t has been added in Table S2 Correction in the manuscript: This tetragonal structure can be regarded as the shear strain of the oxygen sublattice distorted along fourfold axis (Fig. 2c inset), which leads to a smaller tetragonal unit cell than the cubic one. Based on the conversion relation between Fm-3m and P4 2 /nmc (Fig. S7), the obtained tetragonal structure (a t , c t ) can be expanded to a cubic-like unit cell (a = √2 a t , c = c t ). This expanded unit cell has been adopted in the present study to describe the tetragonal phase, so as to unify its lattice constants towards the cubic structure. Table S2:   Table S2.

Correction in
Lattice constants of 5 nm ceria extracted from the low-r nPDF refinement.  Figure S8 claims to be displaying the a-parameter -yet once again, it is unclear whether this is the cubic or tetragonal parameter.

A7:
The lattice parameters in Fig. S8 (Fig. S9 in the revised version) were extracted from XRD Rietveld refinements. As mentioned in Q4, the tetragonal phase can't be This is rather confusing, and makes it hard to truly evaluate the results presented.

A8:
In order to make the volumes of tetragonal and cubic phase comparable, an expanded unit cell, whose volume is two times larger (see Fig. R5), was utilized for the tetragonal phase in Fig. S6b (Fig. S8b in the revised version). This leads to a better understanding of the volume contraction during the tetragonal-cubic phase transition. The volumes of tetragonal ceria are presented below -25 o C, while the volumes of cubic ceria are presented above 75 o C. In the intermediate temperature range, a volumetric contraction occurs along with the tetragonal-cubic phase transition. We have re-edited the corresponding figure (Fig. S8), and some explanation has also been added in to avoid the misunderstanding. Figure R5. Schematic diagram of the unit cell used in Figure S6b expanded from the tetragonal unit cell.

Correction in the manuscript:
During the tetragonal-cubic phase transition from -25 o C to 75 o C, c-axis contracts towards the length of a-axis (√2 a t ), while the oxygen coordinate along c-axis (O z ) approaches to 0.75 (Fig. S8a). This makes a volume contraction of 1.3 ‰ excluding the general thermal expansion (Fig. S8b).

Added in Supplementary Materials:
In Fig. S8b, an expanded unit cell, whose volume is two times larger than the original P4 2 /nmc structure, has been utilized for the tetragonal phase at low temperature. This makes the volumes of the tetragonal and cubic phase comparable. From the PDF results, a volume contraction is observed during the phase transition, which is consistent with the XRD result.

Question 9:
The authors need to clarify which phase is present at what temperature and refine/fit ALL data accordingly. E.g., if the material is not cubic at low T -then plotting a cubic lattice parameter as f(T) at low T is not meaningful. If the material forms a 2-phase mixture, obviously major revisions will be necessary to adequately present the results.

A9:
We appreciate the above questions concerning the structural characterization.
Accordingly, we have clarified the nanostructure of ceria explicitly, and revised the unclear descriptions regarding to the XRD and the PDF analysis in the manuscript.
Although the XRD method cannot distinguish the tetragonal structure from the cubic one, we suppose it meaningful to display the lattice parameters extracted from XRD as a function of temperature, because this provides a direct view of the phase transition through a distinct NTE.
The authors have introduced many of my comments in previous correspondence yet I have still some points that deserve further explanations.
1) The U values need to be added to the main text.
2) The use of U in the O2p states although reported it is not of common use and besides some issues in the VASP implementation for this particular case are known. Therefore, more tests on the effect of the U in the calculated properties are needed, like the vacancy formation energy as a function of U. In addition, the English use in the SI needs revision. For instance, "The U value of Ce f states was set to 5 eV, and for O2p states, we employed a U value of 5.5 eV, as suggested in the previous theoretic works" 3) The authors consider that 36 calculations in the 2x2x2 is a high computational burden. Actually with the present computers it is not and as I consider that this particular result is important I recommend again the authors to address my previous Question 2 in a more adequate manner. Figure S14 shows no localized electrons in a single Ce atom but rather the average structure with multiple Ce getting density by reduction. I refer the authors to the extensive literature describing that localization improves the energetics of the systems by significant amounts and thus that a more adequate treatment of the localization needs to be implemented.

4)
Reviewer #3 (Remarks to the Author): The authors have adequately addressed my comments and nicely clarified the open questions. I am only requesting one minor change: The difference in lattice constants with and without internal standard is NOT due to a "zero-shift error", it is due to a change in sample height (which is similar to zero error at low angles, but follows a different angular dependence). Thus it should be called a sample height error, not a zero shift. Also, while the difference may be small, the authors should use the best possible numbers, which means the ones collected with internal standard. There is no need to use/report the uncorrected numbers!
The point-by-point responses to the reviewers' comments are given below. All the corrections in the manuscript are highlighted in blue.

Q1:
The U values need to be added to the main text. Consequently, we set U{O 2p } = 5.5 eV in the present study to correct the SIE associated with the O 2p states.
As referee suggested, we performed some tests on the effects of U{O 2p } on the calculated properties, like the vacancy formation energy as a function of U{O 2p } (Fig.   R1a), as well as the phonon dispersions with and without U{O 2p } (Fig. R1b).
Consistent with the previous study 2 , we found that the result of vacancy formation energy is more reasonable with U{O 2p } = 5.5 eV than that with U{O 2p } = 0. In addition, the distinction is insignificant between the phonon dispersions with and without U{O 2p }. As a consequence, we employ U{O 2p } = 5.5 eV to model the defect structure preferably with minimal effect on the other properties.
As suggested, the English usage has been revised, and more explanations with respect to the U values have been included in SI.

Correction in SI:
All the calculations in the present study were performed with Vienna ab initio simulation package (VASP) 19 , using the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) and the projector augmented wave (PAW) potential 20 .
The energy cutoff of the plane-wave basis is 500 eV. It is known that standard DFT functionals are incapable of correctly modeling O-derived defect states due to the inherent self-interaction error (SIE) [1][2][3]5 . For the reduced CeO 2 system, i.e., CeO 2 that contains intrinsic O vacancies, such problem is acute for both Ce 4f and O 2p states 2,8,12 .

Q3:
The authors consider that 36 calculations in the 2x2x2 is a high computational burden. Actually with the present computers it is not and as I consider that this particular result is important I recommend again the authors to address my previous Question 2 in a more adequate manner.

A3:
We appreciate the question mentioned in this comment, and the issue of concern has been deliberated carefully before we make the response.
First, we would like to make it clear that the 2 × 2 × 2 supercell of CeO 1.75 with different arrangements of oxygen vacancies will lead to 36 configurations (not 36 calculations, we apologize for the mistake in the previous response), most of them have P1 space group based on the symmetry analysis (discussed in Appendix R1).
For each configuration, such a low symmetry will give rise to hundreds of displacement patterns when we use the finite displacement method for the phonon dispersion calculations (i.e. hundreds of POSCAR files with Phonopy software and VASP calculation), so in total there will be over ten thousands POSCAR files for the listed 36 configurations (Table R1). This is a tremendous computing workload if we calculate the phonon dispersions for all the configurations with different arrangement of oxygen vacancies.
In addition, our STEM-ABF image (Fig. 1d) firmly evidences that the oxygen vacancies are distributed dispersively without any ordering in the lattice of the nanoparticles. Therefore, it is not likely that some particular arrangement of the generated vacancies induces the tetragonal distortion. Thus, we consider the stress effect induced by the oxygen vacancies on the structure phase transition rather than the effect of vacancy formation. It is known that the heterogeneously distributed oxygen vacancies in ceria could give rise to the chemically induced stress 21,22 , which, according to our DFT results, leads to the phonon softening that drives the observed tetragonal-cubic phase transition.

Appendix R1:
The typical fluorite structure was applied to build the 2 × 2 × 2 supercell of ceria (96 atoms). From the XPS results (Fig. S12), the concentration of the oxygen vacancies in 5 nm ceria is around 15 %. Consequently, eight vacancies have been included in the 2 × 2 × 2 supercell (corresponding to CeO 1.75 ) in different configurations. We removed one oxygen vacancy for each unit cell (eight in total), and the vacancy sites are numbered from 1 to 8 (Fig. R2a). In addition, the locations of the eight unit cells are also marked from A to H (Fig. R2b). As a result, the location of each oxygen vacancy in the supercell could be marked, such as A1, F7, etc.
We distinguish the configurations by their specific vacancy arrangements (e.g.,  Table R1. The overall number of POSCAR files generated from the listed configurations is 12963.   Q4: Figure S14 shows no localized electrons in a single Ce atom but rather the average structure with multiple Ce getting density by reduction. I refer the authors to the extensive literature describing that localization improves the energetics of the systems by significant amounts and thus that a more adequate treatment of the localization needs to be implemented.

A4:
Thanks the referee for the comment. In the present study, electron paramagnetic resonance (EPR) was employed to directly observe the paramagnetic centers, i.e., excess spins with unpaired electrons, in both tetragonal and cubic ceria. From the EPR results ( Fig. 4a-b), excess spins in both oxygen vacancies and Ce atoms (resulting in Ce 3+ states) were found in tetragonal ceria, whereas only Ce 3+ states were found in cubic ceria.
On the other hand, the above charge transition between oxygen vacancies and Ce (III) states was also verified by the DFT calculation of spin charge density, which has been frequently used to describe the electronic structure associated with defects 5,23 . The

Correction in the manuscript:
As seen in Fig. 4a, the symmetric signal at g = 2.003 is assigned to the unpaired electrons trapped in oxygen vacancies (paramagnetic defect with excess spins, − ) 24 , whereas the axial signals with g⊥= 1.967 and g // = 1.947 are assigned to the paramagnetic Ce 3+ sites with unpaired f electrons 25 . During the phase transition upon heating, a charge transfer occurs from the ( − s) of tetragonal ceria to the Ce f orbitals of cubic ceria (Fig. 4b), suggesting that the charge states of defects in the energy gap could play a key role in the stability of different configurations. In addition, such charge transfer between tetragonal and cubic phase has been also verified by our DFT calculation of spin charge density (Fig. S14, details in in Supplementary Materials).

Correction in SI:
Notably, excess spins were found in the vacancy of tetragonal lattice, while no excess spins were found in the vacancy of cubic lattice. The calculated results accord well with the EPR experiment ( Fig. 4a-b), and we suppose the distinction of charge states could play a key role in the stability of different configurations Responses to Reviewer #3 Q1: The difference in lattice constants with and without internal standard is NOT due to a "zero-shift error", it is due to a change in sample height (which is similar to zero error at low angles, but follows a different angular dependence). Thus it should be called a sample height error, not a zero shift.

A1:
Thank you for the correction. The inaccurate description in SI has been revised.

Correction in SI:
Note that the difference of the lattice constants extracted with or without the internal standard is subtle, which indicates the systematic error, derived from the change in sample height, has been corrected maximally through the Rietveld refinements.
Q2: Also, while the difference may be small, the authors should use the best possible numbers, which means the ones collected with internal standard. There is no need to use/report the uncorrected numbers! A2: Thank you for the suggestion. We have included the lattice constants of 5 nm ceria calibrated with internal standard in the present study. Figure 1:

Correction in SI:
For the 5 nm ceria, the lattice constants extracted from variable temperature data have been calibrated by quartz (SiO 2 ) internal standard (Fig. S5a). Figure S5: show the difference between the raw data and the calculated patterns. Table S1:   Table S1.

Correction in
The unit cell parameters (a-axis) at different temperatures for 5 nm, 9 nm, 18 nm and bulk ceria obtained using XRD Rietveld refinement.