Theoretical insights into the Peierls plasticity in SrTiO3 ceramics via dislocation remodelling

An in-depth understanding of the dislocations motion process in non-metallic materials becomes increasingly important, stimulated by the recent emergence of ceramics and semiconductors with unexpected room temperature dislocation-mediated plasticity. In this work, local misfit energy is put forward to accurately derive the Peierls stress and model the dislocation process in SrTiO3 ceramics instead of the generalized stacking fault (GSF) approach, which considers the in-plane freedom degrees of the atoms near the shear plane and describes the breaking and re-bonding processes of the complex chemical bonds. Particularly, we discover an abnormal shear-dependence of local misfit energy, which originates from the re-bonding process of the Ti-O bonds and the reversal of lattice dipoles. In addition, this approach predicts that oxygen vacancies in the SrTiO3 can facilitate the nucleation and activation of dislocations with improvement of fracture toughness, owing to the reduction of average misfit energy and Peierls stress due to the disappearance of lattice dipole reversal. This work provides undiscovered insights into the dislocation process in non-metallic materials, which may bring implications to tune the plasticity and explore unknown ductile compositions.

In summary, the highly desired correlation to experiment giving this study broader impact cannot be soundly done in the way presented. 3) Please give the Peierls-Nabarro equation for reference. 4) Figure 6 needs a scale bar. 5) Figure 1: The arrows drawn indicate a uniaxial stress but not a shear stress, please correct. 6) Disregistry. What does this word mean? 7) Figure 3b has the unit dislocation density in the surrounding of a dislocation on the y-axis. Dislocation density quantifies the number of dislocations in a volume/are. The unit does not appear to make sense here. 8) While language is generally okay, there are multiple incidents where language is not at 100%. E.g. line 317 says " Fig. 4c shows the distribution of dislocation core." This seems incomplete or grammatically incorrect. A careful revision of the language (maybe with editorial assistance) may remedy this. Also, small things such as line 454 "data availability" not "date availability". 9) Line 101: experimental values are cited with ref. 18 (Nabarro 1947). Please provide the experimental value and try to find a reference specific to SrTiO3 so that your derived value can be better compared. 10) Line 247. How can only half a b-vector be observed? Is this a statistically represeantative case? E.g. Jin et al. (TEM study of < 110 >-type 35.26 degrees dislocations specially induced by polishing of SrTiO3 single crystals) and others found that partial dislocations only dissociated about 5 unit cells.
Reviewer #3 (Remarks to the Author): The submitted manuscript deals with the estimation of the lattice resistance to dislocation motion in SrTiO3 with and without the presence of an oxygen vacancy. The authors propose a new way of calculating the Peierls stress which increases the previous estimates from very few to hundreds of MPa, with the latter much closer to experimental observations.
Overall, I agree with the authors that this is an interesting and still very poorly understood topic and that new approaches that take into account the peculiarities of ordered crystals are needed.
However, I do have several reservations about the presented hypotheses, data from the literature and mechanisms: -the main assumption, i.e. that in the literature the Peierls stress is solely estimated based on a GSF method that does not take into account any relaxation normal to the slip plane, is not a complete view of the literature. For example, Cordier, Carrez et al. have calculated the Peierls stresses of several complex crystals including perovskites and using GSFs on different crystal planes can achieve a good estimate of the core structure and Peierls stress. Among their work there is also an estimate of 350 MPa for the Peierls stress of SrTiO3 (although they did estimate much much less before using a 1D estimate), which is very close to what the authors suggest here. An example of this work can be found here: https://doi.org/10.1016/j.scriptamat.2010.04.045 -there are also other methods which allow the estimation of energy barriers in deformation of complex crystals and which account for the fact that dislocation motion may not be planar or happen along a rigid shear direction. This is the case for example for synchroshear, which was first suggested for Al2O3 but is mainly associated with Laves phases. An example may be found here: https://www.sciencedirect.com/science/article/pii/S1359646219301599 -the authors plot in figure 5 and discuss in the text the acceleration of kink formation in the presence of a vacancy. However, there is no discussion as to what happens as the dislocation line travels beyond the vacancy -pinning is also an option, which may result in the opposite effect overall and I would suggest to discuss this explicitly, hoping that the available data allows this directly already.
-it is not quite clear why the authors relate dislocation mobility primarily to the more indirect measure of crack propagation, although slip bands are clearly indicated in their experimental figures. Is this as the surface obscures slip bands more completely after reduction? Why discuss the measured hardness directly. there is quite a bit of work on SrTiO3, its dislocations and also the role of vacancies in these works that carry out a very careful dislocation etchpit analysis: https://doi.org/10. 1016/j.scriptamat.2020.07.033 & https://doi.org/10.1111/jace.18118 Overall, I am therefore doubtful of the completeness of the work in terms of its comparison with other methods and how it is embedded in the literature, as some very basic claims made in the paper do not appear entirely accurate.

Response Letter
We appreciate the constructive and helpful suggestions from the reviewers and we have revised the manuscript accordingly. The changes are marked in the revised manuscript, and the responses to the comments are listed as follows: Reviewer #1 (Remarks to the Author) Comment_0: The manuscript "Theoretical insights into the Peierls plasticity in SrTiO3 ceramics via dislocation remodelling" by Y. Li et al. reports a modified Peierls-Nabarro model by employing a local misfit energy (gradient) in the model, enabling a high accuracy prediction of the dislocation mobility, such as the Peierls stress, in a model system of SrTiO3, representing the more difficult to model non-metallic material systems using the traditional method. Further, this methodology was applied to the oxygen deficient SrTiO3 and found improved ductility, thus offering new implications to tune the plasticity and explore new ductile compositions.

Response_0: Thanks for the comments.
Comment_1: It is generally believed that adding a local energy gradient term should improve the traditional generalized stacking fault (GSF) based on Peierls-Nabarro model. So the authors' exploration in this regard is certainly valuable. The question usually falls into how effective for any modified approach could achieve in terms of the desired accuracy? The comparison resulting in a close agreement between the calculations in the present manuscript and the experimentally measured range at finite temperatures is still not enough, to warrant it's publication in Nature journals such as Nature Communication, in my view. Nowadays, a direct DFT calculation of the Peierls stress in non-metallic materials is computational workable, in combination with anisotropy elasticity theory to take care of the finite supercell size correction in DFT results. I suggest the authors to work on these calculations, in order to validate the effectiveness of the modified theory. If successful, the modified theory is envisioned to have great impact, for example, paying a way for high prediction power in highthroughput screening of non-metallic materials using the much faster Peierls-Nabarro model, as well as in material design.

Response_1
: Thanks for the comment and suggestion. We attempted to perform a direct DFT calculation of the Peierls stress as the reviewer suggested. According to our calculations in the manuscript, the size of <110>{110} edge dislocation in SrTiO3 is about 5 nm. We built a 10×11×24 supercell, in which two symmetrical dislocations are introduced using the Molecular Dynamics (MD) simulations following the same method as illustrated in the supporting materials. The process of building the dislocations is shown in the figure below. For the purpose of minimizing the amount of calculation, the relaxed structure was cut into a 1×11×24 supercell containing 2525 atoms. We used 15 nodes (64 cores per node) to carry on the DFT simulations on this large model, but the calculation was still difficult to run smoothly. First principles pseudopotential calculations are really computational demanding. In fact, the DFT calculations on the model with more than a thousand of atoms have not been reported by far. The VASP package required about 10 hours to handle a single ionic relaxation step for the dislocation model even though 960 cores were applied. Besides, technical errors were often reported because the supercell was too large for the electronic minimization algorithm in the VASP package. The current scale of computing resource is large enough for the parallel operation. Further increase of nodes has little effect on improving the computing efficiency because of the consumption between the nodes. Thus, unfortunately, it is still early to say the current DFT calculations are able to handle a dislocation model even with the minimum size.  Besides the accuracy in the prediction of Peierls stress, our model also provides physical insights into the Peierls plasticity of SrTiO3, such as the re-bonding process of the Ti-O bonds and the reversal of lattice dipoles, which has not been disclosed before.
In addition, our theory provides a practical approach to deal with the complex model accompanied with defects, which the traditional GSF method fails to handle.

Revision_1:
The relative discussions have been added in page 6, paragraph 1 and page 14, paragraph 1 of article and page 4 of the supporting materials.

Reviewer #2 (Remarks to the Author)
The authors present a study on an improved model to calculation of the Peierls stress for <110>{110} type dislocations in SrTiO3 which should better account for the complex bond nature and the breaking and re-bonding process. The authors also attempt to relate their theoretical findings to experimentation. Progress in better modeling and a broader understanding around the Peierls stress in ceramics is timely needed and the topic merits interest from nature communications.
Unfortunately, point 1 and 2 below may be very impactful concerns to the technical accuracy. Following those two key concerns, I suggest to reject the manuscript in its current form but to consider a re-submission. After that the list of smaller comments may be redone or extended.
Response_0: Thanks for the encouraging evaluation and the valuable comments. Point 1 is a misread for the horizontal axis of Fig. 4a. For point 2, we have renewed our view on the dynamic process of dislocation based on the reviewers' constructive suggestions.
New simulations and experiments were performed and the results were added to support our analysis. All the smaller comments have also been addressed.
Comment_1: Line 299-301 "…, resulting in the singularity at the initial position." This seems strange. When consulting with Figure 4a, the restoring force (blue) is zero at the minimum of the energy landscape which should be okay. This rises the question about the burgers vector. In the rigid model, the spacing between the energy minima is about 5.5 A. This makes sense as this is sqrt(2)*3.905=b. However, in the new model the energy minima are space 6.1 A apart which is larger than the burgers vector of SrTiO3.
This is an unacceptable discrepancy and my be the reason for the unphysical mismatch.
At the moment I do not see a possibility how a model can generate correct values with an incorrect burgers vector. Likely larger revision or detailed discussion is needed here.

Response_1
: Thanks for the comment. It is actually a misunderstanding for Figure 4a.
The magnitudes of horizontal axis were misread. As shown in the figure below, we used 5.5774 Å as the length of Burgers vector for both the rigid model and the new model.
In the rigid model, the pacing between the energy minima is 5.0197 Å. The value is smaller than the length of Burgers vector, which is the origin of the singularity of the restoring force as we discussed in the manuscript.        Fig. 6c to support the analysis on the fracture toughness. Comment_3: Please give the Peierls-Nabarro equation for reference.

Response_3:
The Peierls-Nabarro equation has been added in page 4, paragraph 1 of article.
Comment_4: Figure 6 needs a scale bar

Response_4:
The scale bar has been added in the figure.
Comment_5: Figure 1: The arrows drawn indicate a uniaxial stress but not a shear stress, please correct.

Response_5:
The illustration "shear direction" in Figure 1 has been replaced by "uniaxial displacement". Revision_6: This word may be unfamiliar to readers, so we have replaced it by the phrase "atomic misfit".
Comment_7: Figure 3b has the unit dislocation density in the surrounding of a dislocation on the y-axis. Dislocation density quantifies the number of dislocations in a volume/are. The unit does not appear to make sense here.

Response_7:
In the Peierls-Nabarro model, the dislocation core is regarded as a continuous distribution of shear S(x) or infinitesimal dislocations with density ρ(x) , where x is the coordinate in the glide plane along the direction normal to dislocation line. The dislocation density ρ(x) in Figure 3b refers to the density of infinitesimal dislocations that distribute along the x-direction.

Revision_7: The explanation on ρ(x) has been added in the captions of Figs. 3 and 4.
Comment_8: While language is generally okay, there are multiple incidents where language is not at 100%. E.g. line 317 says " Fig. 4c shows the distribution of dislocation core." This seems incomplete or grammatically incorrect. A careful revision of the language (maybe with editorial assistance) may remedy this. Also, small things such as line 454 "data availability" not "date availability".

Response_8:
The manuscript has been carefully polished. The sentence " Fig. 4c shows the distribution of dislocation core." has been modified as " Fig. 4c shows the xdependence of S(x) and ρ(x)." The spelling mistake "date availability" has been revised.
Other revised grammatical and spelling mistakes have been marked in the manuscript. others found that partial dislocations only dissociated about 5 unit cells.

Response_10:
The HRTEM image is not clear near another partial dislocation, so we overlapped the HRTEM image with its Fourier filtered image in the area encircled by the white rectangle. As shown in the figure below, this partial dislocation is marked by the blue circle. The separation of the two parts is about 2.6 nm.

Reviewer #3 (Remarks to the Author)
The submitted manuscript deals with the estimation of the lattice resistance to dislocation motion in SrTiO3 with and without the presence of an oxygen vacancy. The authors propose a new way of calculating the Peierls stress which increases the previous estimates from very few to hundreds of MPa, with the latter much closer to experimental observations.
Overall, I agree with the authors that this is an interesting and still very poorly understood topic and that new approaches that take into account the peculiarities of ordered crystals are needed.
However, I do have several reservations about the presented hypotheses, data from the literature and mechanisms.
Response_0: Thanks for the evaluation and the comments. An in-depth understanding of the dislocation motion process in non-metallic materials becomes increasingly important. Our work attempts to address this issue by proposing a local-misfit-energy (LME) method based on the Peierls-Nabarro theory. The method can achieve a good estimate of Peierls stress, and more importantly, provides detailed physical insights into the Peierls plasticity of SrTiO3, such as the re-bonding process of the

Response_1
: Thanks for the comment. We didn't assume that the GSF method does not take into account any relaxation normal to the slip plane in literature. On the contrary, our main assumption is that the GSF method only takes into account relaxation normal to the slip plane, but the degrees of freedom parallel to the slip plane are constrained. Accordingly, the GSF energy only includes the inelastic strain energy.
However, in the PN model, the restoring force originates from the strain energies including both elastic and inelastic parts. The ignorance of the elastic part makes the GSF approach fail to account for the bond breaking and re-bonding process in nonmetallic materials, resulting in an inaccurate physical picture of the dislocation motions.
Our method introduces the in-plane freedom degrees for the atoms near the shear plane, so the misfit region is no longer restricted to the shear plane and the elastic energy can be accurately taken into account. where f is a two-dimensional field which is expressed in the normal basis of the slip plane, and E isf is the inelastic stacking fault energy from which all the linear elastic part has been subtracted. As the authors mentioned in the literature, the f field, which provides a displacement jump when crossing the slip plane of dislocation, is purely along <110>. In another word, the slip system of dislocation is the same as their 1D estimate [Phys. Rev. B, 2008, 77: 014106]. Therefore, the major amelioration in the PNG model is to take the elastic strain energy into account. The calculated energy ε is beyond the GSF energy, which leads to a better estimate of 350 MPa for the Peierls stress.
However, Carrez et al. estimated the displacement field u based on an elementfree Galerkin method (an approximation of a continuous field representation), which is a numerical algorithm and do not explicitly consider the atomic scale details of dislocation cores. Although the equivalent of inelastic energy can be roughly estimated, the physical mechanism of the slip process that are closely related with the atomic structure of SrTiO3 cannot be provided. Besides, the PNG method is still based on the GSF model and uses the GSF energy as an input [Denoual C, Phys. Rev. B 2004, 70: 024106]. Therefore, the PNG method has the same problem with the traditional GSF approach that it cannot treat dislocation structures with impurities or defects. In contrast, the estimation of the total strain in our manuscript is based on the first principles calculation. The simulation of atomic displacements is closely related to the properties of SrTiO3 as we analyzed in the manuscript. Our method can achieve a good estimate of Peierls stress, and more importantly, provides unexplored physical insights into the Peierls plasticity of SrTiO3, such as the re-bonding process of the Ti-O bonds and the reversal of lattice dipoles, which is one of the main targets of our manuscript. In addition, our theory provides a practical approach to deal with the complex model accompanied with defects, such as oxygen vacancies, which has been a pending issue for many years.

Revision_1:
The comparison with the PNG method has been added in page 5, paragraph 2 of article.
Comment_2: there are also other methods which allow the estimation of energy barriers in deformation of complex crystals and which account for the fact that dislocation motion may not be planar or happen along a rigid shear direction. This is the case for example for synchroshear, which was first suggested for Al2O3 but is mainly associated with Laves phases. An example may be found here: https://www.sciencedirect.com/science/article/pii/S1359646219301599. The purposes of the NEB method and our work are totally different. The NEB method is applied to find the slip path of dislocation, but our method based on the PN theory aims to calculate the Peierls stress and the structure of dislocation core. Besides, the two methods are based on the different models, so the energy barriers calculated in the NEB approach cannot be used to solving the Peierls-Nabarro equation. Accordingly, the calculations of the Peierls stress and the structure of dislocation core are beyond the ability of the NEB approach. The NEB method may be good at finding the slip path of dislocation, but it is not applicable to investigate the activation and nucleation of dislocation, which are important to the mechanical toughness especially for ceramic materials (the details on dislocation and toughness are shown in Response 3).

Revision_2:
The comparison with the synchroshear has been added in page 5, paragraph 2 of article.
Comment _3: the authors plot in figure 5 and discuss in the text the acceleration of kink formation in the presence of a vacancy. However, there is no discussion as to what happens as the dislocation line travels beyond the vacancy -pinning is also an option, which may result in the opposite effect overall and I would suggest to discuss this explicitly, hoping that the available data allows this directly already.
-it is not quite clear why the authors relate dislocation mobility primarily to the more

Reviewers' comments:
Reviewer #1 (Remarks to the Author): The authors have done substantial work for the revision of the manuscript, and in many places, the changes are in my view, appropriate. Overall, the work represents a good advancement in the field and is expected to have a high impact. Therefore, I recommend it be published in Nature Communications.
Reviewer #3 (Remarks to the Author): Thank you for the very detailed responses and the additional work and wording which was done to strengthen the conclusions of the manuscript.
I would be happy for this manuscript to be published.
Reviewer #4 (Remarks to the Author): The fundamental understanding of dislocation dynamics in oxides is of urgent interest and importance, particularly in light of the recent upsurge in the dislocation mechanics and functionality studies in such materials that exhibit dislocation plasticity (even at room temperature), which make such materials hold potential for technological applications. This work proposes a new simulation protocol using "local misfit energy" to be distinguished from GSF energy to address the Peierls energy for dislocation behavior in SrTiO3 (STO). Although the attempt merits its value, however, due to the following critical aspects, the reviewer cannot recommend this work for publication, not in the current journal nor in other scientific journals in its present form: 1. The room-temperature dislocations in single-crystal STO are dominantly screw type, not edge type, as has been extensively reported in the literature, see e.g., Brunner et al., Acta Mater., 2006 (the authors were also citing this paper for comparing the CRSS with their simulation in Fig. S4 in the Rebuttal letter, which makes little sense to the reviewer). The current simulation focuses on the edge type and attempts to extend the conclusion to the dislocation dynamics at room temperature in STO. This could be a fundamentally misleading effort and the authors should consider extending their approach to screw-type dislocation. It is worth mentioning that, edge dislocations can be dominating at high temperatures, and the simulation effort in the current work still has its merit and the authors may consider a such extension to high-temperature slip system in STO in their future study. 2. Following point 1, screw-type dislocations in STO tend to cross slip and greatly promote the dislocation multiplication hence dislocation density, and this process is way more important than dislocation nucleation and motion in this case, in order to promote the plasticity in STO at room temperature. Similar results are also readily available in the textbook knowledge on LiF (same cubic structure as STO) regarding dislocation plasticity. See the book by Hull & Bacon, Introduction to Dislocations. 3. It is appreciated that the authors attempted to validate their simulation prediction in experimental tests, particularly related to the Vickers hardness and fracture toughness in reference and reduced STO samples. However, such an attempt is poorly justified based on the following points: 3.1. Besides dislocation nucleation and motion, the authors ignored (or maybe were not aware of) the dislocation multiplication in point 2. During the Vickers indentation process (as the Indenter is pressing into the material), consider the dislocation generation, it will be dislocation nucleation, multiplication, while being accompanied by movement of the mobile dislocations. These processes can be greatly modified/affected by the defect chemistry (e.g., oxygen vacancy concentration) of the samples. 3.2. Consider the scenario that higher oxygen vacancy concentration in the reduced samples could have enhanced the dislocation multiplication (which is likely possible), the total input energy from the indentation will be largely dissipated by the generation of more dislocations in the reduced samples, hence reducing the energy available for crack formation (hence crack length could decrease). This point must be considered and checked before one could assess the crack length and indentation fracture toughness using such a method. In combination with the 2nd reviewer's 2nd comment (regarding crack tip dislocation nucleation, etc.) in the last round review, the authors should have a more complete picture now for the deformation process. 3.3. The authors mentioned the term "toughness" and "fracture toughness" several times in the manuscript. These two terms are completely different and must be clarified. Fracture toughness refers to the resistance to crack propagation, while toughness can be described by the area under the stress-strain curves. In this respect, the toughness of the reduced sample is much smaller than the reference sample before reduction treatment (Fig. S6), while the authors are trying to prove the reduced sample was supposed to display higher fracture toughness (KIc in this case). Please check these basic concepts in mechanics textbooks. 4. The authors discussed the oxygen vacancy on the dislocation nucleation, which also agrees with the most recent experimental results such as by Stich et al. J. Am. Ceram. Soc 2022 and Fang et al, Scripta Mater., 2020 as mentioned by the authors. These are very encouraging matches between simulation and experiments. However, it should be pointed out that these works used nanoindentation tests to probe the dislocation nucleation in a very small stressed volume. In the current work, the bulk uniaxial compression tests were performed to validate the oxygen vacancies effect on the Peierls stress: the authors should be aware that the yielding (incipient plasticity in bulk sample) in this case is mostly achieved by dislocation multiplication and motion process (see point 3 above), while dislocation nucleation is not relevant anymore. It is also very alarming to the reviewer that the authors cut the samples first into 2x2x4 mm3 in size and then performed the compression tests: cutting STO samples would immediately induce many surface dislocations (penetrating into the surface by about several micrometers) and these surface dislocations are effective sources to multiply many more new dislocations to promote the plasticity. Plus that the reduction at 1450C for 6 hours would also anneal some surface dislocations, making the two samples have different initial states (reference sample with surface dislocations from cutting and without thermal treatment, and reduced sample with thermal treatment and most likely a different surface dislocation structure). The careful treatment of these cut samples (e.g., to polish away the surface cut induced regions) to remove such surface dislocations must be added to make sure the samples do have similar initial dislocation conditions. The surface dislocations by cutting can be easily checked by the optical microscope to see abundant slip traces. 5. Regarding the reducing treatment in Argon gas, although the authors measured the electrical resistivity to suggest a higher oxygen vacancy, a most robust analysis should be performed using impedance spectroscopy, which is rather a standard technique for such purposes. 6. The overall experimental section seems very weak and poorly defined to the reviewer, and the authors are suggested to check the literature with experimental details in recent years or consult experimental experts to avoid such experimental pitfalls.

Response Letter
We appreciate the constructive and helpful suggestions from the new reviewer and we have revised the manuscript accordingly. The changes are marked in the revised manuscript, and the responses to the comments are listed as follows:

Reviewer #4 (Remarks to the Author)
Comment_0: The fundamental understanding of dislocation dynamics in oxides is of urgent interest and importance, particularly in light of the recent upsurge in the dislocation mechanics and functionality studies in such materials that exhibit dislocation plasticity (even at room temperature), which make such materials hold potential for technological applications. This work proposes a new simulation protocol using "local misfit energy" to be distinguished from GSF energy to address the Peierls energy for dislocation behavior in SrTiO3 (STO). Although the attempt merits its value, however, due to the following critical aspects, the reviewer cannot recommend this work for publication, not in the current journal nor in other scientific journals in its present form.
Response_0: Thanks for your evaluation and the valuable comments. We have responded to the reviewer's concern on the screw-type dislocation in Response_1 and 2. With the help of the reviewer's suggestions, the discussions in the experimental section have been strengthened and new experiments have been added. All the comments have been addressed point by point.

Comment_1:
The room-temperature dislocations in single-crystal STO are dominantly screw type, not edge type, as has been extensively reported in the literature, see e.g., Brunner et al., Acta Mater., 2006 (the authors were also citing this paper for comparing the CRSS with their simulation in Fig. S4 in the Rebuttal letter, which makes little sense to the reviewer). The current simulation focuses on the edge type and attempts to extend the conclusion to the dislocation dynamics at room temperature in STO. This could be a fundamentally misleading effort and the authors should consider extending their approach to screw-type dislocation. It is worth mentioning that, edge dislocations can be dominating at high temperatures, and the simulation effort in the current work still has its merit and the authors may consider a such extension to high-temperature slip system in STO in their future study.

Response_1
: Thanks for your comment. We notice that the major concern of the reviewer is that the room-temperature dislocations in single-crystal STO are dominantly screw-type, and our simulation on edge dislocations does not accord with the experimental fact. However, we did a thorough literature survey on the dislocation type of STO at room temperature. We found all the published papers clearly conclude that  In summary, edge dislocations dominate the plasticity of STO at room temperature and screw dislocations are hardly observed until the later stage of plastic deformation.
The major characteristics of dislocations in STO, including the dislocation structure, average misfit energy and Peierls stress, which are also the main target of our paper, are all determined by the earlier stage of plastic deformation and the edge dislocations.
Therefore, we believe our study on the edge dislocations in STO accords with the experimental fact and can support the main conclusions of this manuscript.
Of course, we should have compared our calculated Peierls stress with the experimental data of edge dislocations, instead of those at the low temperatures (below 150 K) that are mainly governed by screw dislocations. Thanks for pointing out the inappropriateness in Fig. S4.

Revision_1:
The extrapolation of CRSS is recalculated by fitting the experimental data near room temperature. The plastic behavior is mainly governed by edge-type dislocations in this temperature range. The fitting curve in Fig. S4  Although LiF also has the same cubic structure as STO, the cross-glide of screw dislocations in LiF has a dominating effect at room temperature [W. G. Johnston et al., J. Appl. Phys., 1960, 31: 632], which is different from STO.
Comment_3: It is appreciated that the authors attempted to validate their simulation prediction in experimental tests, particularly related to the Vickers hardness and fracture toughness in reference and reduced STO samples. However, such an attempt is poorly justified based on the following points: 3.1. Besides dislocation nucleation and motion, the authors ignored (or maybe were not aware of) the dislocation multiplication in point 2. During the Vickers indentation process (as the Indenter is pressing into the material), consider the dislocation generation, it will be dislocation nucleation, multiplication, while being accompanied by movement of the mobile dislocations. These processes can be greatly modified/affected by the defect chemistry (e.g., oxygen vacancy concentration) of the samples.
3.2. Consider the scenario that higher oxygen vacancy concentration in the reduced samples could have enhanced the dislocation multiplication (which is likely possible), the total input energy from the indentation will be largely dissipated by the generation of more dislocations in the reduced samples, hence reducing the energy available for crack formation (hence crack length could decrease). This point must be considered and checked before one could assess the crack length and indentation fracture toughness using such a method. In combination with the 2nd reviewer's 2nd comment (regarding crack tip dislocation nucleation, etc.) in the last round review, the authors should have a more complete picture now for the deformation process. 3.3. The authors mentioned the term "toughness" and "fracture toughness" several times in the manuscript. These two terms are completely different and must be clarified.

Response_3
Fracture toughness refers to the resistance to crack propagation, while toughness can be described by the area under the stress-strain curves. In this respect, the toughness of the reduced sample is much smaller than the reference sample before reduction treatment (Fig. S6), while the authors are trying to prove the reduced sample was supposed to display higher fracture toughness (KIc in this case). Please check these basic concepts in mechanics textbooks.
Thanks for clarifying the difference between these two terms.

Revision_3.3:
The term "toughness" has been modified as "fracture toughness" in the manuscript. However, it should be pointed out that these works used nanoindentation tests to probe the dislocation nucleation in a very small stressed volume. In the current work, the bulk uniaxial compression tests were performed to validate the oxygen vacancies effect on the Peierls stress: the authors should be aware that the yielding (incipient plasticity in bulk sample) in this case is mostly achieved by dislocation multiplication and motion process (see point 3 above), while dislocation nucleation is not relevant anymore.  Soc. 2011, 94: 3104-3111]. No slip traces were observed in the polished samples, while the surface stress is also introduced by the polish process. In order to release the surface stress and gain similar initial surface conditions, the polished samples were divided into two groups. One is reduced at 1450 ºC for 6 hours in the Argon atmosphere as the oxygen-deficient samples, and the other is also heated at 1450 ºC for 6 hours in the air atmosphere as the reference samples.

Revision_4.2:
The details on the sample preparation for the compression tests have been added in the caption of Fig. S6. The relevant instruction has been added in page 23, paragraph 3 of article: "The uniaxial compression tests were performed by using the SrTiO3 crystals in the shape of quadrangular prisms with dimensions 2×2×4 mm 3 .
The details on the specimen preparation are available in the supporting materials." Comment_5: Regarding the reducing treatment in Argon gas, although the authors measured the electrical resistivity to suggest a higher oxygen vacancy, a most robust analysis should be performed using impedance spectroscopy, which is rather a standard technique for such purposes.

Reviewers' comments:
Reviewer #4 (Remarks to the Author): The Reviewer very much appreciates the Authors´ efforts in systematically investigating and discussing the dislocation types, and (dominating) dislocation mechanisms, particularly at different deformation stages, as well as their new experiments in the response letter and the revised manuscript. The revisions have now clearly stated the novelty as well as its limitations (e.g., focusing on the initial stage of deformation, and not being able to address multiplication, etc.) of this work, which should be very helpful to the potential communities that will later march into this topic `dislocations in ceramics` (which is an old but also new topic). It has been a very helpful interaction between the reviewer and the authors. I am happy to recommend the publication of this work.