A large modulation of electron-phonon coupling and an emergent superconducting dome in doped strong ferroelectrics

We use first-principles methods to study doped strong ferroelectrics (taking BaTiO3 as a prototype). Here, we find a strong coupling between itinerant electrons and soft polar phonons in doped BaTiO3, contrary to Anderson/Blount’s weakly coupled electron mechanism for "ferroelectric-like metals”. As a consequence, across a polar-to-centrosymmetric phase transition in doped BaTiO3, the total electron-phonon coupling is increased to about 0.6 around the critical concentration, which is sufficient to induce phonon-mediated superconductivity of about 2 K. Lowering the crystal symmetry of doped BaTiO3 by imposing epitaxial strain can further increase the superconducting temperature via a sizable coupling between itinerant electrons and acoustic phonons. Our work demonstrates a viable approach to modulating electron-phonon coupling and inducing phonon-mediated superconductivity in doped strong ferroelectrics and potentially in polar metals. Our results also show that the weakly coupled electron mechanism for "ferroelectric-like metals” is not necessarily present in doped strong ferroelectrics.

. "the exactly same electron concentration, indicating that the additional increase in ImΠqν arises solely from the crystal structure difference. Lowering the symmetry of BaTiO3 crystal structure allows more electron-phonon scattering processes that would be forbidden in the cubic structure by symmetry considerations. " Is this a quantitative or qualitative statement? I suggest quoting numbers. Is this general or specific to these calculations? Seems to be too important to pass through so quickly -need more clarification and quantitative argumentation beyond "the plots show". Is there a more general insight to be learned? 16. "a small compressive strain that lowers the crystal symmetry can also increase its superconducting transition temperature, similar to doped SrTiO3" Please specify compressive strain in which direction. Also please see other relevant experiments in ref.
M. N. Gastiasoro, Superconductivity in Dilute SrTiO_3: A Review, Ann. Phys. 168107 (2020). In terms of the polar modes and anisotropies the strain direction is important. 17. "Since our simulation does not consider dopants explicitly, a more desirable doping method is to use electrostatic carrier doping " But then the structure doesn't not change. So how does electrostatic gating of the surface should work in terms of the structural phase transition? Are the authors saying that just strain and surface conductivity enough? Please clarify with more precision. 18. "We hope that our predictions will stimulate new experiments on doped ferroelectrics and, if confirmed, may help shed light on the mysterious origin of the superconductivity in doped SrTiO3." I don't think the projection for STO is that simple, given that the authors say that for their material Migdal's approximation is valid, while for STO it is not that straightforward. I think the overall projected connection to STO is somewhat premature, many more details would need to be covered to make any such conclusion.

Reply to Reviewers (NCOMMS-20-35777)
We would like to thank all the four reviewers for their helpful questions and comments. In the process of answering these questions/comments and revising the text, our manuscript has improved. Below, we address all their concerns in detail. The resulting modifications to the manuscript clarify some of the most important points of our work.
For each reviewer, we address each question/comment by first quoting the question/comment followed by our reply.

Response to Reviewer #1
We thank the reviewer for her/his comments that "the prediction is interesting and the work deserves to be published." 1) First of all, the capacity of first-principles theory to predict accurate T c is far from established. As the case of MgB 2 has shown, calculations of superconducting critical temperature are impressive in postdiction (and not in prediction) because of the crucial role of subtle details in electron-phonon coupling.
We agree with the reviewer that accurately predicting superconducting transition temperature T c has always been a daunting task. Even for conventional phonon-mediated superconductors, there is uncertainty in the Migdal-Eliashberg theory such as Morel-Anderson pseudopotential µ * . As we show in the inset of Fig. 3f in the main text, T c strongly depends on the value of µ * .
However, the calculation of electron-phonon coupling λ is more reliable, as long as Migdal approximation is valid [1]. In our calculations, the main conclusion is that the electronphonon coupling λ of doped BaTiO 3 can be modulated by carrier density n due to the strong coupling between soft polar phonons and itinerant electrons. Around the polarto-centrosymmetric structural phase transition, the polar phonon gets softened and λ is increased. As a consequence of this, a phonon-mediated superconducting dome emerges around the critical concentration n c . Our first-principles calculations support this picture.
As for the accurate value of superconducting transition temperature T c , we agree with the reviewer that there is uncertainty in our estimation due to µ * and other subtle technical details. That is exactly why we plot T c as a function of µ * to get a range of T c in our estimation.
On the other hand, we would like to mention that recently a couple of ab initio predictions of phonon-mediated superconductors have been later confirmed in experiments (see Table R1). While there is still much room for improving the accuracy of the predicted T c , it shows that first-principles calculations of electron-phonon coupling can point to the right direction of searching for new phonon-mediated superconductors.
In the revised version, we add text on Page 9 (highlighted in blue text) to stress that the predicted superconducting temperature T c has uncertainty due to µ * and other technical details, but the picture of an increased electron-phonon coupling around the structural phase transition in doped BaTiO 3 is robust.  4 39 [2] 17.5 [3] H 3 S 191-204 [4] 203 [5] LaH 10 238 [6] 250 [7] 2) In the present case, the authors begin by stating: "previous studies found that in n-doped BaTiO 3 , increasing the carrier density gradually reduces its polar distortions and induces a continuous polar-to-centrosymmetric phase transition... [31,32]." Ref. 31  We apologize for the confusion on this issue and we thank the reviewer for pointing this out to us.
We clarify here that we determine the critical concentration n c of doped BaTiO 3 using our own theoretical calculations, i.e. the concentration at which the polar displacements of BaTiO 3 just vanish. Our result (n c = 0.1e/f.u. 1.6 × 10 21 cm −3 ) is consistent with the previous theoretical studies [8,9] and is also close to the experiment (PRL 104, 147602 (2010)) in which the critical concentration is found to be about 1.9 × 10 21 cm −3 . down to the minimum accessible temperature of 2 K and the resistivity of this sample increases with cooling down to 2K. We explain that the concentration in this sample is larger than the critical value n c = 1.9 × 10 21 cm −3 , so we do expect that this sample has a cubic structure. Our first-principles calculations also find that doped BaTiO 3 at the same carrier density (n = 2.0 × 10 21 cm −3 ) has a cubic ground-state structure. We agree with the reviewer that the resistivity of that sample increases upon cooling, which deviates from a standard metallic behavior. The authors of PRL 104, 147602 (2010) did not explain the origin of this transport anomaly. However, we notice that the resistivity increases slowly upon cooling, implying that an energy gap is not opened (otherwise the resistivity would exhibit an exponential increase). The slow increase of resistivity might be due to weak localization, but high quality samples are needed to further clarify this important point.
In the revised version, we change the text from "consistent with previous studies" to "consistent with the previous theoretical studies" and add text on Page 4 and 5 (highlighted in blue text) to include information about the experimental results of doped BaTiO 3 .
3) It is easy to see that a comparison with Ca-doped strontium titanate has been the driving idea behind this study. However, even in pure strontium  We thank the reviewer for this important comment.
We first explain that pure SrTiO 3 is cubic at room temperature and develops an antiferrodistortive (AFD) rotation below 105 K (an out-of-phase oxygen octahedral rotation about the c-axis, a 0 a 0 c − in the Glazer notation) [10,11]. In Journal of Physics: Con- This figure is adapted from Fig.   3a in Ref. [26].
We thank the reviewer for this comment.
To our best knowledge, the reason why superconductivity in doped SrTiO 3 vanishes above a critical carrier density (sometimes called upper critical carrier density) is still not clear (we acknowledge private communications with Prof. Xiao Lin, the first author of Ref. [16] and [17]). There are a few theories [18][19][20][21][22][23][24][25] [26]. We would also like to mention that Fig. 3f in our main text uses a linear scale for carrier density, while Fig. 6a of Ref. [17], Fig. 3a of Ref. [26] and Fig. 4c of Ref. [27] all use a logarithmic scale for carrier density. This makes the T c dependence on carrier density seem gradual in our figure, but more abrupt in Ref. [17,26,27].
In the revised version, we add text on Page 2 to mention that it is still not clear why superconductivity in doped SrTiO 3 vanishes above the upper critical carrier density, in spite of the increasing density of states [26].

Response to Reviewer #2
We would like to thank the reviewer for commenting that "as far as I know, the e-soft ph In our study, we focus on the soft polar phonon of BaTiO 3 , which mainly involves the movement of Ti and O atoms. While oxygen vacancies can induce free electrons into BaTiO 3 , they inevitably break the TiO 6 oxygen octahedron and may affect the polar phonon. Therefore, we suggest that La-doping and ionic liquid gating are probably better methods of injecting electrons into BaTiO 3 because 1) in La x Ba 1−x TiO 3 samples, phase separation has not been reported in experiments [28,29]; 2) the carriers induced by ionic liquid gating are confined in the surface layers and thus chemical disorder is minimized [30].
In the revised version, we cite the paper PRB 84, 064125 (2011) (now Reference 46 in the main text) and add text on Page 4 and 5 to provide more discussion about oxygen-deficient BaTiO 3 .
2) The authors predicted that BTO may show superconductivity at low temperatures. However, it was never observed experimentally. The authors attributed it to the poor low temperature measurements in previous reports. We thank the reviewer for this important comment.
We agree with the reviewer that in La-doped BaTiO 3 , the randomness of La distribution may affect the transport properties in the normal state (the metallic state above the superconducting transition temperature). However, Anderson's theorem asserts that superconductivity in a conventional superconductor is robust with respect to non-magnetic disorder in the host material [31]. As a consequence, the superconducting transition temperature T c of a conventional superconductor barely depends on the randomness of defects. In our case, the superconductivity in doped BaTiO 3 is phonon-mediated (i.e. conventional) and La is a non-magnetic dopant. Therefore Anderson' theorem applies and we expect that the superconducting phase in doped BaTiO 3 is robust against chemical disorder (though the normal state resistivity of doped BaTiO 3 depends on sample qualities [28,29]).
On the other hand, we perform additional supercell calculations of La-doped BaTiO 3 . We study four representative cases: a 2 × 2 × 2 supercell with one Ba atom replaced by La to 3) The authors also suggested the electrostatic effect as a possible route to achieve uniform electron doping. However, the maximum doping from ion liquid gating is only about 10 20 − 10 21 cm −3 , that is too low to meet the requirement of doping levels in this manuscript.
We thank the reviewer for this comment.
Our calculations find that the optimal doping of BaTiO 3 is 0.1e/f.u., equivalent to a three-dimensional (3D) carrier density of n 3D 1.6 × 10 21 cm −3 . We argue that inducing this doping level by ionic liquid gating is feasible in experiment.
First, the authors of Ref. [30] use ionic liquid gating and achieve a 2D carrier density of Since the permittivity of BaTiO 3 is between that of KTaO 3 and SrTiO 3 , we expect that the itinerant electrons in doped BaTiO 3 will also be confined to the interface within a few nanometers (as mentioned by the reviewer). Ref. [32] shows that electrostatic gating by ionic liquid can achieve a two-dimensional carrier density as high as 8 × 10 14 cm −2 . This means that a 3D carrier density of about 1.6 × 10 21 cm −3 can be achieved by ionic liquid gating in doped BaTiO 3 close to the interface.
In the revised version, we expand the Section Discussion (on Page 11 and 12, highlighted in blue text) in which we provide more details of electrostatic doping and show that our critical electron concentration of 1.6 × 10 21 cm −3 is feasible by this approach. question is that if the superconductivity is induced by doping on the surface of BTO, can it be related to the bulk phonon properties and why?
We thank the referee for this important comment.
We agree with the reviewer that the chemical environment of surface is generally different from bulk. However, we would like to provide two results which show that our calculations of bulk BaTiO 3 can also be used as a guidance to search for superconductivity in the surface area of BaTiO 3 .
First we perform additional calculations to simulate a Pt/BaTiO 3 heterostructure grown on a SrTiO 3 substrate. Pt is a representative electrode. We include 16 unit cells of BaTiO 3 and two Pt/TiO 2 interfaces. Fig. R5a shows the optimized crystal structure. The two interfaces are not equivalent due to the presence of BaTiO 3 polarization. Fig. R5b shows the layer-resolved Ti-O displacement δ in BaTiO 3 thin films. We find that at both interfaces, two or three atomic layers away from either interface a bulk-like region emerges in which δ is almost uniform. Electrostatic doping can induce itinerant electrons into a region of the target material that is a few nanometers from the surface/interface [30].
Our calculations indicate that other than the very few atomic layers in the vicinity of We thank the reviewer for this intriguing comment.
We agree with the reviewer that in Ca-doped oxygen-deficient Sr 1−x Ca x TiO 3−δ , the 'ferroelectric' critical concentration does not coincide with the optimal doping of superconductivity, as shown in Fig. 4  is probably not purely phonon-mediated. Indeed, Ref. [27] suggests that ferroelectricity and superconductivity are two competing phases in Sr 1−x Ca x TiO 3−δ . However, the microscopic origin of superconductivity in doped SrTiO 3 is still under intensive debate [18][19][20][21][22][23][24][25] and warrants further study.
In the revised version, we add text (highlighted in blue text) on Page 9 to mention this intriguing point.

Response to Reviewer #3
We are very grateful to the reviewer for commenting that "while the study will be of relevance to the much-studied superconducting transition in doped SrTiO 3 , it is also interesting because this case corresponds to a much higher critical carrier density" and that "the results from this study are intriguing and should motivate further investigations of the coupling between soft polar modes and itinerant electrons in doped oxides near polar phase instabilities." We thank the reviewer for this comment.
When the carrier concentration n is less than 0.1e/f.u., all the four crystal structures (R, O, T , C) can be stabilized during atomic relaxation by imposing symmetry. When the carrier concentration n is equal to or larger than 0.1e/f.u., only the cubic structure (C) can be stabilized after atomic relaxation.
However, if the carrier concentration n is less than 0.025e/f.u., the rhombohedral structure (R) has the lowest total energy. If the carrier concentration is larger than 0.025e/f.u. but smaller than 0.1e/f.u., the tetragonal structure (T ) has the lowest total energy. To more clearly demonstrate this point, we perform additional calculations around the carrier density of 0.025e/f.u. and show new the results in Fig. R6.
To summarize, the orthorhombic structure of BaTiO 3 can be stabilized in the calculations when the symmetry is imposed and the doping is smaller than 0.1e/f.u.. But at any doping smaller than 0.1e/f.u., the orthorhombic structure is not the structure with the lowest total energy in BaTiO 3 .
In the revised version, we add a new section in the Supplementary Materials (Section VII) to more clearly show the structural transition around the carrier density of 0.025e/f.u..
2) I would like the authors to clarify how the electron doping impacts the p-d hybridization and Born effective charges. It seems that e-doping might screen the effective interaction between Ti-O and tune the instability. Also, it would be great if the author could also discuss the phonon behavior upon doping, especially in the rhombohedral phase.
We thank the reviewer for this comment. This comment has three sub-comments.
The first sub-comment is about p-d hybridization and screening of effective metal-oxygen interactions in doped BaTiO 3 . We first clarify that the change of p-d hybridization and screening of metal-oxygen interactions are two closely related phenomena, but they are not the same thing. Previous calculations show that p-d hybridization in transition metal oxides can be characterized by the occupancy of metal-d states [33,34]. Therefore for doped BaTiO 3 , we calculate the occupancy of Ti-d states as a function of electron dop- where D(E) is the density of states projected onto Ti-d orbitals. These three integrals are illustrated in Fig. R7a and their values are shown in Fig. R7b. We find that while N cond d almost linearly increases with electron doping, the total Ti-d occupancy N tot d changes much more slowly. This phenomenon is called "rehybridization effect" [35,36]. As N cond The second sub-comment is about Born effective charges. We note that with electron doping, the system becomes metallic. The Born effective charge tensor Z * κ,γα of atom κ is defined as: where P γ is the polarization and τ κ,α is the periodic displacement. However, polarization is ill-defined in metals and therefore we can not calculate the Born effective charges in doped BaTiO 3 .
The third sub-comment is about the phonon behavior upon electron doping. We select three representative dopings: 0, 0.07e/f.u. and 0.14e/f.u.. In the un-doped case, the ground state structure is rhombohedral (R); at 0.07e/f.u. doping, the ground state structure is tetragonal (T ) and at 0.14e/f.u. doping, the ground state structure is cubic (C). In particular, with electron doping, the frequency of the highest optical phonon modes decreases while the low-frequency peak (around 100 cm −1 ) increases its height. We thank the reviewer for this comment. To study localization behavior of doped electrons, we perform additional supercell calculations of La x Ba 1−x TiO 3 . We study two representative cases: a 2 × 2 × 2 supercell with one Ba atom replaced by one La atom to simulate a carrier density of 0.125e/f.u.
(see Fig. R9a) and a 3 × 3 × 3 supercell with two Ba atoms replaced by two La atoms to simulate a carrier density of 0.074e/f.u. (see Fig. R9d). In both cases, we show the iso-value surface of conduction electron density of each Ti atom and we find that the conduction electrons are uniformly distributed on Ti atoms throughout the entire simulation cell (Fig. R9b and e). The 'dice-like' shape of the isovalue surface indicates that the conduction electrons occupy Ti d xy , d yz and d yz orbitals (as pointed out by the reviewer). of La x Ba 1−x TiO 3 . The blue and green curves correspond to the total and Ti-d projected density of states. We find that there is a finite density of states around the Fermi level and it is mainly composed of Ti-d states. This indicates that the doped electrons occupy Ti-d orbitals and they are itinerant.
As for possible magnetism, we intentionally break spin symmetry and perform additional LSDA calculations. In both 2 × 2 × 2 supercell and 3 × 3 × 3 supercell calculations of La x Ba 1−x TiO 3 , we do not find any magnetization in our LSDA calculations. Experimentally no long-range magnetic order is observed in La doped BaTiO 3 [28,29]. We note that if we manually increase the correlation strength on Ti d orbitals, we can artificially stabilize magnetization in doped BaTiO 3 from LSDA+U calculations. This is due to Stoner instability [37]. Fig. R10a shows such a calculation of 2 × 2 × 2 supercell of La x Ba 1−x TiO 3 (x = 0.125), in which magnetization emerges when U Ti is larger than 2 eV. However, it is well-known that BaTiO 3 is a band insulator with weak correlation effects [38]. More importantly, if we artificially increase the correlation strength on Ti d orbitals, we find that the polar distortions are suppressed in bulk BaTiO 3 because U Ti changes the Ti-O hybridization [39]. Fig. R10b shows the c/a ratio and Ti-O displacements δ of bulk BaTiO 3 as a function of U Ti . As U Ti is larger than 2 eV, the polar displacements in BaTiO 3 are completely suppressed, which is at odds with the experimentally observed ferroelectric property. Therefore based on our additional calculations and known experimental results, we conclude that long-range magnetic order (i.e. homogeneous magnetization) in doped BaTiO 3 is unlikely.
In the revised version, we add two new sections in the Supplementary Notes. Section VIII carefully studies La-doped BaTiO 3 using the supercell approach in which we demonstrate We explain that as the carrier density is below 0.10e/f.u., all the four crystal structures (rhombohedral R, orthorhombic O, tetragonal T , and cubic C) can be stabilized. However as the carrier density is above 0.1e/f.u. only the cubic (C) structure can be stabilized.
Therefore we include the structural information for R, O, T doped BaTiO 3 from 0 to 0.09e/f.u., and for C doped BaTiO 3 from 0 to 0.14e/f.u.. Table S2 lists the the lattice parameters and ionic coordinates of BaTiO 3 in the rhombohedral crystal structure (R), Table S3 for the orthorhombic structure (O), Table S4 for the tetragonal structure (T ) and Table S5 for the cubic structure (C).

5)
The authors should try to provide a possible explanation for the largeelectron phonon coupling involving the acoustic branches in the tetragonal phase, contrasting with the behavior in the cubic phase.
We thank the referee for this comment.
We provide a possible explanation, which is based on our calculations. We find that in the cubic structure, some electron-phonon vertices g ν ij (k, q) are exactly equal to zero because some atoms are frozen in the acoustic phonons, while in the low-symmetry structure, those g ν ij (k, q) become non-zero. Because ImΠ qν ∝ |g ν ij (k, q)| 2 [40,41], this leads to an increase in ImΠ qν . In addition, the frequencies of acoustic phonon modes ω qν are very small and λ qν ∝ ImΠ qν /ω 2 qν [40,41], therefore even a slight increase in ImΠ qν results in a substantial enhancement in λ qν .
To demonstrate the above point, we study a specific phonon of BaTiO 3 : the acoustic phonon at q = X. Fig. R11 compares the vibrational mode of q = X acoustic phonon of BaTiO 3 at 0.09e/f.u. doping in the tetragonal structure (a), at 0.11e/f.u. doping in the cubic structure (b) and at 0.11e/f.u. doping under 0.8% (001) compressive strain (c). We find that in the cubic structure (b), Ba atoms are strictly frozen in this acoustic phonon, while in the tetragonal structure (a and c), Ba atoms also participate in the phonon mode. Correspondingly, the imaginary part of electron-phonon self-energy ImΠ qν of doped BaTiO 3 is very small in the cubic structure, but becomes sizable in the tetragonal structure. Due to the low frequency of acoustic phonons, the electron-phonon coupling λ qν is substantially larger in the tetragonal structure than in the cubic structure.
In the revised version, we add text on Page 10 (highlighted in blue text) to further explain this point. We also add a new section in the Supplementary Materials (Section XVI) in which we study the q = X acoustic phonon to demonstrate this point.  We thank the reviewer for this good suggestion.
We study tetragonal BaTiO 3 at 0.09e/f.u. concentration as a representative example. the Fermi level can be strongly modulated by the polar phonon. Fig. R12 shows the relative change of the density of states at the Fermi level as a function of phonon amplitude A. We find a substantial change in D A (E f ) as the phonon amplitude A increases. This indicates that the polar phonon is strongly coupled to itinerant electrons in doped BaTiO 3 .
In the revised version, we add a pointer on Page 7 (highlighted in blue text) to refer the reader to a newly added section in the Supplementary Materials (Section XIV) in which we show that the density of states at the Fermi level can be substantially modulated by imposing polar phonon modes on the crystal structure in doped BaTiO 3 .

7)
In the free energy diagram at T=0 K, does the Y-axis represent the internal energy or enthalpy? In the case of complete relaxation, internal energy and enthalpy are nearly equal, while if author performed the fixed volume/structure calculation then the PV term is important.
In the free energy diagram at T = 0, the Y-axis represents the internal energy. In all our calculations except strain engineering (Fig. 4a in the manuscript), we perform a complete relaxation (cell parameters and internal coordinates). In the strain calculations, the inplane lattice constants are fixed to simulate the bi-axial strain.
In the revised version, we add text on Page 12 (highlighted in blue text) to emphasize this point. We thank the reviewer for this comment and the suggestion of crosschecking KTaO 3 .
We agree with the reviewer that chemical disorder may affect the transport properties of the normal state (the metallic state above the superconducting transition temperature). However, according to Anderson's theorem, superconductivity in a conventional superconductor is robust with respect to non-magnetic disorder in the host material [31]. The superconducting transition temperature T c of a conventional superconductor barely depends on material purity. In our case, the superconductivity in doped BaTiO 3 is phononmediated (i.e. conventional) and La is a non-magnetic dopant. Therefore Anderson' theorem applies and we expect that the superconducting phase in doped BaTiO 3 is robust against chemical disorder.
In the revised version, we expand our discussion of potential experiments on Page 11 and 12 (highlighted in blue text). We also add a section in the Supplementary Materials (Section XV) to show the comparison between doped BaTiO 3 and doped KTaO 3 .

Response to Reviewer #4
We thank the reviewer for the comment "I find the paper 'A large modulation of electronphonon coupling and an emergent superconducting dome in doped strong ferroelectrics' to be timely and potentially relevant for generation of interest in the ferroelectrics as a playground for finding possible new superconductors." 1) "previous studies found that in n-doped BaTiO3, increasing the carrier density gradually reduces its polar distortions and induces a continuous polar-tocentrosymmetric phase transition (similar to Sr 1−x Ca x TiO 3−δ )". Is the picture for the phase transition in Ca-STO that simple? I think it is a bit more involved than that.
We thank the reviewer for this helpful comment.
We agree with the reviewer that the quantum phase transition in Sr 1−x Ca x TiO 3−δ [27] is more complicated than what is described for n-doped BaTiO 3 . The reason that we mention Sr 1−x Ca x TiO 3−δ in the introduction is to provide another relevant experimental system, which inspires us to study doped BaTiO 3 . However, as the reviewer commented below (comment #19), a direct comparison between n-doped BaTiO 3 and Sr 1−x Ca x TiO 3−δ is premature and therefore we remove the phrase "(similar to Sr 1−x Ca x TiO 3−δ )" on Page 3 in the revised version.
2) "the critical concentration for the phase transition is about 10 21 /cm 3 , which is high enough so that the electron-phonon coupling can be directly calculated within the Migdals approximation". I dont think the comparison between the Ca case and other cases is that straightforward. Also, this seems like cherry picking in terms of very large doping range in STO for which it superconducts.
With respect, we are a little confused by the correlation between the quote and the reviewer's comment here. The quote "the critical concentration for the phase transition is about 10 21 /cm 3 ..." refers to doped BaTiO 3 (not SrTiO 3 ). That statement says that the critical concentration for doped BaTiO 3 is high enough so that the Migdal's approximation is valid. However, the comment from the reviewer seems to be concerning doped If we misunderstood the reviewer's comment, we kindly request the reviewer to further clarify it and we are happy to address it again. We thank the reviewer for bringing this newly-published comprehensive review to our attention, which expands our knowledge on doped SrTiO 3 . We cite it and other experiment works on SrTiO 3 in our revised version.
In the revised version, we cite the following papers Ann. Phys. 168107 (2020) (now  We thank the reviewer for this suggestion. The energy scale we compare here is the Debye frequency (hω D ) versus Fermi energy ( F ). For the Migdal's approximation to be valid, the Debye frequency is usually much smaller than Fermi energy (hω D / F ∼ 10 −2 − 10 −3 ).
However for doped SrTiO 3 , since the carrier concentration is very low, the Fermi energy varies between 2 and 60 meV. On the other hand, the role of Debye frequency ω D can be replaced by the longitudinal optical phonon frequency ω L , which is on the order of 100 meV for SrTiO 3 [42]. Therefore in doped SrTiO 3 ,hω D / F ∼ 1 − 10 2 , which violates the Migdal criterion.
In the revised version, we add text on Page 3 (highlighted in blue text) to make this point more clear.

5) "
we find that the phonon bands associated with the soft polar optical phonons are strongly coupled to itinerant electrons across the polar-tocentrosymmetric phase transition in doped BaTiO 3 ". Whats the insight for "why" this is the case?
We thank the referee for this important comment.
For a canonical polar metal such as LiOsO 3 , across the polar-to-centrosymmetric phase transition, the coupling between the soft polar optical phonons and itinerant electrons is very weak [43]. This is known as the 'weak coupling' mechanism [43,44]. The weak coupling in LiOsO 3 is due to the fact that the itinerant electrons reside on Os-d orbitals while the polar phonons involve the movement of Li and O [43]. However, in doped BaTiO 3 , itinerant electrons reside on Ti-d orbitals (see Fig. 2a in the main text) and the polar phonons mainly involve Ti and O movement (see Fig. 2c in the main text).
Because the itinerant electrons and polar phonon are associated with the same atoms in doped BaTiO 3 , the coupling is strong, while in LiOsO 3 the itinerant electrons and polar phonon involve different atoms and thus the coupling is weak.
In the revised version, we add text on Page 7 (highlighted in blue text) to further clarify this point. 6) "In addition, we find that close to the critical concentration, lowering the crystal symmetry of doped BaTiO 3 by imposing epitaxial strain further increases the superconducting temperature via a sizable coupling between itinerant electrons and acoustic phonon bands.". What about the ferroelectric phonon to electron coupling change with strain?
We thank the reviewer for this important comment. Since this comment is related to a few other comments (comments #10, #11, #12, #15), we would like to introduce a normalized around-zone-center branch-resolved electron-phonon coupling λ ν as: where q c is a small number within which there are no phonon band crossings. There are two reasons to introduce this definition: 1) exactly at the zone-center Γ point, the acoustic phonon frequency is zero. Since λ qν ∝ 1/ω qν , the contribution from the acoustic mode is ill-defined at Γ point. 2) Because there are no phonon band crossings within |q| < q c , each phonon mode (labelled by ν) is well-defined. For a general q point, it is not trivial to distinguish which phonon band corresponds to polar modes and which to other optical modes. We choose q c = 0.05 π a where a is the lattice constant. The qualitative conclusions do not depend on the choice of q c (as long as no phonon band crossings occur within |q| < q c ). Now we apply the above definition of λ ν to cubic BaTiO 3 at 0.11e/f.u. doping. Close to Γ point, the three lowest phonon bands (ν = 1 − 3) are acoustic modes, while the next three lowest phonon bands (ν = 4 − 6) are polar modes (this is not true for a general q-point in the Brillouin zone). We find that without epitaxial strain: Under 0.8% (001) compressive strain, we find: Therefore under 0.8% (001) compressive strain, λ polar remains substantial (albeit reduced by 40%), but λ acoustic is increased by one order of magnitude.
With these new numbers of λ ν , we revise our text and make a more quantitative comparison on Page 6, 7 and 10 (highlighted in blue text).

7) "
Our results show that the weakly coupled electron mechanism in 'ferroelectric-like metals' is not necessarily present in doped strong ferroelectrics and as a consequence, the soft polar phonons can be utilized to induce phonon-mediated superconductivity across a structural phase transition.". This needs more explanation for why "is not necessary present. Right now, its just a statement but not an explanation, which would be helpful to have here or to come back to this point later in the text to explain "why".
We thank the referee for this comment. We explain this in our reply to the comment #5.
In the revised version, we add text on Page 7 (highlighted in blue text) to explain this point.
8) " Fig. 1a shows that as electron doping concentration n increases from 0 to 0.15e/f.u., BaTiO 3 transitions from the rhombohedral structure to the tetragonal structure, and finally to the cubic structure.". Is the role of disorder due to dopants considered or derived here? How is its relevance in real materials ruled out? Any experimental study that could be cited?
We thank the referee for this comment.
In our calculations of Fig. 1, we do not explicitly consider disorder due to dopants. We assume that the crystal structure of BaTiO 3 is homogeneous upon chemical doping and manually change the number of electrons to simulate the doping effects (similar to the previous work [9]). Our assumption is supported by the experiments that bulk La-doped BaTiO 3 were synthesized in single phase ceramics with homogeneous microstructures [45] and high-quality La-doped BaTiO 3 thin films were grown and exhibit metallic behaviors [28,29,46].
More importantly, we would like to note that while chemical disorder may affect transport properties of normal states (the metallic state above superconducting transition temperatures), Anderson's theorem asserts that superconductivity in a conventional superconductor is robust with respect to non-magnetic disorder in the host material [31].
That is, the superconducting transition temperature T c of a conventional superconductor barely depends on material purity. In our case, the superconductivity in doped BaTiO 3 is phonon-mediated (i.e. conventional) and La is a non-magnetic dopant. Therefore Anderson' theorem applies and we expect that even if chemical disorder may arise in actual experiment, it does not affect the superconducting properties of doped BaTiO 3 .
In the revised version, we add text to explain possible effects from chemical disorder in the Discussion Section (on Page 11, highlighted in blue text). We thank the reviewer for pointing this out.
The values compared here are the Debye frequency (hω D ) and the Fermi energy ( F ), which can be converted to the Debye temperature (T D =hω D /k B ) and the Fermi temperature On the other hand, the canonical polar metal LiOsO 3 is known to have a weak coupling between itinerant electrons and polar phonons [43]. Using our definition of branch-resolved electron-phonon coupling, we find that for LiOsO 3 (the zone-center polar modes of LiOsO 3 are ν = 5, 6,9), Thus λ polar of doped BaTiO 3 is one order of magnitude larger than that of LiOsO 3 .
Theoretically, λ polar ≥ 0 and there is no upper limit. Usually if λ polar > 1, we may consider the coupling as strong (of course, this is not a strict criterion).
In the revised version, we add text on Page 6 and 7 (highlighted in blue text) to make a quantitative statement of "weak" and "strong" couplings.  We thank the reviewer for this good suggestion. Following our reply to the comment #6, we calculate each λ ν of BaTiO 3 at 0.09e/f.u. doping and at 0.11e/f.u. doping. The results are shown in Fig. R14. We find that for 0.09e/f.u. doping, In both cases, λ polar is larger than λ acoustic and λ others .
With these new numbers of λ ν , we revise our text and make a more quantitative comparison on Page 6 and 7 (highlighted in blue text).
12) "and therefore can also make non-negligible contribution to the total electron-phonon coupling". Provide a number, its hard to judge the acoustical phonons contribution based on the plots in Fig 3. I cannot really see the amplification effect. Can it be also made more quantitative statement?
We thank the referee for this comment and we apologize for the confusion that is caused by the phrase "amplification effect".
We first clarify that the mode-resolved electron-phonon coupling λ qν is [40,41]: For a low phonon frequency ω qν such as those of acoustic phonon modes, a slight increase in the electron-phonon self-energy ImΠ qν will result in a substantial increase in λ qν because ω qν is in the denominator. This is what we mean by "amplification effect".
Nevertheless, we appreciate the reviewer's confusion and in the revised version, we modify the text on Page 9 (highlighted in blue text) and make this point more clear.
Second, following the comment #6 and #11, we now have well-defined numbers for moderesolved electron-phonon couplings. Within the region |q| < q c in which no phonon crossing occurs and each phonon branch can be well assigned to acoustic and polar modes, we find that at 0.09e/f.u. doping in the T structure λ acoustic λ acoustic + λ polar + λ others = 25% (R14) at 0.11e/f.u. doping in the C structure λ acoustic λ acoustic + λ polar + λ others = 5% This quantitatively shows that around the zone-center Γ point, the acoustic modes make non-negligible contribution to the electron-phonon coupling (the acoustic mode contribution is larger in the tetragonal T structure than in the cubic C structure).
In the revised version, we modify the paragraph about acoustic phonons on Page 9 and 10 (highlighted in blue text) and make more quantitative statements. We also revise Fig.   4c which demonstrates the "amplification effect" of λ qν in the acoustic modes of doped BaTiO 3 under (001) compressive strain.
13) "a thought-experiment" should probably be "a numerical experiment" in several places.
We thank the reviewer for this suggestion. We replace "a thought-experiment" with "a numerical experiment" on Page 9 and 10 in the revised version.
14) "λ increases from 0.50 in the P m3m structure to 0.57 in the P 4mm structure, and the superconducting transition temperature Tc increases from 0.76 K in the P m3m structure to 2.0 K in the P 4mm structure". I dont know if we can trust Eliashberg equation and McMillans formula with such small change in alfa and draw conclusions that 2 K is 2 K, it might be 2 might be 10 in real material. Id like to see some comment about the accuracy of the expectation. These predictions are quite often off by a numerical factor.
We thank the reviewer for this insightful comment.
We agree with the reviewer that there is uncertainty in Eliashberg equations and McMillan's formula, such as the Morel-Anderson pseudopotential µ * which is treated as a parameter. Since the superconducting transition temperature T c strongly depends on µ * (see Fig. 3f in the main text), we only use Eliashberg equations and McMillan's formula to estimate T c . The take-home message here is that when epitaxial strain lowers the crystal symmetry of doped BaTiO 3 , its electron-phonon coupling λ is increased, which leads to an increase in T c .
Nevertheless, we appreciate the reviewer's concern and in the revised version we remove the estimated superconducting transition temperatures on Page 9.
15) "The difference in ImΠ qν is further amplified by the low phonon frequencies ωλqν. Fig. 4c explicitly compares the mode-resolved electron-phonon coupling λq1 for the lowest phonon band of the two doped BaTiO3. It is evident that λq1 is substantially larger in 0.11e/f.u. concentration (space group Pm3m), and then we impose a slight compressive the optimized lattice constant a is 3.972 A; under a 0.8% biaxial compressive strain, the ground state structure becomes tetragonal with the optimized short lattice constant a being electron-phonon coupling λ increases from 0.50 in the P m3m structure to 0.57 in the P 4mm structure, and the superconducting transition temperature Tc increases from 0.76 K in the P 4mm tetragonal structure than in the P m3m cubic structure.". I'd like to see the mode-resolved coupling for the FE modes next to the acoustic ones. How is the FE one changed by strain? What is the scale of this mode-resolved lambda? Is it not the same as that of the averaged one?
We thank the reviewer for this comment.
Please refer to our reply to the comment #6 in which we define λ ν for acoustic and polar modes, and we compare λ ν in the absence of strain versus in the presence of strain.
We note that the around-zone-center branch-resolved electron-phonon coupling λ ν can be substantially larger than the total electron-phonon coupling λ. That is because 1) we normalize λ ν (we divide by |q|<qc dq to make λ ν dimensionless) and 2) λ qν is substantial when q is around the high-symmetry cuts (e.g. Γ → X) and is negligible in other regions of the phonon Brillouin zone.
Following our reply to the comment #12, we modify the paragraph of acoustic phonons and make more quantitative discussions on Page 9 and 10 (highlighted in blue text). We also revise Fig. 4c in which we show the mode-resolved electron-phonon coupling λ qν along Γ to X.
16) "the exactly same electron concentration, indicating that the additional increase in ImΠ qν arises solely from the crystal structure difference. Lowering the symmetry of BaTiO3 crystal structure allows more electron-phonon scattering processes that would be forbidden in the cubic structure by symmetry considerations.". Is this a quantitative or qualitative statement? I suggest quoting numbers. Is this general or specific to these calculations? Seems to be too important to pass through so quickly -need more clarification and quantitative argumentation beyond the plots show. Is there a more general insight to be learned?
We thank the reviewer for this comment.
Following our reply to the comments #6, we find that for BaTiO 3  This result is specific to doped BaTiO 3 . A possible explanation, which is based on our calculations, is that in the cubic structure, some electron-phonon vertices g ν ij (k, q) are exactly equal to zero because some atoms are frozen in the acoustic phonons, while in the low-symmetry structure, those g ν ij (k, q) become non-zero. Because ImΠ qν ∝ |g ν ij (k, q)| 2 (see Eq. R13), this leads to an increase in ImΠ qν . In addition, the frequencies of acoustic phonon modes ω qν are very small and λ qν ∝ ImΠ qν /ω 2 qν (see Eq. R13), therefore even a slight increase in ImΠ qν results in a more substantial enhancement in λ qν .
In the revised version, we modify text on Page 10 (highlighted in blue text) to further explain this point. We also add a new section in the Supplementary Materials (Section XVI) in which we study a specific acoustic phonon to demonstrate this point. We thank the reviewer for this comment and we apologize for this omission.
As Fig. 4 in the main text shows, the epitaxial strain is bi-axial, i.e. the in-plane lattice constants of doped BaTiO 3 (a and b) are either compressed or elongated so as to match the lattice constant of the substrate. The out-of-plane lattice constant of doped BaTiO 3 (c) increases (decreases) when the bi-axial strain is compressive (tensile). This type of strain is known as (001) compressive strain.
In the revised version, we add text on Page 9 and 10 (highlighted in blue text) to explicitly explain that the bi-axial (001) compressive strain is imposed such that the in-plane two lattice constants (a and b) are fixed to a smaller value. This (001) compressive strain can increase the electron-phonon coupling of doped BaTiO 3 and possibly superconducting transition temperature.
18) "Since our simulation does not consider dopants explicitly, a more desirable doping method is to use electrostatic carrier doping". But then the structure doesn't not change. So how does electrostatic gating of the surface should work in terms of the structural phase transition? Are the authors saying that just strain and surface conductivity enough? Please clarify with more precision.
We thank the reviewer for this comment.
When we discuss electrostatic gating or epitaxial strain, we mean that those methods should be applied to BaTiO 3 thin films (not bulk). Experiments show that electrostatic gating can induce electrons into a region of KTaO 3 that is a few nanometers thick from the surface [30]. This leads to an effective three-dimensional carrier density of 2.2 × 10 21 cm −3 (equivalently 0.14e/f.u.) and thus superconductivity of about 50 mK in doped KTaO 3 .
We expect that in BaTiO 3 thin films (a few nanometers thick), electrostatic method or epitaxial strain can modulate the structural properties of the entire film [30,47,48].
Therefore based on our calculations, when the electron-phonon coupling of doped BaTiO 3 is sufficiently enhanced around the critical concentration, conventional superconductivity may emerge in BaTiO 3 thin films.
In the revised version, we add text on Page 11 (highlighted in blue text) to discuss a number of new details about the electrostatic doping method.

19) "
We hope that our predictions will stimulate new experiments on doped ferroelectrics and, if confirmed, may help shed light on the mysterious origin of the superconductivity in doped SrTiO 3 .". I dont think the projection for STO is that simple, given that the authors say that for their material Migdals approximation is valid, while for STO it is not that straightforward. I think the overall projected connection to STO is somewhat premature, many more details would need to be covered to make any such conclusion.
We thank the reviewer for this comment and we agree with the reviewer that a direction connection between superconductivity in doped BaTiO 3 and that in doped SrTiO 3 is premature. In the revised version, we remove the sentence "if confirmed, may help shed light on the mysterious origin of the superconductivity in doped SrTiO 3 ." on Page 12. The manuscript has received very extensive reviews and comments. Overall, the interest from the reviewers has been quite strong. The authors have rather painstakingly responded to all comments/questions, adding a significant amount of results to their manuscript and supplement. They have certainly adequately responded to questions and comments from this reviewer, and I now can recommend the manuscript for publication.

O. Delaire
Reviewer #4 (Remarks to the Author): The responses are satisfactory, I recommend to accept.