Colossal flexoresistance in dielectrics

Dielectrics have long been considered as unsuitable for pure electrical switches; under weak electric fields, they show extremely low conductivity, whereas under strong fields, they suffer from irreversible damage. Here, we show that flexoelectricity enables damage-free exposure of dielectrics to strong electric fields, leading to reversible switching between electrical states—insulating and conducting. Applying strain gradients with an atomic force microscope tip polarizes an ultrathin film of an archetypal dielectric SrTiO3 via flexoelectricity, which in turn generates non-destructive, strong electrostatic fields. When the applied strain gradient exceeds a certain value, SrTiO3 suddenly becomes highly conductive, yielding at least around a 108-fold decrease in room-temperature resistivity. We explain this phenomenon, which we call the colossal flexoresistance, based on the abrupt increase in the tunneling conductance of ultrathin SrTiO3 under strain gradients. Our work extends the scope of electrical control in solids, and inspires further exploration of dielectric responses to strong electromechanical fields.


Responses to the comments of Reviewer #1
We would like to thank the reviewer for reviewing our manuscript and providing critical points for us to consider, e.g., "Nevertheless, the referee thinks that the reasoning and data, particularly the analysis of the strain profile, is insufficient to support their conclusion". We have read the reviewer's comments very carefully and have undertaken further works to answer all of his/her comments. The main concern was with regard to the validity of the simulation results, especially the analysis of the strain profile. For clarity, we compare the strain distribution obtained numerically in our study with those calculated using two analytical contact mechanics models, that is, the Hertz model and Boussinesq's solution. The Hertz model and Boussinesq's solution consider the transversely isotropic elastic half-space in contact with a spherical indenter and with a point contact, respectively. The results are given below in Figure   R1 and Table R1. We provide the reviewer's comments followed by our detailed point-by-point responses (written in blue).

Question #1
Based on their contact mechanics model, the calculated strain gradient is around 1e7 m-1. It is doubtful. The tip diameter is said be around 100nm, which is much larger than the thin film thickness of 3.9 nm with substrate.

Response #1
As pointed out by the reviewer, the tip radius (rtip ~ 100 nm) is much larger than the film thickness (t ~ 4 nm) in the model. However, please note that, when estimating strain gradients, we have to use the contact radius, not the tip radius itself. When the loading force is 15 µN, the diamond tip of rtip ~ 100 nm contacts the SrTiO3 surface with a contact radius of a ~ 13 nm (calculated using the equations in Table 6 of Ref. 43 in the main text), comparable to the film thickness t ~ 4 nm. For the same reason, we adopted the closed-form solution of the stress distribution on the film surface in the "thin-film" limit (e.g., when t << a, based on the equations in Table 6 of Ref. 43 in the main text) as the boundary condition of the mechanical equilibrium equation in our phase-field model. To avoid further confusion, we stress the dimension of the contact radius in Line #84 of the revised manuscript as follows: "Note that the actual contact radius is estimated to be around 13 nm for the case of a 15-N tip loading force, which is much smaller than the tip radius rtip."

Question #2
Considering these dimensions, the strain field in the region under the tip cannot really has large gradient.

Response #2
In Responses #3 & #4, we show that, compared to the longitudinal (i.e., out-of-plane) strain u33, the transverse (i.e., in-plane) strain u11 exhibits much larger variation across the film. As a result, the transverse strain gradient can be huge (>10 7 m −1 ) and then primarily contributes to the flexoelectric effect, consistent with findings from previous studies [see, e.g., "W.

Question #3
Even if one can consider an ideal single concentered force problem, which is more favorable to generate high gradient than the contact problem of round tip, one cannot arrive at such large strain gradient. Based on the Boussinesq's solution, the estimated strain gradient \epsilon_33,3 in the region near the surface (at the depth of 1nm) is only the region of 1e6 m-1.

Response #3
We thank the reviewer for suggesting the use of the classical Boussinesq's solution to estimate the strain gradient. To test this possibility, we adopted the Boussinesq solution of an axisymmetric system for a transversely isotropic half-space elastic solid, to obtain the stress and strain field distributions subject to point force loading. To obtain the numerical values, material-related parameters for SrTiO3 were used, such as Young's modulus E = 264 GPa and Poisson's ratio ν = 0.24. The loading force was set to 15 µN for consistency with the conditions corresponding to Fig. 2 of the main text. The stress distribution expression for Boussinesq's solution is given in classical books of contact mechanics (e.g., page 80 of "Fisher-Cripps, Introduction to Contact Mechanics"). Based on the stress distribution, we can calculate the strain distribution by assuming Hooke's law for a pure elastic material. Furthermore, for comparison, we plot the strain distributions obtained by following the closed-form solution of the Hertz model with a spherical indenter (page 87 of the same book) and the numerical results obtained from our phase-field simulation. For the Hertz analytical model, we keep the same contact radius as in our numerical model, e.g., a = 13.1 nm. The results are given below in Figure R1 and Table R1. represent longitudinal (i.e., out-of-plane) and transverse (i.e., in-plane) strains, respectively. The contact center is at the upper-left corner of each plot. b,c, The strain profiles under the contact center as a function of the distance from the surface calculated using Boussinesq's solution (b) and Hertz's model (solid lines in c) in comparison with numerical results from the present work (dotted lines in c). (averaged over 0-3.5 nm in depth) [*The sign of the strain gradient indicates whether the strain has intensified (+) or decayed (−) from the surface into the film/half space. For Boussinesq's solution, the strain gradients estimated at a depth of 10 nm are shown, but when estimated at a depth of 1 nm, they become much larger (≥10 8 m −1 ).] We can see that all three models (Boussinesq, Hertz, and the present numerical simulation) show huge strain gradients. In particular, the total transverse strain gradients (i.e., ∂ut/∂x3 = ∂u11/∂x3 +∂u22/∂x3) are as large as 3.9 × 10 7 , 2.4 × 10 7 , and 3.1 × 10 7 m −1 for the Boussinesq model, Hertz model, and present numerical simulations, respectively. Importantly, our numerical results compare very well with Hertz's analytical solution. There are some minor differences that may be attributed to the electromechanical interactions, including piezoelectric, ferroelectric, and flexoelectric effects, which were not considered in the analytical models.
Please also note that our results are consistent with those of many previous works [see, e.g., i.e., sufficiently large to cause strong flexoelectric effects.

Question #4
The referee understands that there may be additional in-plane strain gradient due to the interface between the thin film and substrate. But it is hard to estimate the value from the presented results in Fig.2

Response #4
We would like to thank the reviewer for suggesting the importance of the in-plane strain gradient. First, we must consider the gradient of transverse (i.e., in-plane) strain (e.g., ∂u11/∂x3), because it could be much larger than the gradient of longitudinal (i.e., out-of-plane) strain (e.g., ∂u33/∂x3). As shown in Table R1, both Hertz's model and our numerical simulation show that the total transverse strain gradient ∂ut/∂x3 (= ∂u11/∂x3 +∂u22/∂x3) is one order of magnitude larger than the longitudinal strain gradient ∂u33/∂x3. Thus, the transverse strain gradient primarily contributes to the flexoelectric effect, consistent with findings from previous studies ]. This is why we focus on the transverse strain gradients ∂ut/∂x3 in our manuscript.
Also, there exists the in-plane gradient of strains (e.g., ∂u11/∂x1), inducing the in-plane polarization. However, please note that only the out-of-plane polarization should mainly contribute to the band crossing and resistivity change across the dielectric layer ( Fig. 1 of the main text). Following the reviewer's comment, we revised Fig. 2b of the main text by labelling the contour lines. This revision would help the readers estimate strain gradients more easily.

Question #5
Btw, the unit of the transverse strain is missing in the figure.

Response #5
We thank the reviewer for pointing out this. Following the reviewer's comment, we added units (%) for the strain distributions in Fig. 2 of the main text.

Question #6
In the simulation model, periodic boundary conditions are assumed for the two in-plane direction. But this is not reasonable for a problem of force applied by the AFM tip. Moreover, contact mechanics model they used is a typical continuum model. But the grip spacing was set to be 0.1 nanometer, and the validity of a solution based on continuum model is high questionable.

Response #6
We thank the reviewer for pointing out this important issue. The reviewer expressed concern about the periodic boundary conditions imposed along the in-plane directions in our model. We believe that, as long as the lateral size is large enough to exclude interactions between the AFM tip with its neighboring periodical images, the 3D model with in-plane periodical boundary conditions is reasonable. In our simulation, we used nx = ny = 128 nm as the lateral dimension of the simulation box, which is large enough to exclude artificial interactions due to the periodic boundary conditions. We test this by plotting surface displacement under the tip contact along the surface across the tip contact center, as shown in Fig. R2 below. The surface displacement is reduced to a near-constant value at the two ends of the simulated system, which suggests that the AFM tip is "isolated" and has negligible interactions with its periodic images. The reviewer also questioned our choice of grid size. When numerically solving partial differential equations for mechanics problems, a smaller grid size generally gives rise to more accurate solutions, although at the expense of computational resources. We used a size of 0.1 nm along the out-of-plane direction of the film, to ensure sufficient resolution in this direction and accuracy of the numerical results. Thus, the choice of grid size is not relevant to the validity of the continuum model.

Question #7
In addition, the authors do not reason why the charge of AFM tip and the electrostatic of the thin film is not considered in the simulation. The polarization can in fact more dominantly be varied by electric field, since it is a first-order theory. Without excluding this fact, the conclusion made by the authors is hard to accept.

Response #7
We thank the reviewer for pointing out this crucial issue. The reviewer questions the absence of the electrostatic interactions between the AFM tip and the thin film surface. He/she believes that SrTiO3 can become polarized due to the electrostatic field, e.g., generated by the charge in the AFM tip. However, if the polarization in SrTiO3 is caused by the charge of the AFM tip, our main observation (i.e., colossal resistivity change) would be almost independent of the tip radius in the experiments, because the charge density on the tip would not vary much with the tip radius. In contrast, we do observe a strong dependence of the resistivity change on the AFM tip radius (Fig. 3 of the main text). This result suggests that the strain gradient, which becomes much smaller with a larger AFM tip radius, is the primary cause of the observed resistivity change.
Furthermore, we carried out additional experiments for estimating the electrostatic field (e.g., built by the charge in the AFM tip) in our experimental geometry. If there exists a built-in electrostatic field, then we can detect it from the shift in the polarization-switching hysteresis loops of BaTiO3. As shown in Fig. R3 below, we observe that in 10 unit cell-thick (i.e., ~4 nmthick) BaTiO3, the hysteresis loops (measured with a sufficiently low loading force) are shifted by around 0.1 V. Accordingly, the built-in electrostatic field is measured as, at most, 3 × 10 7 V m −1 for our experimental geometry. However, the colossal decrease in resistivity requires a much larger threshold electric field Eth ~ bg et where ∆bg and t represent the bandgap and thickness of the dielectric layer, respectively.
Furthermore, given the dielectric permittivity ε ~ 20ε0 of strained SrTiO3 and BaTiO3 ( Supplementary Fig. 7), the electrostatic field of 3 × 10 7 V m −1 can generate an electric polarization of up to 0.005 C m −2 , which is much smaller than the flexoelectric polarization of around 0.2 C m −2 .

Question #8
Also, based on their tunneling current density model in relation to polarization, the referee wonders why the authors do not study a ferroelectric thin film, which has large polarization and permittivity.

Response #8
We would like to thank the reviewer for pointing out the possibility of using ferroelectric materials. Bulk ferroelectric materials could have a large polarization. However, in the ultrathin limit, the value of ferroelectric polarization would be highly suppressed compared to the bulk Therefore, in the AFM tip-based experiment, a large polarization in ultrathin dielectrics can be generated most effectively via flexoelectricity.

Responses to the comments of Reviewer #2
We would like to thank the reviewer for his/her in-depth review and excellent questions/suggestions regarding our manuscript. We have read the reviewer's comments very carefully and have undertaken further works to answer all of his/her questions. In the pages that follow, we provide our responses to each of the reviewer's comments, in order. The responses are written in blue.
In the paper "colossal flexoresistance in dielectrics", Park et al. have claimed to realize colossal "flexoresistance" behavior in ultrathin SrTiO3 (STO) films, which is very interesting.
We thank the reviewer for stating that our work is very interesting.

Question #1
In a previous work (Ref.: Nat. Commun. 10, 537, 2019), the polarization induced local metallization of STO thin film has been reported by the same group. The main idea of the two papers is very similar. So, they should explain the similarities and differences between these two works in details.

Response #1
Regarding the similarity between the papers, we utilized the depolarization field induced by flexoelectric polarization to modify the tunnel barrier profile of an ultrathin dielectric. Also, for generating a flexoelectric polarization, we adopted the AFM tip-based experimental method in both the previous and current works.
However, the current work is distinctly different from the previous work, in terms of both its purpose and achievement. Our previous work [Nat. Commun. 10, 537 (2019) On the other hand, the current work focuses on how to achieve static, damage-free control of electrical states in dielectrics, which has remained a great challenge. To do this, we considered the regime of larger flexoelectric polarization, where the conduction and valence bands of SrTiO3 could cross each other. As discussed in our manuscript, under these conditions, the "whole" SrTiO3 layer behaves as a conductor, due to the highly enhanced tunnel conductance and/or Zener breakdown. Therefore, for the first time, this work demonstrated static, damagefree control of electrical states in an otherwise highly insulating dielectric. This represents a fundamental breakthrough that provides new insight into, and thus could improve, the electrical control of solids.
To conclude, the purpose and achievement of the current work are obviously different from those of the previous work. We have added text pertaining to the difference between the previous and current work to the revised Supplementary Information, as Supplementary Note 1. We would like to thank the reviewer again for pointing out this issue and believe that our revision has improved the clarity of our work.

Question #2
The authors should pay attention to the flexoelectric coefficient value of STO that they set in the phase field simulations. Should the flexoelectric coefficient in STO thin film equals to that in their single crystal counterpart? In my view, once involving such a high strain gradient between the tip and thin film (1E7 m-1), the value of the average flexoelectric coefficient is questionable, since the induced flexoelectric polarization will be gradually saturated as the strain gradient increases.

Response #2
We completely agree with the reviewer's point that, under high strain gradients, the flexocoupling coefficient of SrTiO3 thin film could be different from that of SrTiO3 bulk. In fact, this is the reason why we first measured the effective flexocoupling coefficient of SrTiO3 under high strain gradients in our previous work [Nat. Commun. 10, 537 (2019)]. Interestingly, in our previous work, we discovered the enhanced flexocoupling coefficient of SrTiO3 under high strain gradients, and attributed it to a nonlinear flexoelectric response and/or a surface contribution. Therefore, as mentioned in Line #320 of the manuscript, our phase field simulation adopted the value of the flexocoupling coefficient measured in our previous work.

Question #3
The authors should also pay attention to the role of the surface charge compensation on the flexoresistance behavior. For the ultrathin STO thin film, the AFM tip motion at high pressures can simply mechanically remove screening charges and led to an unscreened surface. This will contribute significantly to the observed resistance reduction.

Response #3
We thank the reviewer for bringing up the surface (screening) charge issue. First, we would like to point out that our on-off experiments (Fig. 4c) and experiment with a graphene electrode ( Supplementary Fig. S10) can overcome the screening charge removal issue. If the resistance reduction is due to the screening charge removal induced by mechanical loading, the response cannot be reversible in an on-off test.
Furthermore, based on simple electrostatics [Phys. Rev. Lett. 94, 246802 (2005)], we find that the screening charge density σS tends to zero in the ultrathin limit (as in our case): where P, d, and ε indicate polarization, thickness, and the dielectric constant of the dielectric layer, respectively. δ1 and δ2 are the Thomas-Fermi screening lengths in the electrodes.
Therefore, the absolute value of σS should change negligibly by any means, as it is already saturated to almost zero.

Question #4
Much more important is that how to distinguish the triboelectric effect from the flexoelectric effect in this AFM scenario?

Response #4
We would like to thank the reviewer for pointing out this crucial issue. If the observed resistance change originates from the triboelectric effect, it would be larger when using the AFM tip with a larger tip radius, where the contact area between the tip and sample becomes larger. The triboelectric effect is related to the degree of electrical charge transfer between the materials; thus, the larger the contact area, the higher the change in the charge transfer. However, our experimental observation is in opposition to this expectation. As the tip radius increased, the resistivity change was suppressed significantly. This excludes the triboelectric effect as the primary origin of the colossal resistivity change observed.
Furthermore, using Kelvin probe force microscopy (KPFM), we measured the surface potential (directly related to the surface charge state) before and after mechanical loading. As shown in 10, 537 (2019)]. Note that in our previous work, we suggested that the total flexoelectric response in ultrathin dielectrics could be enhanced due to the surface contribution.

Question #6
The "flexoresistance" behavior is still questionable in other dielectrics. The authors find the same "electrical-state switching" in BTO thin film with a lower threshold loading force.
However, it is known that the BTO ultrathin film (<10 nm) deposited on SRO generally has a P-up (point from the substrate to the film) ferroelectric polarization state (Ref.: Adv. Mater. Interfaces 3, 1600737, 2016). This direction of ferroelectric polarization, which should be overcome, is opposite to the AFM tip-induced flexoelectric polarization. So, why the threshold loading force for BTO is lower than that of STO?

Response #6
We thank the reviewer for pointing out this important issue. We have performed PFM experiments to measure the self-polarization of as-grown BaTiO3. Please note that the direction of self-polarization in as-grown BaTiO3 depends on many parameters, such as deposition temperature, pressure, annealing conditions, strain state, etc. As shown in Fig. R5 below, the self-polarization in our BaTiO3 film is directed downward. Thus, the ferroelectric polarization in BaTiO3 could lower the threshold loading force in our experiments. This result was added to the revised Supplementary Information as Supplementary Fig. 11. Furthermore, our experimental geometry induces compressive strain in both the transverse and longitudinal directions, attributable to AFM tip-induced downward bending and pressing. Such three-dimensional compression could weaken ferroelectricity in a dielectric layer (e.g.,

BaTiO3
). Therefore, we do not actually have to be concerned with the ferroelectric polarization of BaTiO3. Instead, as mentioned in our manuscript, the flexocoupling strength of BaTiO3 could be inherently larger than that of SrTiO3 [J. Narvaez et al., Phys. Rev. Lett. 115, 037601 (2015)], leading to the lower threshold loading force for BaTiO3.

Question #7
Finally, to emphasize this concept, the flexoresistance behavior should be clearly demonstrated in other material systems.

Response #7
We would like to thank the reviewer for bringing up the universality issue. We have tested the "flexoresistance" behavior in other dielectric materials, i.e., CaTiO3, as shown in Fig. R6 below.
A 10 unit cell-thick CaTiO3 film was grown on a LaAlO3 substrate buffered by a LaNiO3 conducting electrode. The measured I-V results for incremental loading forces confirmed a linear relationship for loading forces exceeding 18 μN. This threshold loading force (18 μN) for CaTiO3 is slightly higher compared to those of SrTiO3 (15 μN) and BaTiO3 (12 μN). This could be due to the larger band gap (3.8 eV) of CaTiO3 compared to those of SrTiO3 (3.2 eV) and BaTiO3 (3.2 eV). In addition, we also tested the on-off behavior; indeed, the behavior was reversible (Fig. R6). This result was added to the revised Supplementary

Question #8
The author mentioned the growth of SRO as step flow mode in the text. However, the thickness of SRO has not been provided. As we know, the conductivity and the lattice parameters of SRO ultrathin films are strongly dependent on the thickness. The lattice will relax to its bulk value (3.93 Å) when the film is thick enough.

Response #8
We thank the reviewer for pointing out this issue. We have performed an XRD reciprocal space map for the SrTiO3/SrRuO3/SrTiO3 heterostructure to confirm the fully strained state of the SrRuO3 layer, as shown in Fig. R7 below. This result was added to the revised Supplementary Information as Supplementary Fig. 5. Also, we added the thickness (i.e., 20 nm) of the SrRuO3 layer to Line #261 of the revised manuscript. Figure R7. Structural characterizations of STO thin films. a, X-ray diffraction (XRD) 2θ-ω scan of an STO thin film grown on a (001)oriented STO substrate, with a conductive SrRuO3 (SRO) buffer layer. The diffraction peak of SRO is indexed in pseudocubic perovskite notation. b, XRD reciprocal space mapping measured from the STO/SRO/STO (001) film around (103)

a b
Reviewers' comments: Reviewer #1 (Remarks to the Author): The authors have made large effort to reason their conclusion of flexoresistance. The referee appreciate it greatly. Nevertheless, the referee is still concerned with other possible reasons for the observed change in the resistance.
For instance, one is the electrostatic interactions between the AFM tip and the thin film surface. The authors argued that "if the polarization in SrTiO3 is caused by the charge of the AFM tip, our main observation (i.e., colossal resistivity change) would be almost independent of the tip radius in the experiments, because the charge density on the tip would not vary much with the tip radius." However, even if the charge density on the tip does not change, the size variation of the tip still can induce changes in the electric response of the thin film, if the tip size is comparable to the size of the gradient field. It is indeed the case, according to the reported data (the contact area radius around 13nm, the gradient field region is around 30nm).
Moreover, regarding the ferroelectric dielectrics, the answers of the authors to the two referees seem inconsistent. On one hand, the authors argue that "in the ultrathin limit, the value of ferroelectric polarization would be highly suppressed compared to the bulk value due to intrinsic size effects", therefore the use of ferroelectric materials is not favored in their study of flexoresistance. (By the way, in their reply to the comment #8 of referee #1, they argued only w.r.t the polarization, but no comment on the higher permittivities of ferroelectrics, which in fact should also be the reason for larger flexocoupling strength of BaTiO3). On the other hand, in the reply to comment #6 of the referee #2 regarding "why the threshold loading force for BTO is lower than that of STO"? The authors seemingly try very hard to convince that the BTO is more favorable to have flexoresistance than STO. It is quite confusing.
Reviewer #2 (Remarks to the Author): It's my pleasure to find all my comments have been well clarified. Excellent work!

Responses to the comments of Reviewer #1
We would like to thank the reviewer for reviewing our manuscript and greatly appreciate his/her high evaluation of our efforts, e.g., "The authors have made large effort to reason their conclusion of flexoresistance. The referee appreciate it greatly." Also, we would like to thank the reviewer for providing other points to consider. We have read and responded to all of the reviewer's comments very carefully. In the pages that follow, we provide our responses to each of the reviewer's comments, in order. The responses are written in blue.

Question #1
For instance, one is the electrostatic interactions between the AFM tips and the thin film surface. The authors argued that "if the polarization in SrTiO3 is caused by the charge of the AFM tip, our main observation (i.e., colossal resistivity change) would be almost independent of the tip radius in the experiments, because the charge density on the tip would not vary much with the tip radius." However, even if the charge density on the tip does not change, the size variation of the tip still can induce changes in the electric response of the thin film, if the tip size is comparable to the size of the gradient field. It is indeed the case, according to the reported data (the contact area radius around 13nm, the gradient field region is around 30nm).

Response #1
We thank the reviewer for suggesting that even with the same charge density, AFM tips with different tip radii (rtip) could elicit different electric responses from the thin film. The reviewer seems to believe that this may explain the observed rtip-dependence of the resistance change. To assess this possibility, we approximately estimated the electric field generated by a charged AFM tip. Here, we simplify the situation, as shown below in Fig. R1, and then consider the electric field generated by a uniformly charged circular disk with radius R. For this finite-sized charged disk, the out-of-plane electric field will decrease in magnitude as R decreases. [Please note that only the out-of-plane electric field (and polarization) should mainly contribute to the band crossing and resistance change across the dielectric layer ( Fig.   1 of the main text).] For instance, a simple electrostatic calculation yields the out-of-plane electric field EOOP at a distance z from the center [i.e., dashed line in Fig. R1(b)], as follows: where σ is the surface charge density of the circular disk and ε is the dielectric permittivity of the dielectric layer. Since the contact radius R decreases with decreasing rtip, the AFM tip with a smaller rtip should induce a smaller average EOOP (as shown in Fig. R2 below), thereby reducing the resistance change. However, this is opposite to our experimental observations.
We observed colossal resistance change only in the case using a sharp AFM tip (with a small rtip). In addition to this opposite trend, as R varies within the experimentally achievable range, the average EOOP changes by only <10% (Fig. R2), which is too small to be attributable to the colossal resistance change and its rtip-dependence. Here, EOOP(z) is calculated along the center line.] Therefore, the electrostatic interactions between the AFM tip and thin film cannot explain the observed rtip-dependence of the resistance change. Furthermore, importantly, as already demonstrated experimentally in our previous response, such electrostatic interactions can only generate a negligible electric field in our experimental geometry, much smaller than that generated by flexoelectricity. Considering these qualitative and quantitative aspects, we exclude the electrostatic interaction between the AFM tip and the thin film as the primary origin of the observed resistance change.

Question #2
Moreover, regarding the ferroelectric dielectrics, the answers of the authors to the two referees seem inconsistent. On one hand, the authors argue that "in the ultrathin limit, the value of ferroelectric polarization would be highly suppressed compared to the bulk value due to intrinsic size effects", therefore the use of ferroelectric materials is not favored in their study of flexoresistance. (By the way, in their reply to the comment #8 of referee #1, they argued only w.r.t the polarization, but no comment on the higher permittivities of ferroelectrics, which in fact should also be the reason for larger flexocoupling strength of BaTiO3). On the other hand, in the reply to comment #6 of the referee #2 regarding "why the threshold loading force for BTO is lower than that of STO"? The authors seemingly try very hard to convince that the BTO is more favorable to have flexoresistance than STO. It is quite confusing.

Response #2
We thank the reviewer for this comment, but we believe that there are some misunderstandings. In particular, we did not insist that the use of ferroelectric materials is not favored in our study of flexoresistance. Our point is that the contribution of ferroelectric polarization itself would not be so important to our experimental observations; instead, the contribution of flexoelectric polarization is the main reason for the observed resistance change. This means that some of the conventional ferroelectric materials (e.g., BTO) could be favorable for our flexoresistance study, as long as their flexocoupling coefficient is large.
Therefore, we do not believe that our responses are inconsistent. A more detailed argument follows below.
Equations (2) and (3) in the main text approximate the threshold strain gradient (∂u/∂x)th as bg th eff where feff is the effective flexocoupling coefficient, e is the electronic charge, and ∆bg and t are the bandgap and thickness of the dielectric layer, respectively. Thus, the lower threshold loading force for BTO originates from its larger feff compared to that of STO. Indeed, it was experimentally observed that BTO inherently could have a much larger feff than STO,