Experimental demonstration of tunable hybrid improper ferroelectricity in double-perovskite superlattice films

Hybrid improper ferroelectricity can effectively avoid the intrinsic chemical incompatibility of electronic mechanism for multiferroics. Perovskite superlattices, as theoretically proposed hybrid improper ferroelectrics with simple structure and high technological compatibility, are conducive to device integration and miniaturization, but the experimental realization remains elusive. Here, we report a strain-driven oxygen octahedral distortion strategy for hybrid improper ferroelectricity in La2NiMnO6/La2CoMnO6 double-perovskite superlattices. The epitaxial growth mode with mixed crystalline orientations maintains a large strain transfer distance more than 90 nm in the superlattice films with lattice mismatch less than 1%. Such epitaxial strain permits sustainable long-range modulation of oxygen octahedral rotation and tilting, thereby inducing and regulating hybrid improper ferroelectricity. A robust room-temperature ferroelectricity with remnant polarization of ~ 0.16 μC cm−2 and piezoelectric coefficient of 2.0 pm V−1 is obtained, and the density functional theory calculations and Landau-Ginsburg-Devonshire theory reveal the constitutive correlations between ferroelectricity, octahedral distortions, and strain. This work addresses the gap in experimental studies of hybrid improper ferroelectricity for perovskite superlattices and provides a promising research platform and idea for designing and exploring hybrid improper ferroelectricity.

1.Abstract: The authors should write a more comprehensive abstract with the clear-cut idea of their work and output in a quantitative form.In addition, More sentences should be used to describe the results.2.Introduction: In the literature review, authors should briefly present some of the existing relevant work emphasizing on the gap between the current work and the existing literature and also should demonstrate how the current work will fill this gap.3.The novelty of this work was not specified, authors should be discussed about the novelty of their work.4.Please highlight/indicate the objectives and limitations of this study in detail at the last paragraph of the Introduction.5.Conclusion: Presentation of the conclusion is not good.The author should write the output in a quantitative form.

Reviewer #2 (Remarks to the Author):
The found and proof of the room-temperature hybrid improper ferroelectricity in La2NiMnO6/ La2CoMnO6 films is interesting.The authors fabricated high-quality double-perovskite films and characterized the ferroelectricity and piezoresponse of the films successfully.However, the strategy of using different layers of superlattice structure to induce the strain lacks of the innovation.And there are some concerns which need to be addressed before the publication in this high-quality journal: 1) The evolution of the BO6 octahedron tilting/rotation could reflect through the SAED (which is absence in the manuscript or supporting information) or the FFT images (which shows no evidence of the BO6 octahedron in the Fig. 1b).It is possibly that the concepts of SAED and FFT are confused in the article.
2) For the εxx used the vector g1 which is perpendicular to the interface, is it possible the Fig. 2b indicates the situation of out-of-plane strain?Or there may be some mistakes in the annotation of the images.Meanwhile, the stripe-like contrast in the grayscale of Fig. 2c should also be perpendicular to its reciprocal lattice vectors.Thus, all the GPA analysis need to be checked carefully.
3) The conclusion of the tilting/rotation of BO6 octahedron in the superlattice films made by Fig. 2h need to be reconsidered, at least providing the quantitative data.The discrepancy of the displacement is difficult to observe only by the individual atomic-scale HAADF-STEM image.

Below is a list of the main changes to the manuscript:
(1) Introduction: The existing relevant works are presented and summarized to emphasize the gap on HIF study to highlight the novelty of our work and how to fill the gap.The objectives and limitations of this work are indicated in detail at the last paragraph.
(2) Abstract: The related quantitative results and necessary descriptions are added, in response to Reviewer #1's comments.
(3) Main text: There are several important changes in the text to comply with the comments raised by the reviewers, including innovation highlights, mechanism analysis and experimental supports, which have been discussed in the point-to-point response.Conclusion is revised as the output in a quantitative form according to the Reviewer #1's comments.
(4) Figures and captions: we update the calculation results on GPA and add the ABF-STEM measurements in Fig. 2 to accurately visualize the strain and quantitatively discuss the oxygen octahedral rotation (OOR).
(5) Supporting online materials: we add the results of SAED and EDS mappings in Supplementary Figures S2 and S3, respectively, in order to double-perovskite superstructures.The local HAADF image and strain visualizations are added in Supplementary Figures S6, S7 and S9.The details and discussions of ABF-STEM measurements are displayed in Supplementary Figures S10 to S12.
(6) References: we add six highly relevant references to improve our manuscript on the comparison of ferroelectric properties and quantitative analysis of OOR.Response: Thanks for the reviewer's constructive comments.We fully agree with these viewpoints.A brief presentation of relevant previous work is necessary to emphasize the gap in HIF research and highlight the importance and innovation of our research contents.The necessary details of the brief presentation and analysis on the existing literature are as follows: So far, studies of hybrid improper ferroelectricity (HIF) in perovskite superlattices are still stagnant at the theoretical level [9][10][11] , and there is still no experimental case reported for perovskite superlattice systems with well-defined HIF.We have summarized two main factors on experiment that hinder the design of HIF in perovskite superlattices.
On the one hand, the fabrication of high-quality double-perovskite superlattices is challenging but necessary for identifying HIF 18,19 .On the other hand, strain-induced structural distortions are quite complex and difficult in tailoring 24 , and the epitaxial strain in thin films not only depends on the excitation of large lattice mismatch 25,26 , but also decays rapidly with increasing film thickness 27,28 .In our revised manuscript, the above existing relevant work have been briefly presented to emphasize the research gap on HIF in the Introduction.
For the first factor, by using ozone-assisted depositing and annealing processes, we synthesized double-perovskite La2NiMnO6/La2CoMnO6 superlattice films with high oxidation states and ordering of B-site cations.For the second factor, the epitaxial growth mode with mixed crystalline orientations slows down the rapid release of epitaxial strain with film thickness and increases the strain critical thickness as large as 90 nm.Meanwhile, film thickness-dependent epitaxial strain ensures the stability of the oxygen octahedral distortion pattern in the sustainable long-range modulation.
Such strain regulation effectively precludes serious interference in the determination of HIF by changes in octahedral distortion pattern raised from various lattice mismatches commonly for changing the epitaxial strain.The relevant demonstration of filling the current gap has been added before the last paragraph of the Introduction.
In brief, we overcome the difficulties in tuning the epitaxial strain and oxygen octahedral distortion for achieving HIF in experimental system of perovskite superlattices, i.e., a robust room-temperature HIF with remnant polarization of ~0.16 μC cm -2 and piezoelectric coefficient of ~2.0 pm V -1 in La2NiMnO6/La2CoMnO6 superlattice films.This work addresses the gap in HIF experimental studies of perovskite superlattices and demonstrates a promising strategy of strain-driven oxygen octahedral distortions for inducing HIF in double-perovskite superlattices.
Thanks again for the reviewer's valuable suggestion which highlights the research content and innovation of our work.

Revision:
The revised sentences "In general, there are two main factors on experiment that hinder the design of HIF in double-perovskite superlattices.Firstly, the growth window of B-site ordered double perovskites is quite narrow 18,19 , and during the film growth, the incomplete oxidation state of transition metal cations inevitably leads to various lattice defects [20][21][22][23] ."were added in page 4/lines 9-12.
The revised sentence "Secondly, strain-induced structural distortions are quite complex and difficult in tailoring 24 ; and the epitaxial strain in thin films not only depends on the large lattice mismatch 25,26 , but also decays rapidly with increasing film thickness 27,28 ."was added in page 4/lines 14-17.
The new sentence "Notably, for determining HIF, the thickness-dependent epitaxial strain can maintain a single mode of octahedral distortion during strain regulation, which facilitates the demonstration of non-monotonic ferroelectric contributions from the coupling of OOR and OOT." was added in page 4/lines 24-27.
Comment 3: The novelty of this work was not specified, authors should be discussed about the novelty of their work.
Response: Thanks for the reviewer's comments.According to the reviewer's advice, we have highlighted the innovation discussion in our revised manuscript.The discussion details of the comprehensive innovation analysis are as follows: In theoretical calculations, hybrid improper ferroelectricity (HIF) has been predicted in several double-perovskite superlattice systems 10,11 .However, it is quite difficult to fabricate high-quality double-perovskite superlattices in experiments.Even worse, the strain-induced structural distortion is difficult in tailoring 24 , and the non-continuous epitaxial strain derived from large lattice mismatch decays rapidly with the film thickness 25,28 .Therefore, the difficulties in stability of the sustainable long-range modulation on octahedral distortion hinder the induction and exploration of HIF in perovskite superlattices.The relevant statements have been also revised and added in the Introduction.
In our work, the above common difficulties in experiment for studying HIF in perovskite superlattices are effectively overcame.We synthesized double-perovskite La2NiMnO6/La2CoMnO6 superlattice films with high oxidation states and ordering of B-site cations by ozone-assisted depositing and annealing processes, which is illustrated in the structure and XPS analysis.By the epitaxial growth mode with mixed crystalline orientations, a large strain critical thickness (＞90 nm) is achieved to permit the sustainable long-range modulation of oxygen octahedral rotation/tilting (OOR/OOT).Therefore, the thickness-dependent epitaxial strain maintains the same mode of octahedral distortion during strain modulation, which facilitates the demonstration of non-monotonic ferroelectric contributions from the strain-regulated coupling of octahedral rotation and tilting.It is a crucial feature for identifying HIF in experimental measurements.
These are the important novelty of the long-range modulation of OOR without changing octahedral distortion mode.According to the reviewer's advice, we have specified the novelty in the revised manuscript.The relevant statements on the novelty of the sustainable long-range modulation for octahedral distortion had been discussed and added in the X-ray diffraction analysis, Introduction, and Conclusion.
Furthermore, we have definitively confirmed the HIF and regulated it in LNMO/LCMO superlattices through experiments and structural optimization.The obtained robust room-temperature HIF with remnant polarization of ~0.16 μC cm -2 and piezoelectric coefficient of 2.0 pm V -1 is comparable to that in the current excellent magnetic unconventional ferroelectrics 32,33  In this work, we overcame the universal difficulties of the sustainable long-range modulation for octahedral distortion in designing HIF by regulating thickness-dependent epitaxial strain, and achieved a tunable HIF in LNMO/LCMO double-perovskite superlattices.The obtained room-temperature HIF behaves a remnant polarization of ~0.16 μC cm -2 and a piezoelectric coefficient of 2.0 pm V -1 .
Although these properties still have a non-negligible shortfall with conventional ferroelectrics [29][30][31] , it is comparable to the current excellent magnetic unconventional ferroelectrics 32,33 .This limitation of our work has been indicated in the manuscript.
The objectives and highlights of this work on HIF were summarized as follows: (i) synthesizing double-perovskite La2NiMnO6/La2CoMnO6 superlattices with high cationic oxidation states and B-site ordering by ozone-assisted depositing and annealing processes; (ii) achieving a large strain critical thickness (＞90 nm), greatly increasing the strain propagation distances for thin films only with a small lattice mismatch (＜1%); (iii) first confirming and regulating the room-temperature HIF in LNMO/LCMO double-perovskite superlattice films by the strain-driven oxygen octahedral distortions in experiments; (iv) revealing the constitutive correlations between octahedral distortions, epitaxial strain, and hybrid improper ferroelectricity.
In brief, although the properties of HIF in LNMO/LCMO superlattices still have a non-negligible shortfall with conventional ferroelectrics, these findings and results first complement the experimental cases of perovskite superlattices in HIF studies.We thank the reviewer again for the valuable suggestion.The added details of the objectives and limitations on our HIF study at the last paragraph of the Introduction have improved our manuscript more comprehensively.
The new sentences "A large strain critical thickness is achieved by the epitaxial growth mode with mixed crystalline orientations, which permits the sustainable long-range modulation of oxygen octahedral rotation and tilting.Such strain-driven octahedral distortion strategy in determining HIF, without changing the lattice mismatch, can effectively preclude serious interference from the changes of octahedral distortion pattern."were added page 5/lines 4-9.Response: Thanks for the reviewer's comments.We agree with the reviewer's suggestion on quantitative form of Conclusion, which will make material properties and study innovations much clearer.We have revised the Conclusion in our manuscript, especially for the results and properties output in a quantitative form.

Revision:
The new sentences "In summary, by a strain-driven oxygen octahedral distortion strategy, we report experimental demonstration of a tunable hybrid improper ferroelectricity (HIF) at room temperature in La2NiMnO6/La2CoMnO6 double-perovskite superlattices.In the optimized films with ~60 nm thickness and 0.72% out-of-plane compressive strain, the remnant polarization (Pr) and piezoelectric coefficient (d33) are ~0.16μC cm -2 and ~2.0 pm V -1 , respectively, which compare to the current excellent magnetic unconventional ferroelectrics.The epitaxial growth mode with mixed crystalline orientations relieves the release of epitaxial strain and achieves a large strain critical thickness (> 90 nm) even within small lattice mismatch (< 1%).Such thickness-dependent strain overcomes the universal difficulties of the sustainable long-range modulation for octahedral distortion in designing HIF under a single octahedral distortion pattern.A hybrid improper mechanism coupling octahedral rotation/tilting and Jahn-Teller distortions is determined.The constitutive correlations between HIF, octahedral distortions, and strain are revealed by a ferroelectric phase transition model based on the Landau-Ginsburg-Devonshire theory.
This study confirms the effectiveness of the strain-driven oxygen octahedral distortion strategy for inducing and regulating the hybrid improper ferroelectricity in double-perovskite superlattices, and provides an experimental platform and a reliable strategy for overcoming the incompatibility of electronic mechanism in multiferroics." were added as Conclusion (page 19/line 19 to page 20/line 7).

Reviewer #2
Comments: The found and proof of the room-temperature hybrid improper ferroelectricity in La2NiMnO6/La2CoMnO6 films is interesting.The authors fabricated high-quality double-perovskite films and characterized the ferroelectricity and piezoresponse of the films successfully.However, the strategy of using different layers of superlattice structure to induce the strain lacks of the innovation.And there are some concerns which need to be addressed before the publication in this high-quality journal.
Response: Thanks for the reviewer's positive comments, especially for elaborating on the related issues and providing helpful suggestions in the comment report.According to the Comments 1 to 3, we have responded to all the questions and revised certain inaccurate expressions accordingly, including the added SAED measurement, the accurate analysis of GPA and the ABF-STEM characterization for quantitative octahedral rotations.
Before the point-to-point responses, we have to apologize for the inadequate statements of the innovations on regulating and determining HIF in the LNMO/LCMO superlattices by changing film thicknesses.It seems that the strategy of only using different layers of superlattice structure to regulate strain is common and lacks of innovation.Indeed, the epitaxial strain regulated by the film thickness is critical for inducing and confirming the HIF in LNMO/LCMO superlattices with specific growth modes.
On the one hand, the thickness-dependent epitaxial strain maintains a single mode of octahedral distortion during strain modulation, which facilitates the demonstration of non-monotonic ferroelectric contributions from the strain-regulated coupling of octahedral rotation and tilting.It is a crucial feature for identifying HIF in experimental measurements.In general, strain-induced structural distortions are quite complex and difficult in tailoring 24 , and the epitaxial strain in thin films usually depends on the excitation of large lattice mismatch 25,26 .Therefore, a great stability of the sustainable long-range modulation on octahedral distortion within a single distortion pattern is the precondition for inducting and identifying HIF in perovskite superlattices.However, replacing/changing the substrates or its orientations will inevitably alter the epitaxial growth modes and octahedral target rotation pattern, since the phase/lattice mismatch between film materials and substrates is fixed.The non-monotonicity of HIF with strain will not be confirmed for the system with different distortion modes.In this work, the thickness-dependent epitaxial strain does well solve the above problem, since such strain regulation does not change any mismatches of the system, thereby ensuring a single octahedral rotation pattern for identifying HIF.We have added the relevant discussion and analysis in Introduction and the section of HIF test.
On the other hand, this epitaxial strain with a large critical thickness provides the sustainable long-range modulation of octahedral rotation/tilting for regulating HIF.
Commonly, the epitaxial strain decays rapidly with the increasing of film thickness 27,28 .However, in our work, a epitaxial growth mode with mixed crystalline orientations for double perovskites slows down the rapid release of epitaxial strain with film thickness and greatly increases the strain critical thickness (＞90 nm) even in a small lattice mismatch (< 1%).Therefore, the strategy of thickness-dependent epitaxial strain effectively overcomes the difficulties in the long-range regulation of the strain-driven oxygen octahedral rotation/tilting within a single distortion pattern to achieve and confirm HIF.We have added the relevant emphasis and discussion in Introduction, Conclusion and the analysis section of epitaxial strain and HIF.
In brief, the thickness-dependent epitaxial strain solved the difficulties in stabilizing the long-range modulation of octahedral rotation and achieving a large critical thickness of strain transmission, thereby inducing and determining HIF in the LNMO/LCMO superlattices.Thanks for the reviewer's comments again.The corresponding innovation statements on strain-driven octahedral distortion regulated by film thickness for achieving and confirming HIF has been carefully revised and added in our manuscript as shown in the following Revision.

Revision:
The revised sentences "The epitaxial growth mode with mixed crystalline orientations maintains a large strain transfer distance more than 90 nm in the superlattice films with lattice mismatch less than 1%.Such epitaxial strain permits sustainable long-range modulation of oxygen octahedral rotation and tilting, thereby inducing and regulating HIF." were added in the Abstract (page 2/lines 8-12).
The revised sentence "Secondly, strain-induced structural distortions are quite complex and difficult in tailoring 24 ; and the epitaxial strain in thin films not only depends on the large lattice mismatch 25,26 , but also decays rapidly with increasing film thickness 27,28 ."was added in page 4/lines 14-17.
The new sentence "Notably, for determining HIF, the thickness-dependent epitaxial strain can maintain a single mode of octahedral distortion during strain regulation, which facilitates the demonstration of non-monotonic ferroelectric contributions from the coupling of OOR and OOT." was added in page 4/lines 24-27.
The new sentence "More importantly, such thickness-dependent epitaxial strain in the films does not change the lattice mismatch, thus the stability of the oxygen octahedral distortion pattern in the sustainable long-range modulation can be greatly ensured." was added in page 7/lines 24-27.
The new sentences "Generally, HIF can be theoretically predicted in A/B-site ordered double-perovskite superlattice systems, but the experimental difficulties in controlling epitaxial strain and octahedral distortion hinder the inducing of HIF in perovskite superlattices.For the thickness-dependent strain without changing the lattice mismatch, the stability of octahedral distortion pattern in the sustainable long-range modulation is greatly ensured in LNMO/LCMO superlattices." were added in page 15/lines 25-30.
The new sentence "In this work, we overcame the universal difficulties of the sustainable long-range modulation for octahedral distortion in designing HIF by regulating thickness-dependent epitaxial strain, and achieved a tunable HIF at room temperature in La2NiMnO6/La2CoMnO6 double-perovskite superlattices." was added in page 4/lines 28-31.
The new sentences "A large strain critical thickness are achieved by the epitaxial growth mode with mixed crystalline orientations, which permits the sustainable long-range modulation of oxygen octahedral rotation and tilting.Such strain-driven octahedral distortion strategy in determining HIF, without changing the lattice mismatch, can effectively preclude serious interference from the changes of octahedral distortion pattern."were added in page 5/lines 4-9.
The new sentences "For designing HIF in experiments, driving oxygen octahedral rotation/tilting usually requires large octahedral rotation phase mismatch between the films and substrates.Thus, epitaxial strain plays a very important role in maintaining and enhancing octahedral rotation.The thickness-dependent epitaxial strain maintains a single mode of octahedral distortion during strain regulation, which facilitates the demonstration of non-monotonic ferroelectric contributions, as a crucial feature for identifying HIF in experimental measurements, from the strain-modulated coupling of octahedral rotation and tilting."were added in page 19/lines 8-16.
Comment 1: The evolution of the BO6 octahedron tilting/rotation could reflect through the SAED (which is absence in the manuscript or supporting information) or the FFT images (which shows no evidence of the BO6 octahedron in the Fig. 1b).It is possibly that the concepts of SAED and FFT are confused in the article.
Response: Thanks for the reviewer's professional comments.We have performed the SAED tests.According to the combination of the extinction law and the ordered close-packed structures, we supplemented the corresponding discussion of epitaxial structures from the diffraction pattern.
As the reviewer pointed out, the concepts of SAED and FFT are different.The SAED pattern is the reciprocal space image for lattice structures which is measured directly by transmission electron microscope (TEM) through rotating the selected diaphragm.
Thus, the SAED pattern contains the separable information about the amplitude zone axis for generating diffraction, the second close-packed plane is (0 k 0).Thus, the simplest extra diffraction spot originates from (0 1 0).These discussions have been added to the section of structure analysis both in main text and Supplementary Information.Furthermore, the good uniformity and stoichiometry of B-site cations in double perovskites are quite crucial for generating extra diffraction which is dependent on the close-packed plane with ordered B-site cations.We then tested EDS images for superlattice films to indicate the good uniformity and stoichiometry for B-site cations.As shown in Supplementary Figure S3, the elemental distribution is homogeneous and the ratio of B-site atoms close to 2:1:1.The related analysis on EDS mapping was added in Supplementary Information.
In addition, since the rotation/tilting of the oxygen octahedra belongs to the fine evolution inside the crystal and the atom mass of oxygen is much too light, it is quite difficult to reflect octahedra rotations by lattice-scale diffraction spots measurements.
By investigating the literature, we adopted annular bright-field (ABF) STEM images with atomic resolution to demonstrate the octahedral rotations in superlattice films, and the related results and discussions are shown in Comment 3.
Thanks for the reviewer's comments again.According to the reviewer's comments, As shown in Supplementary Figure S3, the elemental distribution is homogeneous and the ratio of B-site atoms close to 2:1:1.These results contribute to generate extra diffraction spots from the close-packed plane with B-site orderings in superlattices." were added in Supplementary Information page 1/line 22 to page 2/line 4.
The new sentence "High-resolution lattice images, SAED and EDS mappings were measured by a field emission transmission electron microscopy (TEM, JEOL, JEM-2100F)." was added in Methods (page 20/lines 26-28).
Comment 2: For the εxx used the vector g1 which is perpendicular to the interface, is it possible the Fig. 2b indicates the situation of out-of-plane strain?Or there may be some mistakes in the annotation of the images.Meanwhile, the stripe-like contrast in the grayscale of Fig. 2c should also be perpendicular to its reciprocal lattice vectors.
Thus, all the GPA analysis need to be checked carefully.
Response: Thanks for the reviewer's professional comments.We have carefully checked all the GPA analyses in our manuscript and apologize for the confusions caused by the inaccurate expressions and interpretations on GPA.The epitaxial strain within LNMO/LCMO superlattices are systematically analyzed by using the newest generation of GMS-3.5.3 with the FRWRtools plugin.According to the comments, we have responded to the questions and revised the manuscript.The details of the related analysis and description of GPA are as follows.
As suggested by the reviewer, the correlation between strain field, phase contrast, and reciprocal lattice vectors should be unified and standardized with each other.For the accuracy of the strain analysis, we comprehensively investigated the basic principles and operating procedures of GPA to ensure this correlation.GPA is an effective approach for processing HRTEM images by combining real-space and Fourier-space information to estimate and visualize the spatial distribution of strain 1 .By measuring the displacement of lattice fringes of HRTEM image with respect to an unstrained reference area, the local Fourier components of lattice fringes is calculated so that the information concerning the strain of lattice can be extracted by analyzing interference fringes 1 .The GPA method is based upon centering an aperture around the assigned reflection in the Fourier transformation of an HRTEM image and subsequently performing an inverse Fourier transformation.The phase of image, namely geometric phase Pg(r), is related to the component of displacement field u(r) in the direction of the reciprocal lattice vector g: where r is the position in the image 1 .A two-dimensional lattice is defined in real space basis vectors a1 and a2, which correspond to the reciprocal lattice vectors g1 and g2, respectively.Through calculating two sets of lattice fringes, the displacement field is given by 1 : The information of local strain can be obtained by analyzing the gradient of the displacement field, which is defined as ε and described as matrix form: = (∂ux/∂x ∂ux/∂y) (∂uy/∂x ∂uy/∂y) (3) In the above matrix, the value of each element can be converted to an image.More details about GPA method can be found in Hÿtch's work 1 .All in all, in an ideal situation, the selected reciprocal lattice vector is perpendicular to the orientation of crystal planes in real space, and the contrast of geometric phase is perpendicular to the reciprocal lattice vector.In fact, however, lattice defects, deviations of zone axes, multiphase coexistence, as well as different reference areas, can cause significant changes in the strain field and geometrical phases.Thus, during the operation for GPA, two non-colinear reciprocal space vectors (g1 and g2) with large intensities of power spectra should be selected to reach an excellent signal-to-noise ratio 2 .Furthermore, a proper mask size in Fourier space is regulated to smooth the image around the peaks and reduce the initial noise during Fourier filtering 2 .
As shown in Supplementary Figure S8, the calculated GPA displays a standard correlation between strain field (εxx and εyy ), phase contrast (Pg1 and Pg2) and reciprocal lattice vectors (g1 and g2).Epitaxial strain is released through the dislocations and lattice distortions along the boundaries of different regions.Notably, the phase contrast is significantly intermittent, indicating the influence caused by the lattice defects in the superlattice films.Similarly, Figure 2a-2c behaves a standard geometric correlation in GPA calculation for the TEM image of the SL30 at interface.
The strain fields are well visualized at the macroscopic lattice scale.Response Figure 1 presents the full primary files and operating panel for the GPA calculation.However, as shown in Figure 2d-2f, the GPA calculation of the film surface is slightly different.
The phase contrasts (Pg1 and Pg2) are not perfectly perpendicular to the reciprocal lattice vectors (g1 and g2).This is attributed to the fact that, at the atomic scale, the atomic columns corresponding to the multiple epitaxial modes do not form a single focused diffraction spot, but are accompanied by satellite peaks, which are reflected in the macroscopic expression of the phase contrasts.In addition, to some extent, it may also be related to the oxygen octahedral distortion.Even so, it should be reminded that this phase changes do not affect the results of the strain analysis, which is mainly carried out in only one phase period (from - to  ) and the atoms in the reference region are well aligned.The standard operation and corresponding primary files for GPA are presented in Response Figure 2. We estimated the local strain using a statistical approach that corresponds to the epitaxial structure (Supplementary Figure S7).
We have updated our GPA calculations, and the epitaxial strain were thoroughly clarified and discussed in the manuscript.Thanks for the reviewer's professional comments again.These revisions make the strain visualization more accurate and greatly improve the quality of our manuscript.Response Figure 2. The process of GPA on the surface of SL30 films performed by using the FRWRtools plugin in GMS-3.5.3.Two non-collinear reciprocal lattice vectors g1 and g2 are selected for GPA.The IP and OP strains are output as εxx and εyy, respectively.The corresponding strain and phase variations are visualized by the calibration scales from -1% to 1% and from - to , respectively.The resolution and smoothing are set to 1 nm and 10, respectively.The final analysis files include diffraction image, and IP and OP displacement fields (uxx and uyy), strain fields (εxx, εyy, εxy and rotation) and phase (Pg1 and Pg2).
Response References:

Revision:
The new figures of GPA (Figure 2a-2g and Supplementary Figure S7 and S8) were updated in Main Text and Supplementary Information, respectively.
The new sentences "However, the phase contrasts (Pg1 and Pg2) are not perfectly perpendicular to the reciprocal lattice vectors (g1 and g2).This is attributed to the diffraction spots with satellite peaks raised from the atomic columns under the multiple epitaxies, reflecting in the macroscopic expression of the phase contrasts.
Even so, it should be noted that this phase reflection does not affect the results of the strain analysis because it is mainly carried out in only one phase period (from - to ) and the atoms in the reference region are well aligned."were added in page 9/lines 11-17.
The new sentences "The calculated GPA displays a standard geometric correlation between strain field (εxx and εyy ), phase contrast (Pg1 and Pg2) and reciprocal lattice vectors (g1 and g2).The FFT of three different regions clearly demonstrates the changing process of the lattice parameters.Epitaxial strain is released through the dislocations and lattice distortions along the boundaries of different regions.Notably, the phase contrast is significantly intermittent, indicating the influence caused by the lattice defects in the superlattice films."were added in Supplementary Information page 4/lines 6-12.
The new sentences "We utilized an ozone-assisted growth method to achieve the growth of high-quality double-perovskite superlattices.As shown in Supplementary Figure S6, the local HAADF images reveal fully epitaxial structures with the coherent growth for the LNMO and LCMO superlattice layers.Moreover, for the local IP and OP displacement fields (uxx and uyy) of atomic columns, the displacement distributions of A-site cations are almost unchanged along the IP and OP directions, while changing slightly for B-site cations.This difference in the atomic projection, to some extent, indicates the titling/rotation of BO6 octahedron in the superlattice films, which can usually affect the ferroic order parameters in the system.The reliable rotation or tilting of the oxygen octahedron is analyzed by the changes in B-O-B bond angles, according to the ABF STEM images for oxygen distribution (see Figure 2h).
Supplementary Figure S7 shows the strain maps processed in grayscale to reflect the strain distribution clearly.According to the practical in-plane and out-of-plane crystal axes, the appropriate cutting lines were employed to obtain the intensity profiles of strain."were added in Supplementary Information page 3/lines 1-15.The new sentence "The average of calculated bond angles for SL60 is similar to the Finally, thanks to the editor and reviewers again for all the excellent and professional comments, which give us a great chance to improve our work better.

REVIEWER COMMENTS
Reviewer #1 (Remarks to the Author): This work presents some interesting and important results, and the authors have revised it well.

Reviewer #2 (Remarks to the Author):
There are still some concerns which need to be addressed before the publication in this high-quality journal: 1) The authors should check the use of "in-plane" and "out-of-plane" again to avoid the misleading of the readers.
2) Does there exist OOR or OOT in the bulk of La2NiMnO6 or La2CoMnO6?
3) The tool/method of simulating the oxygen positions should be added in the part of methods in the article.
Below is a list of the main changes to the manuscript: (1) Results: We defined the directions of "in-plane" (IP) and "out-of-plane" (OP).
(3) References: Three highly relevant references were added to expound the mechanism of HIF.
(4) Methods: The details of determining oxygen positions for BO6 octahedral distortions are added.
In available from Qz and Qx in RSM tests, respectively (Fig. 1d).Microscopically, the distribution of local strains is visualized by GPA, according to the HR-TEM images (Fig. 2 and Supplementary Fig. S8).The changes of OP and IP interlayer spacing indicate the microscopic strain relaxations within the superlattice films (Supplementary Fig. S9).We have added well-defined terms for "in-plane (IP)" and "out-of-plane (OP)" in the manuscript.
Thanks again for the reviewer's suggestion.The related revisions have effectively avoided the misleading of the readers on "in-plane" and "out-of-plane".

Revision:
The revised sentences "Figure 1c exhibits the XRD θ-2θ scans of superlattice films with different thicknesses.The macroscopic out-of-plane (OP) epitaxial strain of thin films is usually determined by this diffraction collected perpendicular to the film surface."were added in page 7/lines 5-7.
The revised sentences "The reciprocal space mapping (RSM) around the asymmetric reflection (103)pc further determines the macroscopic epitaxial strain and the dependence of strain transfer on film thicknesses.As shown in Fig. 1d, Qx of the SL10 for the direction parallel to the film surface, i.e. in-plane (IP), is aligned with that of the substrates, which indicates the coherent epitaxy of superlattices with full strain.
According to the lattice parameters calculated from the Qz (OP) and Qx (IP), the superlattice films behave macroscopic IP tensile and OP compressive strain, and the strain relaxation occurs obviously due to the increasing film thickness."were added in page 8/lines 1-8.
Comment 2: Does there exist OOR or OOT in the bulk of La2NiMnO6 or La2CoMnO6?
Response: Thanks for the reviewer's professional comments.As mentioned in the manuscript (page 18/line 18), the BO6 octahedron in the bulk of LNMO and LCMO does rotate or tilt indeed, which can be indexed as a -a -c + pattern in Glazer's notation 9 .
superlattices, it seems that the double-perovskite superlattice systems of (A2BB'O6)n/(A2BB''O6)n (n=1) could be a novel platform for achieving HIF through the unique structural design.The related analysis of HIF in B-site ordered perovskite superlattices has been further highlighted in Discussion.
Thanks for the reviewer's constructive comments again.We explained the reason why Finally, thanks to the editor and reviewers again for all the excellent and professional comments, which give us a great chance to improve our work better.

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Methods: The details of film growth, SAED patterns, EDS mappings and ABF-STEM measurements are added.In the following, the reviewer's original comments are shown by black italic characters.Our Responses are shown by blue characters, and Revisions are shown by red characters.and the DFT calculations and Landau-Ginsburg-Devonshire theory reveal the constitutive correlations between HIF, octahedral distortions, and strain."were added in the Abstract (page 2/lines 8-15).Comment 2: Introduction: In the literature review, authors should briefly present some of the existing relevant work emphasizing on the gap between the current work and the existing literature and also should demonstrate how the current work will fill this gap.

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intensity) of the electron wave.While the FFT image is derived from a mathematical transformation for high-resolution TEM image of the lattice, which contains the information both of amplitude and phase of the electron wave.Therefore, the FFT reflecting phase information from HR-TEM image can be used to study the computational simulations on structural image, such as GPA, whereas the SAED pattern cannot.In general, from the crystal structure point of view, the distributions of lattice planes reflected from SAED and FFT in the reciprocal space are the same, corresponding to the same set of diffraction spot patterns.In this work, we use diffraction spot patterns to determine the epitaxial growth mode of the double-perovskite superlattices, which is consistent for FFT and SAED.As shown in Supplementary Figure S2, the SAED patterns indicate three growth modes of LNMO/LCMO superlattices grown on (001)-Nb:STO substrates, i.e., type-I, -II and -III; [001](001)SL//[001](001)Sub, [001](110)SL//[010](001)Sub and [001](110)SL//[100](001)Sub, respectively.Notably, the additional diffraction spots of type-Ⅱ and -Ⅲ are more obvious than that of type-Ⅰ, which indicates a relatively few proportions of type-Ⅰ epitaxial structures within the tested region.These extra diffraction spots originate from the superstructures of B-site ordered double perovskites.In fact, the extinction law of space group P21/n is (h 0 l) and (0 k 0), where h + l = odd number; k = odd number, respectively.For the type-I and -II, the simplest extra diffraction spot originates from (00l), where l is odd number, because (001) is the close-packed plane for ordered B-site cations except the extinction of primary diffraction.While for the type-Ⅲ, the emitted electron beam is along [001] we have added the necessary measurements and supplements on electron diffraction and epitaxial structures.The corresponding discussions and revisions greatly improve the quality of our manuscript.SupplementaryFigure S2.Selected area electron diffraction of LNMO/LCMO superlattices with different epitaxial modes.a, SAED pattern of double-perovskite superlattice films SL90.b, The local magnification image of additional diffraction spots corresponding to three types of growth modes, marked by marked by circles and arrows in different colors, respectively.The additional diffraction spots of Ⅱ-and Ⅲ-type are more obvious than that of Ⅰ-type, which indicates a relatively few proportions of Ⅰ-type epitaxial structures within the tested region.The new sentences "Notably, the additional diffraction spots of type-Ⅱ and -Ⅲ are more obvious than that of type-Ⅰ, which indicates a relatively few proportions of type-Ⅰ epitaxial structures within the tested region.These extra diffraction spots originate from the superstructures of B-site ordered double perovskites.In fact, the extinction law of space group P21/n is (h 0 l) and (0 k 0), where h + l = odd number; k = odd number, respectively.For the type-I and -II, the simplest extra diffraction spot originates from (00l), where l is odd number, because (001) is the close-packed plane for ordered B-site cations except the extinction of primary diffraction.While for the type-Ⅲ, the emitted electron beam is along [001] zone axis for generating diffraction, the second close-packed plane is (0 k 0).Thus, the extra diffraction spot originates from (0 k 0), where k is odd number.Furthermore, EDS mappings were tested to indicate the good uniformity and stoichiometry of B-site cations in double perovskites.

Fig. 2 |
Fig. 2 | Geometric phase analysis (GPA) and octahedral distortions of the LNMO/LCMO superlattice films.a Low magnification TEM image of the SL30 at interface and the FFT image of the films.b, c Corresponding GPA analysis of (a) along in-plane direction εxx (b) and out-of-plane direction εyy (c), respectively.d HAADF STEM image of the SL60 near surface and the inset of FFT image.e, f Corresponding GPA analysis of εxx (e) and εyy (f) on local atomic image (d), respectively.The insets attached to εxx and εyy are corresponding phase images with normalized phase variation from -π to π (black to white).White circles in FFT images mark the non-collinear reciprocal lattice vectors g1 and g2 for GPA.Yellow squares show the reference region for GPA.The color scale indicates the relative difference of local strain in the films.g Average intensity profiles of the red and blue lines in the grayscale images of εxx (e) and εyy (f), respectively.h Local annular bright-field (ABF) STEM images of the SL60 films and STO.The schematic shows the corresponding BO6 octahedral distortions.i The ABF-STEM image of the cross-sectional interface of SL60 on STO and the tilting angle (B-O-B') of oxygen octahedrons by collecting 19 layers of perovskite unit cells.

Comment 3 :
The conclusion of the tilting/rotation of BO6 octahedron in the superlattice films made by Fig.2hneed to be reconsidered, at least providing the quantitative data.The discrepancy of the displacement is difficult to observe only by the individual atomic-scale HAADF-STEM image.Response: Thanks for the reviewer's constructive comments on the quantitative analysis of oxygen octahedral rotation/tilting.We agree with the reviewer that the atomic displacement of BO6 octahedral distortions is difficult to be observed only by the individual atomic-scale HAADF-STEM image and the necessary quantitative analysis should be provided.The quantitative analysis of octahedral rotation/tilting can be performed from the detected distributions of oxygen coordination by using the STEM annular bright-field (ABF) technique which is sensitive to the atoms with light masses such as oxygen 42 .As shown in Fig.2h, the oxygen atomic columns of STO in cross-sectional ABF-STEM image are arranged in straight chains, which is consistent with the 180°b ond angle of Ti-O-Ti in Pm-3m symmetry.While for the LNMO/LCMO superlattices, the elongated oxygen sublattices are arranged in a zigzag-like pattern.This sharp contrast in the oxygen coordination indicates the obvious OOR/OOT in SL60 films.Compared to STO substrates with a single crystalline orientation, the partial oxygen sublattices in superlattices are relatively ambiguous due to the phase interfering from multiple growth modes.Since the apical O atom overlaps with A-site La atom, we can only determine the bond angles (B-O-B') associated with OOR by measuring the oxygen positions on the left and right sides of B-site atoms.The quantitative bond angles of B-O-B' are counted layer by layer within a large region as shown in Fig. 2i.The fluctuation of the bond angle fully reflects the modulation of the octahedral rotation by the epitaxial strain.In addition, the dramatic changes within the six perovskite layers exhibit the clamping effect of the STO substrates on OOR.This limitation of octahedral structures originates from the lattice and symmetry mismatch between the films and substrates, remaining a coherent interface and lattice connectivity 43 .According to the approximately equal values of median and average of bond angles within a single layer, OOR tends to be stable as the clamping effect disappears.The corresponding bond angle is estimated as ~157°on average, i.e., the octahedron is rotated by approximately 11.5°.The distribution sites of oxygen atom columns are displayed in Response Figure 1 for counting the B-O-B' bond angles, and the statistical data of B-O-B' bond angles are listed in Response Table 1.In brief, the epitaxial strains in LNMO/LCMO superlattices have the ability to achieve the sustainable long-range modulation of the oxygen octahedral distortions, especially for OOR/OOT.The related discussions are added to the section of structure analysis.Notably, the ABF image recorded along the STO-[110] axis is the best visualization to analyze OOR.Supplementary Figure S10 shows annular bright-field (ABF) STEM images of the SL60 films along different crystal axis directions of SL-[110] and -[0 -1 1].According to the corresponding schematic of oxygen distribution, the layered structure of oxygen columns measured along SL-[110] axis cannot be observed clearly.The stacking of A-and B-site atoms in close-packed plane does not provide the sufficient atomic gaps for the oxygen atoms with lighter masses to reflect the contrast of the sublattice projections.Consequently, despite the fact that the octahedron is in a simple geometric perspective, the tilting or rotation of the oxygen octahedron cannot be determined.While, along SL-[0 -1 1] axis, the elongated and misaligned oxygen sublattices are arranged in a zigzag-like pattern.This result significantly indicates the qualitative octahedral distortions in the superlattice films.The layered structure of oxygen columns forms a close-packed plane which is highlighted both in the ABF image and structural schematic by the red dashed line and blue rectangular box, respectively.However, the quantitative analysis of BO6 octahedral distortions cannot be proceeded because the geometrical position of octahedron is too complex to measure any bond angles of B-O-B' for OOR/OOT.Supplementary Figure S11 shows the illustrations of the geometric structure for the sample preparation by using focused ion beam (FIB) and the STO-[110] axis for the ABF measurements.The films are sliced along the diagonal of the STO substrates (4 5°) for FIB preparation.The distribution features of oxygen columns for both superlattice films and STO substrates are clearly displayed in the schematic along the STO-[110] axis for ABF measurements.More importantly, the quantitative analysis of the BO6 distortions can be further performed by measuring the bond angle of B-O-B' Response Figure 1.The distribution sites of oxygen atom column for the statistics of the B-O-B' bond angles.
value measured in ABF-STEM image, indicating the reliability of the calculations."was added in page 17/line 30 to page 18/line 2. The new sentences "Supplementary Figure S10 shows annular bright-field (ABF) STEM images of the SL60 films along different crystal axis directions of SL-[110] and -[0 -1 1].According to the corresponding schematic of oxygen distribution, the layered structure of oxygen columns measured along SL-[110] axis cannot be observed clearly.The stacking of A-and B-site atoms in close-packed plane does not provide the sufficient atomic gaps for the oxygen atoms with lighter masses to reflect the contrast of the sublattice projections.Consequently, despite the fact that the octahedron is in a simple geometric perspective, the tilting or rotation of the oxygen octahedron cannot be determined.While, along SL-[0 -1 1] axis, the elongated and misaligned oxygen sublattices are arranged in a zigzag-like pattern.This result significantly indicates the qualitative octahedral distortions in the superlattice films.The layered structure of oxygen columns forms a close-packed plane which is highlighted both in the ABF image and structural schematic by the red dashed line and blue rectangular box, respectively.However, the quantitative analysis of BO6 octahedral distortions cannot be proceeded because the geometrical position of octahedron is too complex to measure any bond angles of B-O-B' for OOR/OOT." were added in Supplementary Information page 4/line 19 to page 5/line 4. The new sentences "The best visualization of recording ABF-STEM image to analyze OOR is along the STO-[110] axis, i.e., SL-[100] or [010] axes.Supplementary Figure S11 shows the illustrations of the geometric structure for the sample preparation by using focused ion beam (FIB) and the STO-[110] axis for the ABF measurements.The films are sliced along the diagonal of the STO substrates (~45°) for FIB preparation.The distribution features of oxygen columns for both superlattice films and STO substrates are clearly displayed in the schematic along the STO-[110] axis for ABF measurements.More importantly, the quantitative analysis of the BO6 distortions can be further performed by measuring the bond angle of B-O-B' under a simple geometrical perspective of oxygen octahedron.The basic information of the superlattice films and FIB sample for ABF-STEM measurement are shown in Supplementary Figure S12." were added in Supplementary Information page 5/lines 6-16.Added Reference: 40. S. Li, et al.Strong ferromagnetism achieved via breathing lattices in atomically thin cobaltites.Adv.Mater.33, 2001324 (2020).42.Ishikawa, R. et al.Direct imaging of hydrogen-atom columns in a crystal by annular bright-field electron microscopy.Nat.Mater.10, 278-281 (2011).43.X. Ding, et al.Crystal symmetry engineering in epitaxial perovskite superlattices.Adv.Funct.Mater.31, 2106466 (2021).
the following, the reviewer's original comments are shown by black italic characters.Our Responses are shown by blue characters, and Revisions are shown by red characters.
a -a -c + octahedral rotations cannot induce HIF in the bulk of LNMO and LCMO, and the analysis of HIF in LNMO/LCMO superlattices has been further emphasized in Discussion.These revisions greatly improve the quality of the article on HIF research.experiment.Based on the optimization results of the strained crystal structures, the oxygen positions can be determined, and the BO6 octahedral deformation and OOR/OOT are further determined by measuring the B-O bond lengths and B-O-B bond angles, respectively."were added in Methods (page 22/line 29 to page 23/line 5).