Orthogonal interlayer coupling in an all-antiferromagnetic junction

In conventional ferromagnet/spacer/ferromagnet sandwiches, noncollinear couplings are commonly absent because of the low coupling energy and strong magnetization. For antiferromagnets (AFM), the small net moment can embody a low coupling energy as a sizable coupling field, however, such AFM sandwich structures have been scarcely explored. Here we demonstrate orthogonal interlayer coupling at room temperature in an all-antiferromagnetic junction Fe2O3/Cr2O3/Fe2O3, where the Néel vectors in the top and bottom Fe2O3 layers are strongly orthogonally coupled and the coupling strength is significantly affected by the thickness of the antiferromagnetic Cr2O3 spacer. From the energy and symmetry analysis, the direct coupling via uniform magnetic ordering in Cr2O3 spacer in our junction is excluded. The coupling is proposed to be mediated by the non-uniform domain wall state in the spacer. The strong long-range coupling in an antiferromagnetic junction provides an unexplored approach for designing antiferromagnetic structures and makes it a promising building block for antiferromagnetic devices.

The submitted manuscript addresses the interlayer coupling between antiferromagnetic layers, which distinguishes this work from the previous studies where the interlayer coupling was observed between ferromagnetic layers. In my view, the authors provided compelling arguments based on their SMR and XMLD data that the Néel vectors of the two FeO3 layers are orthogonal in the ground state. These results are interesting and should be published in some form.
Having said this, I am not convinced at all that their model of a quasi-long range order in the antiferromagnetic spacer layer correctly explains their results. While the Cr2O3 (0001) moments are orthogonal to the interfaces, they are capable to transmit the exchange coupling between the two Fe2O3 layers. While this coupling is expected to be bilinear (despite the likely small canting of the magnetic moments), the presence of roughness can produce a biquadratic coupling according to Slonczewski's mechanism. This explanation seems to me more plausible than that provided by the authors.
There are few other comments which need to be taken into account by the authors.
1. The authors need to place their work properly in the context of the previous knowledge in the field. The references given above my serve as a guide to the authors. 2 The manuscript needs to be significantly improved in terms of the English and the style. There are many unclear or incorrect statements. Below I list just a couple of them in the abstract.
-The first sentence in the abstract is erroneous and misleading.
-The third sentence in the abstract is unclear.
-The firth sentence in the abstract is incorrect (see comment 3).
3. In the last sentence on page 2, the authors say that "collinear parallel/antiparallel arrangements of Néel vectors in antiferromagnets are identical." They are not identical by symmetry. It is not clear what the authors want to say. Also, in the same sentence they say "the small noncollinear interaction would become dominant in an AFM/spacer/AFM junction." The bilinear interlayer coupling occurs across an AFM layer. This interaction can cause small canting of the magnetic moments, but it is still bilinear.
4. In their model, the authors introduce the Dzyaloshinskii-Moriya interaction (DMI) in Fe2O3. It is not clear what its role is. On page 4, they say that "the hysteresis is due to the existence of Dzyaloshinskii-Moriya interaction (DMI)…" However, normally the hysteresis can be well explained by the magnetic anisotropy… 5. On page 5, the authors say; "The resistance peak appears before H = 0, which violates the principle of thermodynamics, indicating the existences of coupling effect." It is not clear why the principle of thermodynamics is violated in their system. Probably, the authors meant something else.
Reviewer #3: Remarks to the Author: The manuscript by Zhou et al. reports an orthogonal coupling of two Fe2O3 layers separated by a Cr2O3 layer. Firstly, the results of magneto-transport measurements on Fe2O3/Cr2O3/Fe2O3/Pt structure as well as two reference samples with only Fe2O3/Pt layers and Cr2O3/Fe2O3/Pt layers are presented. The results are well explained by assuming an emergence of an orthogonal coupling for the two Fe2O3 layers. XMLD results support this picture. Then, a temperature dependence of the magneto-transport properties is presented, from which the origin of the orthogonal coupling is discussed. Finally, a theoretical model is put forward, with which the experiment is semiquantitatively explained. The interlayer magnetic coupling is an important phenomenon for functional magnetic and spintronic devices as well as a fundamental interest in condensed-matter physics, and collinear antiferromagnetic coupling is widely used in hard-disc drive and nonvolatile memory. Meanwhile, as far as I know, the orthogonal interlayer coupling has not been well studied so far. In this regard, I think this work would potentially give a high impact to the community of magnetics and spintronics. However, I think the authors should consider the following three points and revise the manuscript.
(1) The authors assume that the signal of the magneto-transport measurement arises from the spin-Hall magnetoresistance (SMR), but its evidence is not presented in this manuscript. For example, what about the contribution of anisotropic magnetoresistance and other magnetoresistive effects. I think the authors should perform the same measurement for samples without the Pt cap layer or with a cap layer comprised by a low spin Hall material.
(2) The authors attribute the orthogonal coupling to a quasi-long range order (QLRO), but I could not find any solid evidence of the QLRO in Cr2O3 layer. I think the authors should somehow investigate the spin structure of the Cr2O3 layer as a function of the temperature and/or the thickness of Cr2O3 layer; otherwise, the proposed scenario is just a speculation. In addition, I cannot understand well why the orthogonal coupling between the Fe2O3 is stabilized when the QLRO emerges in Cr2O3. More detailed explanation is desirable.
(3) English should be improved. There are many sentences containing grammatical error.

Response Letter of NCOMMS-21-38921A-Z
We very much appreciate the positive evaluations of our manuscript (NCOMMS-21-38921A-Z) by Reviewer #1 ("In my view, the authors provided compelling arguments based on their SMR and XMLD data" and "These results are interesting and should be published in some form") and Reviewer #3 ("I think this work would potentially give a high impact to the community of magnetics and spintronics"). We address the issues raised by them point by point below. And their comments are helpful for our improvements further. Leilei Qiao and Yunfeng You are added as coauthor for the help in the physical model and English writing improvement. Amendments of our revised manuscript are summarized below in bold face style. The main revisions include: 1. We revised our theory model as suggested by Reviewer #1. 2. We revised the statements in the abstract, introduction as well as the main text as pointed out by Reviewer #1. 3. We performed magnetoresistance measurement for samples with a cap layer comprised by a low spin Hall material titanium (Ti) as suggested by Reviewer #3. 4. We added more detailed explanation about our theory as pointed out by Reviewer #3. 5. We carefully revised the English writing throughout the manuscript.

Response to Reviewer #1
The manuscript is devoted to the interlayer coupling between two antiferromagnetic (AFM) Fe 2 O 3 layers across a thin layer of insulating antiferromagnet Cr 2 O 3 . The authors observe a spin Hall magnetoresistance (SMR) effect exhibiting features which indicate the orthogonal alignment of the Néel vectors of the two Fe 2 O 3 layers in the ground state. This interpretation is supported by their x-ray magnetic linear dichroism (XMLD) data. The authors argue that they observe an orthogonal interlayer coupling driven by a quasi-long range order in the AFM spacer layer.
Historically, the interlayer coupling has been a hot topic of research since its discovery in 1986 by P. Grünberg et al., Phys. Rev. Lett. 57, 2442(1986, due to its relevance to the giant magnetoresistance (GMR) effect discovered later. This bilinear (~m1•m2) coupling across a metallic spacer layer, oscillating as a function of its thickness, is now well understood within a model of the standing wave in the metallic spacer layer (e.g., M. D. Stiles, Interlayer exchange coupling, in Ultrathin Magnetic Structures III p. 99, (2005)). In addition to the bilinear coupling, a biquadratic (~(m1•m2)^2) coupling has been discovered (see, e.g., S. O. Demokritov, J. Phys. D: Appl. Phys. 31, 925 (1998), for review). While the bilinear coupling aligns the magnetic moments collinear (parallel or antiparallel), the biquadratic coupling tends to align the moments orthogonal. The widely accepted explanation of the biquadratic coupling is Slonczewski's model (J. C. Slonczewski, Phys. Rev. Lett. 67, 3172 (1991)), where the interface roughness in conjunction with the oscillatory bilinear coupling drives the effect.
The submitted manuscript addresses the interlayer coupling between antiferromagnetic layers, which distinguishes this work from the previous studies where the interlayer coupling was observed between ferromagnetic layers. In my view, the authors provided compelling arguments based on their SMR and XMLD data that the Néel vectors of the two Fe2O3 layers are orthogonal in the ground state.
These results are interesting and should be published in some form.
A: We sincerely thank the Reviewer for the positive evaluation such as "the authors provided compelling arguments", "These results are interesting and should be published in some form" and recommendation for the publication of our paper. We have addressed the Reviewer's comments as shown below. Q1) Having said this, I am not convinced at all that their model of a quasi-long range order in the antiferromagnetic spacer layer correctly explains their results. While the Cr 2 O 3 (0001) moments are orthogonal to the interfaces, they are capable to transmit the exchange coupling between the two Fe 2 O 3 layers. While this coupling is expected to be bilinear (despite the likely small canting of the magnetic moments), the presence of roughness can produce a biquadratic coupling according to Slonczewski's mechanism. This explanation seems to me more plausible than that provided by the authors.
3 A: Thanks for the helpful suggestion from the Reviewer. We revised our model based on Slonczewski's mechanism and we found that the interfacial roughness plays an important role in forming the orthogonal interlayer coupling.
Based on this analysis, we think that the quasi-long range order state in the Cr 2 O 3 spacer may be better described as an in-plane non-uniform domain wall state. In fact, this picture is still similar to our previous model, in that there is in-plane magnetic order in the Cr 2 O 3 spacer, and this magnetic order is non-uniform in the film plane. The moments form magnetic order in a short range, but the whole Cr 2 O 3 spacer cannot be described by a single Néel vector. Meanwhile, the original Slonczewski model in ferromagnet/normal metal/ferromagnet is insensitive to temperature, which cannot be used to described the unique temperature dependence of the coupling in our antiferromagnetic trilayer. The temperature evolution of the magnetic order in the Cr 2 O 3 spacer is an additional degree of freedom that has to be taken into account. Hence, we think that the combination of the Slonczewski model and our previous model that focus on the temperature evolution of the magnetic order in the Cr 2 O 3 spacer provides the thorough description for the coupling phenomena.
We rewrote the modelling part on Page 11 as following: "Non-uniform domain wall state mediated interlayer coupling. Having excluded the magnetic ordering which is uniform in the film plane, we consider magnetic ordering, which is non-uniform in the film plane, as the origin of the interlayer coupling. It is known that orthogonal interlayer coupling could exist in FM/NM/FM trilayers due to the interfacial roughness and oscillating collinear exchange coupling 13 . The collinear interlayer coupling mediated by antiferromagnets also oscillates as a function of the antiferromagnetic layer thickness due to the antiparallel alignment of the magnetic moments of the adjacent monolayers in the antiferromagnet 7,8 . Hence, the preferred Néel vector orientation of the top and bottom Fe 2 O 3 could be either parallel or antiparallel. When a parallel-preferred and an antiparallel-preferred area are close enough to each other, the Fe 2 O 3 cannot form a 180° domain wall to relax the Cr 2 O 3 magnetic order in both areas. Assuming that the Néel vector in each Fe 2 O 3 layer is uniform, the parallel state would induce a 180° domain wall over the Cr 2 O 3 thickness t in the antiparallel-preferred area (Fig. 4a). The orthogonal state, however, would induce two 90° domain walls in both areas, which is equal to a 180° domain wall over 2t in energy. The 180° domain wall over 2t has a lower energy than the 180° 4 domain wall over t, hence the orthogonal state has a lower energy, resulting in the orthogonal interlayer coupling. Considering the further relaxation of the Fe 2 O 3 Néel vector and the distance L between the parallel-preferred and the antiparallel-preferred areas, the order of the coupling energy can be estimated as E  We added sentences on Page 16, Line 1: "The unique temperature dependence of the orthogonal interlayer coupling strength in the antiferromagnetic trilayers is dramatically different from its counterpart in ferromagnet/normal metal/ferromagnet trilayers, which is insensitive to temperature 10 . We attribute this temperature dependence to the evolution of the magnetic order of Cr 2 O 3 caused by the temperature-dependent anisotropy." We added sentences at Supplementary Note 17: "We next compare this energy with Eq. (3) in the main text. Since the Néel temperature of Fe 2 O 3 (956 K) S13 is approximately three times of that of Cr 2 O 3 (307 K) S14 , we use A Fe ≈ 3 A Cr in the estimation. Considering L ≈ t, t t = 18 a, where a is the monolayer distance, and the exchange coefficient J = 8 meV for the Cr 2 O 3 S15 , A Cr ≈ Ja, we get E c ~ 0.049 meV (per unit cell), in the same order compared with the estimated value from the experiment. The relatively smaller experimental value could be due to the relaxation in the Fe 2 O 3 in the out-of-plane direction, and the fact that the interfacial Fe and Cr moments are not identically parallel or antiparallel S16 ." There are few other comments which need to be taken into account by the authors.
Q2) The authors need to place their work properly in the context of the previous knowledge in the field. The references given above my serve as a guide to the authors.
A: The Reviewer's comment is reasonable. We revised the sentences at the beginning of our manuscript to cover previous knowledge in the research field as the Reviewer pointed out. "The most well-established example is the giant magnetoresistance system, ferromagnet/transition metal/ferromagnet 1-5 , where the electron standing wave state 6 in spacer is induced by the magnetizations in two ferromagnets (FMs) and leads to the collinear interlayer coupling. Apart from metallic spacer, the interlayer coupling can exist across antiferromagnetic 7,8 and non-magnetic 9 insulators, providing more material alternatives for devices. Noncollinear coupling may also exist in FMs/spacer/FMs 10-12 , which is caused by interface roughness and the oscillatory collinear coupling 13 . But such a coupling is usually overshadowed by collinear coupling 10 , which is much easier for detection than the noncollinear coupling in FMs." Q3) The manuscript needs to be significantly improved in terms of the English and the style. There are many unclear or incorrect statements. Below I list just a couple of them in the abstract.
-The first sentence in the abstract is erroneous and misleading.
-The third sentence in the abstract is unclear.
-The fifth sentence in the abstract is incorrect (see comment 3).
A: We have carefully revised the English writing throughout the whole manuscript.
We revised the first sentence in the abstract "The collinear interlayer coupling between magnetic moments in ferromagnet/spacer/ferromagnet sandwich is widely studied and accelerates the development of spintronics, while the noncollinear coupling is usually absent because of the low coupling energy and large magnetization in ferromagnets." The coupling effect can be detected as a magnetic field where the coupling energy equals Zeeman energy (M· H). Therefore, in antiferromagnets with a 7 small net moment, the low coupling energy can be reflected as a large effective magnetic field. We modified the third sentence "However, the small net moment in AFMs can embody a low coupling energy as a sizable coupling field, enabling the detectability of noncollinear interlayer coupling." As the Reviewer pointed out in comment 3, the AFM spacer can mediate interlayer coupling between two ferromagnets. However, this is not our case here. In our all-AFM junction, the uniform magnetic ordering (net moment) in Cr 2 O 3 cannot generate orthogonal interlayer coupling ("Analysis on the magnetic ordering" section). In addition, the net moment in Cr 2 O 3 mediated coupling is also excluded by comparison between SMR and M-H loop. We revised the fifth sentence "From the energy and symmetry analysis, the direct coupling via uniform magnetic ordering in Cr 2 O 3 spacer in our junction here is excluded." Q4) In the last sentence on page 2, the authors say that "collinear parallel/antiparallel arrangements of Néel vectors in antiferromagnets are identical." They are not identical by symmetry. It is not clear what the authors want to say. Also, in the same sentence they say "the small noncollinear interaction would become dominant in an AFM/spacer/AFM junction." The bilinear interlayer coupling occurs across an AFM layer. This interaction can cause small canting of the magnetic moments, but it is still bilinear.
A: The Reviewer's comment is reasonable. In ideal conditions, the collinear parallel/antiparallel arrangements of Néel vectors in antiferromagnets are not identical. But in multi-domain state, such arrangements of Néel vectors are identical. In addition, the collinear arrangements (parallel and antiparallel) of Néel vectors usually cannot be detected by resistance owing to the 180° period of magnetoresistance. Therefore, the collinear arrangements are usually identical and cannot be distinguished in resistance measurement.
In contrast, the noncollinear coupling of Néel vectors can be readout in magnetoresistance measurement because of the different resistance states. Thus the noncollinear interaction can be detected in an AFM/spacer/AFM junction, although the coupling energy is not very large.
Accordingly, we revised the sentence to eliminate ambiguity "Moreover, collinear parallel/antiparallel arrangements of Néel vectors in antiferromagnets are usually identical in magnetoresistance measurements, so that the small noncollinear interaction can be clearly 8 detected in an AFM/spacer/AFM junction."

Q5) In their model, the authors introduce the Dzyaloshinskii-Moriya interaction (DMI)
in Fe2O3. It is not clear what its role is. On page 4, they say that "the hysteresis is due to the existence of Dzyaloshinskii-Moriya interaction (DMI)…" However, normally the hysteresis can be well explained by the magnetic anisotropy… A: As the Reviewer pointed out, the hysteresis in our sample can be explained by Zeeman energy and anisotropy energy. The strong Dzyaloshinskii-Moriya interaction (DMI) causes the canting of Néel vector and induces a net moment in Fe 2 O 3 . The hysteretic behavior suggests that the DMI induced net moment is 180° switched when the Zeeman energy overcomes the anisotropy energy. The Néel vector, which is locked to the DMI induced net moment, is also 180° switched, leading to the observed SMR hysteresis loop. Without this DMI induced net moment, when sweeping the magnetic field, the Néel vector would not be 180° switched, but rather be canted to opposite direction under opposite magnetic field, so cannot generate an SMR hysteresis loop. Therefore, the DMI is introduced into our model. Q6) On page 5, the authors say; "The resistance peak appears before H = 0, which violates the principle of thermodynamics, indicating the existences of coupling effect." It is not clear why the principle of thermodynamics is violated in their system.
Probably, the authors meant something else.
A: Generally speaking, the resistance peak, which means the switching of magnetic ordering, should appear after H = 0 (hysteresis behavior). This is decided by the energy competition between Zeeman energy and anisotropy energy. Therefore, the magnetic ordering switching phenomenon occurs before H = 0 suggests that there should exists another effect contributing to the system energy. Accordingly, we revised the sentence "One resistance peak appears before H = 0, indicating the existence of coupling effect." to eliminate misunderstanding. layers. XMLD results support this picture. Then, a temperature dependence of the magneto-transport properties is presented, from which the origin of the orthogonal coupling is discussed. Finally, a theoretical model is put forward, with which the experiment is semi-quantitatively explained.
The interlayer magnetic coupling is an important phenomenon for functional magnetic and spintronic devices as well as a fundamental interest in condensed-matter physics, and collinear antiferromagnetic coupling is widely used in hard-disc drive and nonvolatile memory. Meanwhile, as far as I know, the orthogonal interlayer coupling has not been well studied so far. In this regard, I think this work would potentially give a high impact to the community of magnetics and spintronics.
However, I think the authors should consider the following three points and revise the manuscript.
A: We very much appreciate the Reviewer's positive evaluation such as "The interlayer magnetic coupling is an important phenomenon" and "I think this work would potentially give a high impact to the community of magnetics and spintronics". We have addressed issues pointed out by the Reviewer below.
Q1) The authors assume that the signal of the magneto-transport measurement arises from the spin-Hall magnetoresistance (SMR), but its evidence is not presented in this manuscript. For example, what about the contribution of anisotropic magnetoresistance and other magnetoresistive effects. I think the authors should perform the same measurement for samples without the Pt cap layer or with a cap layer comprised by a low spin Hall material.
A: Thanks for the Reviewer's helpful suggestion. To clearly verify that the SMR contribute to the signal, we performed measurement with a cap layer comprised by a low spin Hall material titanium (Ti), as the Reviewer suggested. The corresponding data are added to Supplementary Note 4.
Note 4. Magnetoresistance measurement with low spin Hall material as the cap layer. Fig. S4. It can be seen that there exists negligible magnetoresistance signal when the magnetic field is applied, excluding other magnetoresistive effect such as anisotropic magnetoresistance.

Fig. S4. Negligible magnetoresistance in Fe 2 O 3 /Cr 2 O 3 /Fe 2 O 3 /Ti.
Accordingly, we added "Note that the signal disappears in the junction with Ti as the cap layer, (Supplementary Note 4) which has negligible spin Hall effect, suggesting that the signal is caused by SMR." on Page 5, Line 2 in the main-text.
Q2) The authors attribute the orthogonal coupling to a quasi-long range order (QLRO), but I could not find any solid evidence of the QLRO in Cr2O3 layer. I think the authors should somehow investigate the spin structure of the Cr2O3 layer as a function of the temperature and/or the thickness of Cr2O3 layer; otherwise, the proposed scenario is just a speculation. In addition, I cannot understand well why the orthogonal coupling between the Fe2O3 is stabilized when the QLRO emerges in Cr2O3. More detailed explanation is desirable.
A: The Reviewer's comment is reasonable. According to the Reviewer's suggestions, we theoretically investigate the spin structure of the Cr 2 O 3 spacer using the Slonczewski model, where orthogonal interlayer coupling is stabilized by oscillating bilinear coupling and interfacial roughness. When a parallel-preferred area and an antiparallel-preferred area are closed to each other, the Fe 2 O 3 cannot form a 180° domain wall to relax the Cr 2 O 3 magnetic order in both areas. Hence, there would be a 180° Cr 2 O 3 domain wall in the parallel state, and two 90° Cr 2 O 3 domain walls in the orthogonal state. The two 90°domain walls can be regarded as a 180°domain wall with twice as much width, which has a smaller energy than the 180°domain wall with original width. 11 Hence, the orthogonal state has a smaller total energy, leading to the orthogonal coupling.
We rewrote the modelling part on Page 11 and added Fig. 4: "Non-uniform domain wall state mediated interlayer coupling. Having excluded the magnetic ordering which is uniform in the film plane, we consider magnetic ordering, which is non-uniform in the film plane, as the origin of the interlayer coupling. It is known that orthogonal interlayer coupling could exist in FM/NM/FM trilayers due to the interfacial roughness and oscillating collinear exchange coupling 13 . The collinear interlayer coupling mediated by antiferromagnets also oscillates as a function of the antiferromagnetic layer thickness due to the antiparallel alignment of the magnetic moments of the adjacent monolayers in the antiferromagnet 7,8 . Hence, the preferred Néel vector orientation of the top and bottom Fe 2 O 3 could be either parallel or antiparallel. When a parallel-preferred and an antiparallel-preferred area are close enough to each other, the Fe 2 O 3 cannot form a 180° domain wall to relax the Cr 2 O 3 magnetic order in both areas. Assuming that the Néel vector in each Fe 2 O 3 layer is uniform, the parallel state would induce a 180° domain wall over the Cr 2 O 3 thickness t in the antiparallel-preferred area (Fig. 4a). The orthogonal state, however, would induce two 90° domain walls in both areas, which is equal to a 180° domain wall over 2t in energy. The 180° domain wall over 2t has a lower energy than the 180° domain wall over t, hence the orthogonal state has a lower energy, resulting in the orthogonal interlayer coupling. Taking  where q is the volume percentage of the in-plane Néel vector that can form this NUDW state. Note that a L 2 factor is subtracted to get the coupling energy per area.

Response Letter of NCOMMS-21-38921B
We very much appreciate the positive evaluations of our manuscript (NCOMMS-21-38921B) by Reviewer #1 ("The authors have adequately responded to my comments, and I recommend publication of their manuscript.") and Reviewer #3 ("I think the authors reasonably addressed the comments by both reviewers and the revised manuscript is suitable for publication in Nature Communications"). We address the issues raised by them point by point below.
And the comments are helpful for our improvements further. Chong Chen are added as co-author for the help on the physical model and English polishing.
Amendments of our revised manuscript are summarized below in bold face style.
The main revisions include: 1. We polish the English writing carefully.
2. The roughness data are added.
3. Fig. 4 is modified to be more informative and clear.

Response to Reviewer #1:
The authors have adequately responded to my comments, and I recommend publication of their manuscript. Before publication, I suggest to the authors revisit their text to improve the English. Also, I recommend improving the abstract to better articulate the motivation and findings.
A: We sincerely thank the Reviewer for his/her positive evaluation. We have addressed the Reviewer's comments and further improved the English writing.

Response to Reviewer #3:
I think the authors reasonably addressed the comments by both reviewers and the revised manuscript is suitable for publication in Nature Communications. However, I think the quality of this manuscript can be further improved by addressing the following two points.
A: We really appreciate the positive evaluations of the Reviewer. We have addressed the Reviewer's comments as shown below. 2 Q1) The authors attribute the orthogonal coupling to the co-existence of parallelpreferred and antiparallel-preferred areas due to an interfacial roughness and oscillating collinear exchange coupling, and based on this scenario, they put forward a theoretical model to describe their experimental results. The explanation appears reasonable, but it still remains a matter of speculation since the magnitude of interfacial roughness of their samples is not clear, while a relatively large variation of Cr2O3 layer thickness of the order of several angstroms is required for the co-existence of the parallel-and antiparallel-preferred areas according to literatures such as Ref. 7.
Thus, I think it is better that the authors present the information of thickness variation and interfacial roughness in their samples.
A: The interfacial roughness of our sample is measured and the corresponding data are added to Supplementary Note 13:

Note 13. Thickness variation and interfacial roughness in Cr2O3.
The interfacial roughness in Cr2O3 layer is important in our theoretical model. Therefore, the thickness variation and interfacial roughness are measured via atomic force microscope, and the corresponding data are shown in Fig. S14. It can be seen that the variation of Cr2O3 thickness is at the order of several angstroms (Fig. S14a). To further support the thickness variation, the roughness data along two diagonal lines are shown in Fig. S14b and c. The thickness variation is large enough (the order of several angstroms) for the co-existence of parallel-and antiparallel-preferred areas in Fe2O3 layers S13 , further supporting our theory model. Q2) Secondly, the newly-added Fig. 4 can help the readers to understand their scenario, but this figure can become further helpful by adding a few more words to guide the readers. For example, what is t, where are 90-degree and 180-degree DWs, and where does the magnetic energy increase and why. In addition, the direction of some arrows is not accurate and confusing.
A: Thanks for the Reviewer's helpful suggestion. As the Reviewer pointed out, more information such as t, 90° and 180° domain walls are added in Fig. 4.
To exhibit the enhancement of the magnetic energy more clearly, the equivalent domain walls are added in Fig. 4. 180° domain wall is formed within t in the case of parallel/antiparallel state (Fig. 4a), leading to the energy E = ACr/t. In the case of orthogonal state, the equivalent 180° domain wall is formed within 2t (Fig. 4b), resulting in the lower energy E = ACr/2t. Therefore, the orthogonal state is stabilized. In addition, to show the rotation of the direction of arrows more accurately, guidelines for eyes are added.