Coupled multiferroic domain switching in the canted conical spin spiral system Mn2GeO4

Despite remarkable progress in developing multifunctional materials, spin-driven ferroelectrics featuring both spontaneous magnetization and electric polarization are still rare. Among such ferromagnetic ferroelectrics are conical spin spiral magnets with a simultaneous reversal of magnetization and electric polarization that is still little understood. Such materials can feature various multiferroic domains that complicates their study. Here we study the multiferroic domains in ferromagnetic ferroelectric Mn2GeO4 using neutron diffraction, and show that it features a double-Q conical magnetic structure that, apart from trivial 180o commensurate magnetic domains, can be described by ferromagnetic and ferroelectric domains only. We show unconventional magnetoelectric couplings such as the magnetic-field-driven reversal of ferroelectric polarization with no change of spin-helicity, and present a phenomenological theory that successfully explains the magnetoelectric coupling. Our measurements establish Mn2GeO4 as a conceptually simple multiferroic in which the magnetic-field-driven flop of conical spin spirals leads to the simultaneous reversal of magnetization and electric polarization.

On the other hand, from the extensive ND study under E and B applications, to fully understand the microscopic coupling between 8 different domains is experimentally and theoretically challenging, which the authors made significant progresses in my view. It reveals the key role of the canted conical spin order and its flop with magnetic field to have both reversal of M and P with magnetic field. As the manuscript is well written (although the readers might have to carefully follow all the previous publications to fully understand due to the complicated nature of the subjects and rather concise writings of the author's), I believe the present work might be publishable in nature comm.
I have to still mention a problem in the author's view reflected in the manuscript though. As the MF research gets more and more ample, we all know that the understanding of the mechanism in each rare case of ferromagnetic ferroelectric MF becomes increasingly important. The authors seems to address well this issue in the abstract and introduction while they explain the cases of the hexaferrites and CoCr2O4. However, the authors seem to describe only the author's works without unbalanced citations of the other progresses in the hexaferrites, particulaly the ones with Al doped compounds. I hope this can be remedied in their manuscript before being published.
Reviewer #3 (Remarks to the Author): The article by T. Honda et al. entitled "Coupled multiferroic domain switching in the canted conical spin spiral system Mn2GeO4" describes a combination of high-quality polarized neutron diffraction work and a phenomenological theory on Mn2GeO4 that explains how the application of a magnetic field allows to simultaneously switch magnetic and ferroelectric domains. In particular, the presented data and corresponding analysis is of very high quality. As such, I recommend that the manuscript is published in Nature Communications. However, I have several comments that need to be addressed before the manuscript may be published. They are described below: 1. Section Phenomenological coupling theory: The authors write "We construct coupling terms that the reader can check are invariant under these transformation properties, and this leads to three such terms given by…". While I have no doubt that the terms are indeed invariant under these transformation properties, I don't think it is the reader's responsibility to verify this. I strongly suggest that the authors include a section in the Supplementary Material that demonstrates that this is the case.
2. Section Electric-field-cooling effect on spin-helicity: To the expert reader, familiar with the SNP technique, it is immediately clear that measuring the (z, x) and (z, -x) polarization channels provides a direct way of determining the spin helicity. However, to the broader audience, and even to most researchers familiar with neutron scattering this is not clear. After mentioning that these polarized channels were determined for varying the strength of the electrical field applied during cooling, the authors go on to describe further technical details, and only on the next page, explain that measuring these cross-sections, and eventually the full polarization matrix allows to determine the population of the two spin-helicity domains. To highlight the importance of this measurement to the non-expert reader and to improve the flow of the text, I recommend adding a statement at the beginning of the paragraph that explains that measuring the (z, x) and (z, -x) polarization channels is a direct way of determining spin helicity and is, thus, the method of choice to understand the nature of the U coupling term. I note that, for example, in the section Correlation between the C and Q domains the authors have done a much better job of explaining at the beginning of the section what kind of information can be obtained via the complimentary unpolarized neutron diffraction measurements.
3. In Fig. 2a, as well as the corresponding discussion in the text, the polarized scans of the magnetic Bragg peaks are called "k-scan profiles". I assume that the authors want to imply that they are scanning the over the peaks keeping the h and l components for the momentum transfer constant while varying k component. I think that for someone experienced in neutron diffraction this is clear, however, for a non-expert "k-scan profiles" is a confusing name, notably, because they will not be familiar with the fact that in scattering techniques the three components of the momentum transfer in reciprocal lattice units are typically called (h, k, l). I would very much prefer to call the scans Q-scans (because in the method section the momentum transfer is called Q) or even better momentum transfer scans along the k direction. I would further replace the label in the upper left corner of Fig. 2a to be (2-qh k 0). 4. The authors highlight twice in the manuscript that no clear microscopic explanation for the simultaneous reversal of the magnetization and polarization observed in Mn2GeO4 and several other multiferroic materials exists to date. The discussion is written in a way that suggests that the authors established a microscopic theory that explains such switching for Mn2GeO4. In contrast, the theory presented in the paper is only phenomenological and based on symmetry considerations. The "microscopic" explanation provided in the discussion of the paper is based on the magnetic structure, which consists of two coupled canted antiferromagnetic conical spinchains. The superposition of those two chains leads to net polarization and magnetization along the c axis. The neutron diffraction results described in the results part of the text show that application of a magnetic field along c leads to a flop of the cone axis in each chain, and thus simultaneous reversal of both polarization and magnetization. While the authors observe the microscopic magnetic structure of Mn2GeO4 and its change as function of applied magnetic and electrical field, this does not provide a microscopic explanation based on a microscopic Hamiltonian. Basically, the microscopic interactions that lead to the observed magnetic structure and its behavior in applied fields is not provided by the authors. The authors say that the polarization in the individual chains is well-explained within inverse Dzyaloshinskii-Moriya mechanism. This indeed makes sense in terms of their experimental observations. However, in that sense there is no new insights on how this simultaneous reversal of polarization and magnetization works, as this has been already proposed earlier for other materials. What is distinct about Mn2GeO4 seems to be rooted in details of its magnetic structure, namely that it is composed in a special superposition of two canted antiferromagnetic conical spin-chains. I thus feel that the manuscript needs some fine-tuning that is less suggestive of the authors identifying a new level of microscopic understanding of the simultaneous reversal of the magnetization and polarization in multiferroic materials. 5. For the summary paragraph, I wonder if there a some "lessons" or "guidelines" that can be extracted from the more detailed understanding of the coupling of the magnetization and electrical polarization in Mn2GeO4. For example, is it expected that such coupling based on coupled spin chains is found in more materials? Or is this a specific case?

Response to the reviewers' comments
We would like to sincerely thank all the reviewers for their careful reading of our manuscript as well as for constructive suggestions to improve the quality of the manuscript. In the following, we reply to the respective reviewers' comments.
Hereinafter, the comments from each reviewer appear in black italic letters while our reply is written with blue letters. Significant revisions and newly added descriptions are indicated with red letters in the manuscript.

. In these examples, the materials present both (weak) polarization and (weak) magnetization.
Responding to the reviewer's comment, we replaced the term "single phase multiferroics" with "spin-driven ferroelectrics" in the sentence. We don't want use the term "type-II multiferroics" which is less familiar to non-expert readers. In the list raised by the reviewer includes some type-II multiferroics (or spin-driven ferroelectrics). Among them, we agree with the reviewer that some orthoferrites such as GdFeO 3 exhibits both spontaneous magnetization and polarization. However, the others such as TbMnO 3 , RMn 2 O 5 , and CuO are antiferromagnets with no "spontaneous" magnetization in the absence of a magnetic field. Thus, we consider the revised sentence properly describe the current situation of multiferroic research field.
In addition, we replaced the description "still quite rare" by "not common" in the third sentence of the introduction section (p. 2).
We also italicize the word "same" in the first sentence of the last paragraph in p. 2 in order to emphasize the rare condition with P||M in spin-driven ferroelectrics.
( This comment is related to the comment (3-1) raised by Reviewer 3. Please see also our reply to the comment (3-1).
We considered the reviewer's suggestion about a schematic illustration. However, the spin functions we are dealing with are very complicated, and we think that the algebraic exposition is the most definitive. To state that taking z into z takes f(x,y,z) into f(x,y,z) is easier for the reader than diagrams to illustrate this. Therefore, we prefer not to have to try to implement this idea.
(1-3) In Fig. 1c, I suggest the authors to mark the directions of polarization and magnetization, respectively. If possible, the (magnetic) space group or point group resulted by the corresponding magnetic configurations can also be added in Fig. 1c.
The magnetic structure of Mn 2 GeO 4 is composed of both incommensurate and commensurate orders, and therefore is very complicated. Due to the magnetic structure, only the operation 2 1 || c can remain as a symmetry element, which means that the point group symmetry lowers from mmm to 2. Indeed, this satisfies the symmetry requirement for magnetic point groups of ferromagnetic and ferroelectric structure with M||z and P||z ("2" or "1" for the present case) [Cracknell, A. P. Magnetism in crystalline materials (Pergamon Press, Oxford 1975)].
To the best of our knowledge, incommensurate magnetic structures are not described by a magnetic space group, but with a magnetic superspace group in four dimensions. Therefore, we have not determined the magnetic space group.
Following the reviewer's suggestion, we marked the directions of polarization and magnetization in the revised Fig. 1c, and added a description about the magnetic point group "(the most probable magnetic point group: 2 with 2 1 axis along c)" in the figure caption of Fig. 1c.
(1-4) Will it be possible to provide a schematic illustration to show how the inversion center is broken by the magnetic configurations in Fig 1c? In the Pnma crystal structure of Mn 2 GeO 4 , the inversion center is located at Mn1 site. Thus, it is apparent that the magnetic configurations break the inversion symmetry. We consider that schematic illustrations of the inversion centers in Fig. 1c make the figure confused, and don't want to add such an illustration.
( Thank the reviewer for his/her critical reading and finding the typo. We corrected the typo in the revised manuscript.
( Following the reviewer's suggestions, we added the following descriptions at the caption of Supplementary Table 1. "The transformation properties of the order parameters, given below (S1-S7), were developed by considering how the magnetization distribution transforms under the symmetry operations of the space group, similar to the procedure used in ref. 29. The mirror operations, m x , m y , and m z , are the minimum symmetry elements for point group Pnma, because all the other operations (I, 2 x , 2 y , and 2 z ) can be generated by the product of these mirror operations. Note that the transformations of axial vectors such as the magnetization, magnetic order parameters, or the angular momentum r × p are different from those of polar vectors such as position r and momentum p. For axial vectors, the result for transformations involving a change in handedness (e. g. mirror or inversion operation) includes an extra factor of (−1)."   (2012). As pointed out by the reviewer, it may be better for readers' understanding to describe some more details how the phenomenological coupling theory was developed. In this reply, we attach, for the reviewer's eyes only, a preprint written by one of the authors (Harris_preprint.pdf). This preprint provides complete details on the derivation of the transformation properties of the order parameter and will be submitted to as soon as the present paper is available to the public. From the preprint, the reviewer might be able to judge the reliability of our phenomenology. However, we consider that it is impractical to give the full details of the derivation in the present paper. Instead, we inserted the following subsection in Methods section (p. 11).
"Phenomenological coupling theory. In order to develop the phenomenological coupling theory, we first determined the symmetry properties of the magnetic and ferroelectric order parameters found experimentally. The symmetry properties of the magnetic order parameters are determined by their irreducible representation, and leads directly to the transformation properties of the order parameters given in equations (S2)-(S7) of Supplementary  (1)-(3). A full account of the phenomenological coupling theory will be published elsewhere." In addition, we revised the sentence just above equation (1) on p. 4, as follows.
"We construct coupling terms that the reader can check are invariant under these transformation properties by using equations (S1)-(S7) in Supplementary Please see our reply to the comment (3-3).
(3-3) In Fig. 2a, as well as the corresponding discussion in the text, the polarized scans of the magnetic Bragg peaks are called "k-scan profiles". I assume that the authors want to imply that they are scanning the over the peaks keeping the h and l components for the momentum transfer constant while varying k component. I think that for someone experienced in neutron diffraction this is clear, however, for a non-expert "k-scan profiles" is a confusing name, notably, because they will not be familiar with the fact that in scattering techniques the three components of the momentum transfer in reciprocal lattice units are typically called (h, k, l). I would very much prefer to call the scans Q-scans (because in the method section the momentum transfer is called Q) or even better momentum transfer scans along the k direction. I would further replace the label in the upper left corner of Fig. 2a to be (2-qh k 0).
Following the reviewer's suggestions, we revised the following parts.
The third sentence of the section "Electric-field-cooling effect on spin-helicity" (p. 5) was revised as follows.
"The intensity profiles of the ( 2 q h 1−q k 0) peaks were measured in both the non-spin-flip (z,x) and spin-flip (z,−x) polarization channels. From the difference in the intensities observed in each polarization channel, the relative proportions of the two helicity domains within each Q domain can be determined directly (see Methods)." In the fifth sentence of the section "Electric-field-cooling effect on spin-helicity" (p. 5), we replaced the term "k-scan profiles obtained" with "the wave-vector dependence of the scattering".
In the figure caption of Fig. 2a, we replaced the description "Polarized neutron scattering measurements of the k-scan profiles" with "Polarized neutron scattering as a function of wave-vector transfer along the (0,k,0) wave-vector direction".
(3-4) The authors highlight twice in the manuscript that no clear microscopic explanation for the simultaneous reversal of the magnetization and polarization observed in Mn2GeO4 and several other multiferroic materials exists to date. The discussion is written in a way that suggests that the authors established a microscopic theory that explains such switching for Mn2GeO4. In contrast, the theory presented in the paper is only phenomenological and based on symmetry considerations. The "microscopic" explanation provided in the discussion of the paper is based on the magnetic structure, which consists of two coupled canted antiferromagnetic conical spin-chains. The reviewer considers that some descriptions about the microscopic understanding of the observed M-P coupling is a bit too strong. We basically agree to the reviewer's comment on this point. Therefore, following the reviewer's suggestion, we fine-tuned some descriptions.
At the end of the first paragraph of introduction (p. 2), we added the following sentence.
"An important result of the present paper is to show that the phenomenological model we introduce here explains the switching mechanisms we observe in the double-Q conical-spiral multiferroic, Mn 2 GeO 4 ." At the third last sentence of the first paragraph on p. 3 ("Although phenomenological symmetry…"), the term "no clear microscopic mechanism" was replaced by "no clear explanation".
The word "microscopic" at the last sentence of the second paragraph on p. 9 ("The term W describes…") was removed.
The word "microscopically" at the last sentence of the summary paragraph in the previous manuscript ("Furthermore, we revealed…") was removed.
By these revisions, the manuscript becomes less suggestive of our identifying a new level of microscopic understanding of the simultaneous reversal of the magnetization and polarization in multiferroic materials.
(3-5) For the summary paragraph, I wonder if there a some "lessons" or "guidelines" that can be extracted from the more detailed understanding of the coupling of the magnetization and electrical polarization in Mn2GeO4. For example, is it expected that such coupling based on coupled spin chains is found in more materials? Or is this a specific case?
Responding to the reviewer's comment, we added the following statement at the end of summary paragraph (p. 10).
The present study conclusively describes how magnetization and ferroelectric polarization can be coupled on a phenomenological level and stands out as a text-book case of how complex and seemingly unexpected coupling terms can be engineered in spin-driven ferroelectrics with double-Q conical magnetic order. The essence of the present phenomenological discussion can be also applied to other conical-spin-driven ferroelectrics.